id	sid	tid	token	lemma	pos
ejde-636	1	1	electronic	electronic	ADJ
ejde-636	1	2	journal	journal	NOUN
ejde-636	1	3	of	of	ADP
ejde-636	1	4	differential	differential	ADJ
ejde-636	1	5	equations	equation	NOUN
ejde-636	1	6	,	,	PUNCT
ejde-636	1	7	vol	vol	NOUN
ejde-636	1	8	.	.	NOUN
ejde-636	1	9	2024	2024	NUM
ejde-636	1	10	(	(	PUNCT
ejde-636	1	11	2024	2024	NUM
ejde-636	1	12	)	)	PUNCT
ejde-636	1	13	,	,	PUNCT
ejde-636	1	14	no	no	INTJ
ejde-636	1	15	.	.	NOUN
ejde-636	1	16	29	29	NUM
ejde-636	1	17	,	,	PUNCT
ejde-636	1	18	pp	pp	ADJ
ejde-636	1	19	.	.	PUNCT
ejde-636	2	1	1–20	1–20	PROPN
ejde-636	2	2	.	.	PUNCT
ejde-636	3	1	issn	issn	PROPN
ejde-636	3	2	:	:	PUNCT
ejde-636	3	3	1072	1072	NUM
ejde-636	3	4	-	-	SYM
ejde-636	3	5	6691	6691	NUM
ejde-636	3	6	.	.	PUNCT
ejde-636	4	1	url	url	PROPN
ejde-636	4	2	:	:	PUNCT
ejde-636	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-636	4	4	,	,	PUNCT
ejde-636	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-636	4	6	doi	doi	PROPN
ejde-636	4	7	:	:	PUNCT
ejde-636	4	8	10.58997	10.58997	NUM
ejde-636	4	9	/	/	SYM
ejde-636	4	10	ejde.2024.29	ejde.2024.29	NOUN
ejde-636	4	11	normalized	normalize	VERB
ejde-636	4	12	ground	ground	NOUN
ejde-636	4	13	state	state	NOUN
ejde-636	4	14	of	of	ADP
ejde-636	4	15	a	a	DET
ejde-636	4	16	mixed	mixed	ADJ
ejde-636	4	17	dispersion	dispersion	NOUN
ejde-636	4	18	nonlinear	nonlinear	NOUN
ejde-636	4	19	schrödinger	schrödinger	NOUN
ejde-636	4	20	equation	equation	NOUN
ejde-636	4	21	with	with	ADP
ejde-636	4	22	combined	combined	ADJ
ejde-636	4	23	power	power	NOUN
ejde-636	4	24	-	-	PUNCT
ejde-636	4	25	type	type	NOUN
ejde-636	4	26	nonlinearities	nonlinearitie	NOUN
ejde-636	4	27	zhouji	zhouji	PROPN
ejde-636	4	28	ma	ma	PROPN
ejde-636	4	29	,	,	PUNCT
ejde-636	4	30	xiaojun	xiaojun	PROPN
ejde-636	4	31	chang	chang	PROPN
ejde-636	4	32	,	,	PUNCT
ejde-636	4	33	zhaosheng	zhaosheng	PROPN
ejde-636	4	34	feng	feng	PROPN
ejde-636	4	35	abstract	abstract	PROPN
ejde-636	4	36	.	.	PUNCT
ejde-636	5	1	we	we	PRON
ejde-636	5	2	study	study	VERB
ejde-636	5	3	the	the	DET
ejde-636	5	4	existence	existence	NOUN
ejde-636	5	5	of	of	ADP
ejde-636	5	6	normalized	normalize	VERB
ejde-636	5	7	ground	ground	NOUN
ejde-636	5	8	state	state	NOUN
ejde-636	5	9	solutions	solution	NOUN
ejde-636	5	10	to	to	ADP
ejde-636	5	11	a	a	DET
ejde-636	5	12	mixed	mixed	ADJ
ejde-636	5	13	dispersion	dispersion	NOUN
ejde-636	5	14	fourth	fourth	ADJ
ejde-636	5	15	-	-	PUNCT
ejde-636	5	16	order	order	NOUN
ejde-636	5	17	nonlinear	nonlinear	NOUN
ejde-636	5	18	schrödinger	schrödinger	NOUN
ejde-636	5	19	equation	equation	NOUN
ejde-636	5	20	with	with	ADP
ejde-636	5	21	combined	combined	ADJ
ejde-636	5	22	power	power	NOUN
ejde-636	5	23	-	-	PUNCT
ejde-636	5	24	type	type	NOUN
ejde-636	5	25	nonlinearities	nonlinearitie	NOUN
ejde-636	5	26	.	.	PUNCT
ejde-636	6	1	by	by	ADP
ejde-636	6	2	analyzing	analyze	VERB
ejde-636	6	3	the	the	DET
ejde-636	6	4	subadditivity	subadditivity	NOUN
ejde-636	6	5	of	of	ADP
ejde-636	6	6	the	the	DET
ejde-636	6	7	ground	ground	NOUN
ejde-636	6	8	state	state	NOUN
ejde-636	6	9	energy	energy	NOUN
ejde-636	6	10	with	with	ADP
ejde-636	6	11	respect	respect	NOUN
ejde-636	6	12	to	to	ADP
ejde-636	6	13	the	the	DET
ejde-636	6	14	prescribed	prescribed	ADJ
ejde-636	6	15	mass	mass	NOUN
ejde-636	6	16	,	,	PUNCT
ejde-636	6	17	we	we	PRON
ejde-636	6	18	employ	employ	VERB
ejde-636	6	19	a	a	DET
ejde-636	6	20	constrained	constrain	VERB
ejde-636	6	21	minimization	minimization	NOUN
ejde-636	6	22	method	method	NOUN
ejde-636	6	23	to	to	PART
ejde-636	6	24	establish	establish	VERB
ejde-636	6	25	the	the	DET
ejde-636	6	26	existence	existence	NOUN
ejde-636	6	27	of	of	ADP
ejde-636	6	28	ground	ground	NOUN
ejde-636	6	29	state	state	NOUN
ejde-636	6	30	that	that	PRON
ejde-636	6	31	corresponds	correspond	VERB
ejde-636	6	32	to	to	ADP
ejde-636	6	33	a	a	DET
ejde-636	6	34	local	local	ADJ
ejde-636	6	35	minimum	minimum	NOUN
ejde-636	6	36	of	of	ADP
ejde-636	6	37	the	the	DET
ejde-636	6	38	associated	associated	ADJ
ejde-636	6	39	functional	functional	NOUN
ejde-636	6	40	.	.	PUNCT
ejde-636	7	1	under	under	ADP
ejde-636	7	2	certain	certain	ADJ
ejde-636	7	3	conditions	condition	NOUN
ejde-636	7	4	,	,	PUNCT
ejde-636	7	5	by	by	ADP
ejde-636	7	6	studying	study	VERB
ejde-636	7	7	the	the	DET
ejde-636	7	8	monotonicity	monotonicity	NOUN
ejde-636	7	9	of	of	ADP
ejde-636	7	10	ground	ground	NOUN
ejde-636	7	11	state	state	NOUN
ejde-636	7	12	energy	energy	NOUN
ejde-636	7	13	as	as	SCONJ
ejde-636	7	14	the	the	DET
ejde-636	7	15	mass	mass	NOUN
ejde-636	7	16	varies	vary	VERB
ejde-636	7	17	,	,	PUNCT
ejde-636	7	18	we	we	PRON
ejde-636	7	19	apply	apply	VERB
ejde-636	7	20	the	the	DET
ejde-636	7	21	constrained	constrain	VERB
ejde-636	7	22	minimization	minimization	NOUN
ejde-636	7	23	arguments	argument	NOUN
ejde-636	7	24	on	on	ADP
ejde-636	7	25	the	the	DET
ejde-636	7	26	nehari	nehari	NOUN
ejde-636	7	27	-	-	PUNCT
ejde-636	7	28	pohozaev	pohozaev	NOUN
ejde-636	7	29	manifold	manifold	ADJ
ejde-636	7	30	to	to	PART
ejde-636	7	31	prove	prove	VERB
ejde-636	7	32	the	the	DET
ejde-636	7	33	existence	existence	NOUN
ejde-636	7	34	of	of	ADP
ejde-636	7	35	normalized	normalize	VERB
ejde-636	7	36	ground	ground	NOUN
ejde-636	7	37	state	state	NOUN
ejde-636	7	38	solutions	solution	NOUN
ejde-636	7	39	.	.	PUNCT
ejde-636	8	1	1	1	X
ejde-636	8	2	.	.	X
ejde-636	8	3	introduction	introduction	NOUN
ejde-636	8	4	and	and	CCONJ
ejde-636	8	5	main	main	ADJ
ejde-636	8	6	results	result	NOUN
ejde-636	8	7	consider	consider	VERB
ejde-636	8	8	the	the	DET
ejde-636	8	9	mixed	mixed	ADJ
ejde-636	8	10	dispersion	dispersion	NOUN
ejde-636	8	11	nonlinear	nonlinear	NOUN
ejde-636	8	12	schrödinger	schrödinger	NOUN
ejde-636	8	13	equation	equation	NOUN
ejde-636	8	14	with	with	ADP
ejde-636	8	15	combined	combined	ADJ
ejde-636	8	16	power	power	NOUN
ejde-636	8	17	-	-	PUNCT
ejde-636	8	18	type	type	NOUN
ejde-636	8	19	nonlinearities	nonlinearitie	NOUN
ejde-636	8	20	i∂tψ	i∂tψ	NOUN
ejde-636	8	21	−	−	PROPN
ejde-636	9	1	ϵ∆2ψ	ϵ∆2ψ	X
ejde-636	9	2	+	+	CCONJ
ejde-636	9	3	γ∆ψ	γ∆ψ	VERB
ejde-636	9	4	+	+	NUM
ejde-636	9	5	µ|ψ|q−2ψ	µ|ψ|q−2ψ	PROPN
ejde-636	10	1	+	+	CCONJ
ejde-636	11	1	|ψ|p−2ψ	|ψ|p−2ψ	NOUN
ejde-636	11	2	=	=	SYM
ejde-636	11	3	0	0	NUM
ejde-636	11	4	,	,	PUNCT
ejde-636	11	5	(	(	PUNCT
ejde-636	11	6	1.1	1.1	NUM
ejde-636	11	7	)	)	PUNCT
ejde-636	12	1	where	where	SCONJ
ejde-636	12	2	n	n	PRON
ejde-636	12	3	≥	≥	NOUN
ejde-636	12	4	1	1	NUM
ejde-636	12	5	,	,	PUNCT
ejde-636	12	6	µ	µ	PRON
ejde-636	12	7	≥	≥	NOUN
ejde-636	12	8	0	0	NUM
ejde-636	12	9	,	,	PUNCT
ejde-636	12	10	ϵ	ϵ	PRON
ejde-636	12	11	≥	≥	NOUN
ejde-636	12	12	0	0	NUM
ejde-636	12	13	,	,	PUNCT
ejde-636	12	14	γ	γ	PROPN
ejde-636	12	15	∈	∈	PROPN
ejde-636	12	16	r	r	NOUN
ejde-636	12	17	,	,	PUNCT
ejde-636	12	18	ψ	ψ	NOUN
ejde-636	12	19	∈	∈	PROPN
ejde-636	12	20	r	r	NOUN
ejde-636	12	21	×	×	PROPN
ejde-636	12	22	rn	rn	PROPN
ejde-636	12	23	→	→	PROPN
ejde-636	12	24	c	c	PROPN
ejde-636	12	25	and	and	CCONJ
ejde-636	12	26	2	2	NUM
ejde-636	12	27	<	<	X
ejde-636	12	28	q	q	X
ejde-636	12	29	<	<	X
ejde-636	12	30	p	p	X
ejde-636	12	31	≤	≤	NUM
ejde-636	12	32	4∗.	4∗.	NUM
ejde-636	12	33	note	note	NOUN
ejde-636	12	34	that	that	SCONJ
ejde-636	12	35	equation	equation	NOUN
ejde-636	12	36	(	(	PUNCT
ejde-636	12	37	1.1	1.1	NUM
ejde-636	12	38	)	)	PUNCT
ejde-636	12	39	becomes	become	VERB
ejde-636	12	40	the	the	DET
ejde-636	12	41	well	well	ADV
ejde-636	12	42	-	-	PUNCT
ejde-636	12	43	known	know	VERB
ejde-636	12	44	schrödinger	schrödinger	NOUN
ejde-636	12	45	equation	equation	NOUN
ejde-636	12	46	when	when	SCONJ
ejde-636	12	47	ϵ	ϵ	PROPN
ejde-636	12	48	=	=	SYM
ejde-636	12	49	0	0	NUM
ejde-636	12	50	and	and	CCONJ
ejde-636	12	51	γ	γ	X
ejde-636	12	52	=	=	SYM
ejde-636	12	53	1	1	NUM
ejde-636	12	54	.	.	PUNCT
ejde-636	13	1	this	this	DET
ejde-636	13	2	equation	equation	NOUN
ejde-636	13	3	has	have	AUX
ejde-636	13	4	been	be	AUX
ejde-636	13	5	extensively	extensively	ADV
ejde-636	13	6	studied	study	VERB
ejde-636	13	7	as	as	ADP
ejde-636	13	8	a	a	DET
ejde-636	13	9	partial	partial	ADJ
ejde-636	13	10	differential	differential	NOUN
ejde-636	13	11	equation	equation	NOUN
ejde-636	13	12	,	,	PUNCT
ejde-636	13	13	presenting	present	VERB
ejde-636	13	14	various	various	ADJ
ejde-636	13	15	mathematical	mathematical	ADJ
ejde-636	13	16	challenges	challenge	NOUN
ejde-636	13	17	from	from	ADP
ejde-636	13	18	the	the	DET
ejde-636	13	19	perspective	perspective	NOUN
ejde-636	13	20	of	of	ADP
ejde-636	13	21	mathematical	mathematical	ADJ
ejde-636	13	22	physics	physics	NOUN
ejde-636	14	1	[	[	X
ejde-636	14	2	4	4	NUM
ejde-636	14	3	,	,	PUNCT
ejde-636	14	4	6	6	NUM
ejde-636	14	5	]	]	PUNCT
ejde-636	14	6	.	.	PUNCT
ejde-636	15	1	over	over	ADP
ejde-636	15	2	the	the	DET
ejde-636	15	3	past	past	ADJ
ejde-636	15	4	decades	decade	NOUN
ejde-636	15	5	,	,	PUNCT
ejde-636	15	6	a	a	DET
ejde-636	15	7	lot	lot	NOUN
ejde-636	15	8	of	of	ADP
ejde-636	15	9	attention	attention	NOUN
ejde-636	15	10	has	have	AUX
ejde-636	15	11	been	be	AUX
ejde-636	15	12	paid	pay	VERB
ejde-636	15	13	to	to	PART
ejde-636	15	14	normalized	normalize	VERB
ejde-636	15	15	solutions	solution	NOUN
ejde-636	15	16	of	of	ADP
ejde-636	15	17	the	the	DET
ejde-636	15	18	nonlinear	nonlinear	ADJ
ejde-636	15	19	schrödinger	schrödinger	NOUN
ejde-636	15	20	equation	equation	NOUN
ejde-636	15	21	with	with	ADP
ejde-636	15	22	both	both	CCONJ
ejde-636	15	23	pure	pure	ADJ
ejde-636	15	24	and	and	CCONJ
ejde-636	15	25	mixed	mixed	ADJ
ejde-636	15	26	nonlinearities	nonlinearitie	NOUN
ejde-636	15	27	[	[	X
ejde-636	15	28	1	1	NUM
ejde-636	15	29	,	,	PUNCT
ejde-636	15	30	7	7	NUM
ejde-636	15	31	,	,	PUNCT
ejde-636	15	32	10	10	NUM
ejde-636	15	33	,	,	PUNCT
ejde-636	15	34	11	11	NUM
ejde-636	15	35	,	,	PUNCT
ejde-636	15	36	12	12	NUM
ejde-636	15	37	,	,	PUNCT
ejde-636	15	38	13	13	NUM
ejde-636	15	39	,	,	PUNCT
ejde-636	15	40	17	17	NUM
ejde-636	15	41	,	,	PUNCT
ejde-636	15	42	18	18	NUM
ejde-636	15	43	,	,	PUNCT
ejde-636	15	44	19	19	NUM
ejde-636	15	45	,	,	PUNCT
ejde-636	15	46	22	22	NUM
ejde-636	15	47	,	,	PUNCT
ejde-636	15	48	23	23	NUM
ejde-636	15	49	,	,	PUNCT
ejde-636	15	50	26	26	NUM
ejde-636	15	51	,	,	PUNCT
ejde-636	15	52	34	34	NUM
ejde-636	15	53	,	,	PUNCT
ejde-636	15	54	35	35	NUM
ejde-636	15	55	,	,	PUNCT
ejde-636	15	56	38	38	NUM
ejde-636	15	57	]	]	PUNCT
ejde-636	15	58	and	and	CCONJ
ejde-636	15	59	the	the	DET
ejde-636	15	60	references	reference	NOUN
ejde-636	15	61	therein	therein	ADV
ejde-636	15	62	.	.	PUNCT
ejde-636	16	1	for	for	ADP
ejde-636	16	2	the	the	DET
ejde-636	16	3	specific	specific	ADJ
ejde-636	16	4	case	case	NOUN
ejde-636	16	5	µ	µ	X
ejde-636	16	6	=	=	SYM
ejde-636	16	7	0	0	NUM
ejde-636	16	8	,	,	PUNCT
ejde-636	16	9	when	when	SCONJ
ejde-636	16	10	2	2	NUM
ejde-636	16	11	<	<	X
ejde-636	16	12	p	p	X
ejde-636	16	13	<	<	X
ejde-636	16	14	2	2	NUM
ejde-636	16	15	+	+	SYM
ejde-636	16	16	4	4	NUM
ejde-636	16	17	n	n	NOUN
ejde-636	16	18	,	,	PUNCT
ejde-636	16	19	all	all	DET
ejde-636	16	20	solutions	solution	NOUN
ejde-636	16	21	to	to	ADP
ejde-636	16	22	(	(	PUNCT
ejde-636	16	23	1.1	1.1	NUM
ejde-636	16	24	)	)	PUNCT
ejde-636	16	25	with	with	ADP
ejde-636	16	26	ϵ	ϵ	PROPN
ejde-636	16	27	=	=	SYM
ejde-636	16	28	0	0	NUM
ejde-636	16	29	exist	exist	VERB
ejde-636	16	30	globally	globally	ADV
ejde-636	16	31	,	,	PUNCT
ejde-636	16	32	and	and	CCONJ
ejde-636	16	33	the	the	DET
ejde-636	16	34	associated	associated	ADJ
ejde-636	16	35	standing	standing	NOUN
ejde-636	16	36	waves	wave	NOUN
ejde-636	16	37	are	be	AUX
ejde-636	16	38	orbitally	orbitally	ADV
ejde-636	16	39	stable	stable	ADJ
ejde-636	16	40	.	.	PUNCT
ejde-636	17	1	however	however	ADV
ejde-636	17	2	,	,	PUNCT
ejde-636	17	3	for	for	ADP
ejde-636	17	4	p	p	PRON
ejde-636	17	5	≥	≥	NUM
ejde-636	17	6	2	2	NUM
ejde-636	17	7	+	+	CCONJ
ejde-636	17	8	4	4	NUM
ejde-636	17	9	n	n	NOUN
ejde-636	17	10	,	,	PUNCT
ejde-636	17	11	the	the	DET
ejde-636	17	12	solutions	solution	NOUN
ejde-636	17	13	to	to	ADP
ejde-636	17	14	equation	equation	NOUN
ejde-636	17	15	(	(	PUNCT
ejde-636	17	16	1.1	1.1	NUM
ejde-636	17	17	)	)	PUNCT
ejde-636	17	18	can	can	AUX
ejde-636	17	19	exhibit	exhibit	VERB
ejde-636	17	20	singularity	singularity	NOUN
ejde-636	17	21	within	within	ADP
ejde-636	17	22	a	a	DET
ejde-636	17	23	finite	finite	ADJ
ejde-636	17	24	time	time	NOUN
ejde-636	17	25	.	.	PUNCT
ejde-636	18	1	to	to	PART
ejde-636	18	2	address	address	VERB
ejde-636	18	3	regularization	regularization	NOUN
ejde-636	18	4	and	and	CCONJ
ejde-636	18	5	stabilization	stabilization	NOUN
ejde-636	18	6	of	of	ADP
ejde-636	18	7	these	these	DET
ejde-636	18	8	solutions	solution	NOUN
ejde-636	18	9	,	,	PUNCT
ejde-636	18	10	karpman	karpman	NOUN
ejde-636	18	11	-	-	PUNCT
ejde-636	18	12	shagalov	shagalov	NOUN
ejde-636	18	13	[	[	X
ejde-636	18	14	21	21	NUM
ejde-636	18	15	,	,	PUNCT
ejde-636	18	16	20	20	NUM
ejde-636	18	17	]	]	PUNCT
ejde-636	18	18	proposed	propose	VERB
ejde-636	18	19	the	the	DET
ejde-636	18	20	inclusion	inclusion	NOUN
ejde-636	18	21	of	of	ADP
ejde-636	18	22	a	a	DET
ejde-636	18	23	small	small	ADJ
ejde-636	18	24	fourth	fourth	ADJ
ejde-636	18	25	-	-	PUNCT
ejde-636	18	26	order	order	NOUN
ejde-636	18	27	dispersion	dispersion	NOUN
ejde-636	18	28	term	term	NOUN
ejde-636	18	29	ϵ∥∆u∥22	ϵ∥∆u∥22	NUM
ejde-636	18	30	in	in	ADP
ejde-636	18	31	the	the	DET
ejde-636	18	32	model	model	NOUN
ejde-636	18	33	.	.	PUNCT
ejde-636	19	1	through	through	ADP
ejde-636	19	2	a	a	DET
ejde-636	19	3	combination	combination	NOUN
ejde-636	19	4	of	of	ADP
ejde-636	19	5	stability	stability	NOUN
ejde-636	19	6	analysis	analysis	NOUN
ejde-636	19	7	and	and	CCONJ
ejde-636	19	8	numerical	numerical	ADJ
ejde-636	19	9	simulations	simulation	NOUN
ejde-636	19	10	,	,	PUNCT
ejde-636	19	11	they	they	PRON
ejde-636	19	12	demonstrated	demonstrate	VERB
ejde-636	19	13	the	the	DET
ejde-636	19	14	stable	stable	ADJ
ejde-636	19	15	outcomes	outcome	NOUN
ejde-636	19	16	for	for	ADP
ejde-636	19	17	2	2	NUM
ejde-636	19	18	<	<	X
ejde-636	19	19	p	p	X
ejde-636	19	20	<	<	X
ejde-636	19	21	2	2	NUM
ejde-636	19	22	+	+	SYM
ejde-636	19	23	8	8	NUM
ejde-636	19	24	n	n	NOUN
ejde-636	19	25	,	,	PUNCT
ejde-636	19	26	while	while	SCONJ
ejde-636	19	27	noting	note	VERB
ejde-636	19	28	the	the	DET
ejde-636	19	29	instability	instability	NOUN
ejde-636	19	30	phenomena	phenomenon	NOUN
ejde-636	19	31	for	for	ADP
ejde-636	19	32	p	p	PRON
ejde-636	19	33	≥	≥	NUM
ejde-636	19	34	2	2	NUM
ejde-636	19	35	+	+	SYM
ejde-636	19	36	8	8	NUM
ejde-636	19	37	n	n	NOUN
ejde-636	19	38	.	.	PUNCT
ejde-636	20	1	consequently	consequently	ADV
ejde-636	20	2	,	,	PUNCT
ejde-636	20	3	p	p	X
ejde-636	20	4	=	=	SYM
ejde-636	20	5	2	2	NUM
ejde-636	20	6	+	+	SYM
ejde-636	21	1	8	8	NUM
ejde-636	21	2	n	n	NUM
ejde-636	21	3	2020	2020	NUM
ejde-636	21	4	mathematics	mathematic	NOUN
ejde-636	21	5	subject	subject	ADJ
ejde-636	21	6	classification	classification	NOUN
ejde-636	21	7	.	.	PUNCT
ejde-636	22	1	35q55	35q55	NUM
ejde-636	22	2	,	,	PUNCT
ejde-636	22	3	31b30	31b30	NUM
ejde-636	22	4	,	,	PUNCT
ejde-636	22	5	35j30	35j30	NUM
ejde-636	22	6	.	.	PUNCT
ejde-636	23	1	key	key	ADJ
ejde-636	23	2	words	word	NOUN
ejde-636	23	3	and	and	CCONJ
ejde-636	23	4	phrases	phrase	NOUN
ejde-636	23	5	.	.	PUNCT
ejde-636	24	1	normalized	normalize	VERB
ejde-636	24	2	solutions	solution	NOUN
ejde-636	24	3	;	;	PUNCT
ejde-636	24	4	schrödinger	schrödinger	NOUN
ejde-636	24	5	equation	equation	NOUN
ejde-636	24	6	;	;	PUNCT
ejde-636	24	7	lagrange	lagrange	NOUN
ejde-636	24	8	multiplier	multiplier	NOUN
ejde-636	24	9	;	;	PUNCT
ejde-636	24	10	ground	ground	NOUN
ejde-636	24	11	states	state	NOUN
ejde-636	24	12	;	;	PUNCT
ejde-636	24	13	nehari	nehari	NOUN
ejde-636	24	14	-	-	PUNCT
ejde-636	24	15	pohozaev	pohozaev	NOUN
ejde-636	24	16	manifold	manifold	NOUN
ejde-636	24	17	.	.	PUNCT
ejde-636	25	1	©	©	PROPN
ejde-636	25	2	2024	2024	NUM
ejde-636	25	3	.	.	PUNCT
ejde-636	26	1	this	this	DET
ejde-636	26	2	work	work	NOUN
ejde-636	26	3	is	be	AUX
ejde-636	26	4	licensed	license	VERB
ejde-636	26	5	under	under	ADP
ejde-636	26	6	a	a	DET
ejde-636	26	7	cc	cc	NOUN
ejde-636	26	8	by	by	ADP
ejde-636	26	9	4.0	4.0	NUM
ejde-636	26	10	license	license	NOUN
ejde-636	26	11	.	.	PUNCT
ejde-636	27	1	submitted	submit	VERB
ejde-636	27	2	november	november	PROPN
ejde-636	27	3	18	18	NUM
ejde-636	27	4	,	,	PUNCT
ejde-636	27	5	2023	2023	NUM
ejde-636	27	6	.	.	PUNCT
ejde-636	28	1	published	publish	VERB
ejde-636	28	2	april	april	PROPN
ejde-636	28	3	1	1	NUM
ejde-636	28	4	,	,	PUNCT
ejde-636	28	5	2024	2024	NUM
ejde-636	28	6	.	.	PUNCT
ejde-636	29	1	1	1	NUM
ejde-636	29	2	2	2	NUM
ejde-636	29	3	z.	z.	PROPN
ejde-636	29	4	ma	ma	PROPN
ejde-636	29	5	,	,	PUNCT
ejde-636	29	6	x.	x.	PROPN
ejde-636	29	7	chang	chang	PROPN
ejde-636	29	8	,	,	PUNCT
ejde-636	29	9	z.	z.	PROPN
ejde-636	29	10	feng	feng	PROPN
ejde-636	29	11	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	29	12	appears	appear	VERB
ejde-636	29	13	as	as	ADP
ejde-636	29	14	a	a	DET
ejde-636	29	15	new	new	ADJ
ejde-636	29	16	mass	mass	ADJ
ejde-636	29	17	critical	critical	ADJ
ejde-636	29	18	exponent	exponent	NOUN
ejde-636	29	19	.	.	PUNCT
ejde-636	30	1	despite	despite	SCONJ
ejde-636	30	2	the	the	DET
ejde-636	30	3	significance	significance	NOUN
ejde-636	30	4	of	of	ADP
ejde-636	30	5	the	the	DET
ejde-636	30	6	mixed	mixed	ADJ
ejde-636	30	7	dispersion	dispersion	NOUN
ejde-636	30	8	fourth	fourth	ADJ
ejde-636	30	9	-	-	PUNCT
ejde-636	30	10	order	order	NOUN
ejde-636	30	11	nonlinear	nonlinear	NOUN
ejde-636	30	12	schrödinger	schrödinger	NOUN
ejde-636	30	13	equation	equation	NOUN
ejde-636	30	14	in	in	ADP
ejde-636	30	15	physical	physical	ADJ
ejde-636	30	16	contexts	contexts	NOUN
ejde-636	30	17	,	,	PUNCT
ejde-636	30	18	it	it	PRON
ejde-636	30	19	remains	remains	AUX
ejde-636	30	20	inadequately	inadequately	ADV
ejde-636	30	21	understood	understand	VERB
ejde-636	30	22	,	,	PUNCT
ejde-636	30	23	as	as	SCONJ
ejde-636	30	24	addressed	address	VERB
ejde-636	30	25	in	in	ADP
ejde-636	30	26	[	[	X
ejde-636	30	27	4	4	NUM
ejde-636	30	28	,	,	PUNCT
ejde-636	30	29	8	8	NUM
ejde-636	30	30	,	,	PUNCT
ejde-636	30	31	15	15	NUM
ejde-636	30	32	,	,	PUNCT
ejde-636	30	33	29	29	NUM
ejde-636	30	34	,	,	PUNCT
ejde-636	30	35	30	30	NUM
ejde-636	30	36	,	,	PUNCT
ejde-636	30	37	32	32	NUM
ejde-636	30	38	]	]	PUNCT
ejde-636	30	39	.	.	PUNCT
ejde-636	31	1	in	in	ADP
ejde-636	31	2	this	this	DET
ejde-636	31	3	article	article	NOUN
ejde-636	31	4	,	,	PUNCT
ejde-636	31	5	we	we	PRON
ejde-636	31	6	are	be	AUX
ejde-636	31	7	concerned	concern	VERB
ejde-636	31	8	with	with	ADP
ejde-636	31	9	equation	equation	NOUN
ejde-636	31	10	(	(	PUNCT
ejde-636	31	11	1.1	1.1	NUM
ejde-636	31	12	)	)	PUNCT
ejde-636	31	13	and	and	CCONJ
ejde-636	31	14	its	its	PRON
ejde-636	31	15	standing	stand	VERB
ejde-636	31	16	waves	wave	NOUN
ejde-636	31	17	solutions	solution	NOUN
ejde-636	31	18	of	of	ADP
ejde-636	31	19	the	the	DET
ejde-636	31	20	form	form	NOUN
ejde-636	31	21	ψ(t	ψ(t	PROPN
ejde-636	31	22	,	,	PUNCT
ejde-636	31	23	x	x	NOUN
ejde-636	31	24	)	)	PUNCT
ejde-636	31	25	=	=	SYM
ejde-636	31	26	eiωtu(x	eiωtu(x	PROPN
ejde-636	31	27	)	)	PUNCT
ejde-636	31	28	,	,	PUNCT
ejde-636	31	29	where	where	SCONJ
ejde-636	31	30	ω	ω	PROPN
ejde-636	31	31	∈	∈	PROPN
ejde-636	31	32	r	r	NOUN
ejde-636	31	33	is	be	AUX
ejde-636	31	34	a	a	DET
ejde-636	31	35	lagrange	lagrange	NOUN
ejde-636	31	36	multiplier	multipli	ADJ
ejde-636	31	37	and	and	CCONJ
ejde-636	31	38	u(x	u(x	NOUN
ejde-636	31	39	)	)	PUNCT
ejde-636	31	40	satisfies	satisfie	NOUN
ejde-636	31	41	ϵ∆2u−	ϵ∆2u−	PROPN
ejde-636	31	42	γ∆u+	γ∆u+	PUNCT
ejde-636	31	43	ωu−	ωu−	NUM
ejde-636	31	44	µ|u|q−2u−	µ|u|q−2u−	NUM
ejde-636	31	45	|u|p−2u	|u|p−2u	NOUN
ejde-636	32	1	=	=	NOUN
ejde-636	32	2	0	0	NUM
ejde-636	32	3	in	in	ADP
ejde-636	32	4	rn	rn	PROPN
ejde-636	32	5	.	.	PUNCT
ejde-636	33	1	(	(	PUNCT
ejde-636	33	2	1.2	1.2	NUM
ejde-636	33	3	)	)	PUNCT
ejde-636	33	4	when	when	SCONJ
ejde-636	33	5	we	we	PRON
ejde-636	33	6	consider	consider	VERB
ejde-636	33	7	solutions	solution	NOUN
ejde-636	33	8	to	to	ADP
ejde-636	33	9	(	(	PUNCT
ejde-636	33	10	1.2	1.2	NUM
ejde-636	33	11	)	)	PUNCT
ejde-636	33	12	,	,	PUNCT
ejde-636	33	13	a	a	DET
ejde-636	33	14	possible	possible	ADJ
ejde-636	33	15	choice	choice	NOUN
ejde-636	33	16	is	be	AUX
ejde-636	33	17	to	to	PART
ejde-636	33	18	consider	consider	VERB
ejde-636	33	19	a	a	DET
ejde-636	33	20	fixed	fix	VERB
ejde-636	33	21	value	value	NOUN
ejde-636	33	22	ω	ω	NUM
ejde-636	33	23	∈	∈	PROPN
ejde-636	33	24	r	r	NOUN
ejde-636	33	25	and	and	CCONJ
ejde-636	33	26	search	search	NOUN
ejde-636	33	27	for	for	ADP
ejde-636	33	28	solutions	solution	NOUN
ejde-636	33	29	as	as	ADP
ejde-636	33	30	the	the	DET
ejde-636	33	31	critical	critical	ADJ
ejde-636	33	32	points	point	NOUN
ejde-636	33	33	of	of	ADP
ejde-636	33	34	the	the	DET
ejde-636	33	35	action	action	NOUN
ejde-636	33	36	functional	functional	ADJ
ejde-636	33	37	aω,µ(u	aω,µ(u	NOUN
ejde-636	33	38	)	)	PUNCT
ejde-636	34	1	=	=	PUNCT
ejde-636	34	2	ϵ	ϵ	ADP
ejde-636	34	3	2	2	NUM
ejde-636	34	4	∥∆u∥22	∥∆u∥22	X
ejde-636	34	5	+	+	CCONJ
ejde-636	34	6	γ	γ	X
ejde-636	34	7	2	2	NUM
ejde-636	34	8	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	34	9	+	+	CCONJ
ejde-636	34	10	ω	ω	NUM
ejde-636	34	11	2	2	NUM
ejde-636	34	12	∥u∥22	∥u∥22	PROPN
ejde-636	35	1	−	−	PROPN
ejde-636	35	2	µ	µ	X
ejde-636	35	3	q	q	X
ejde-636	35	4	∥u∥qq	∥u∥qq	ADP
ejde-636	35	5	−	−	PROPN
ejde-636	35	6	1	1	NUM
ejde-636	35	7	p	p	NOUN
ejde-636	35	8	∥u∥pp	∥u∥pp	NOUN
ejde-636	35	9	.	.	PUNCT
ejde-636	36	1	in	in	ADP
ejde-636	36	2	this	this	DET
ejde-636	36	3	case	case	NOUN
ejde-636	36	4	,	,	PUNCT
ejde-636	36	5	we	we	PRON
ejde-636	36	6	focus	focus	VERB
ejde-636	36	7	on	on	ADP
ejde-636	36	8	the	the	DET
ejde-636	36	9	existence	existence	NOUN
ejde-636	36	10	of	of	ADP
ejde-636	36	11	minimal	minimal	ADJ
ejde-636	36	12	action	action	NOUN
ejde-636	36	13	solutions	solution	NOUN
ejde-636	36	14	,	,	PUNCT
ejde-636	36	15	namely	namely	ADV
ejde-636	36	16	,	,	PUNCT
ejde-636	36	17	solutions	solution	NOUN
ejde-636	36	18	minimizing	minimize	VERB
ejde-636	36	19	aω,µ	aω,µ	PUNCT
ejde-636	36	20	among	among	ADP
ejde-636	36	21	all	all	DET
ejde-636	36	22	non	non	ADJ
ejde-636	36	23	-	-	ADJ
ejde-636	36	24	trivial	trivial	ADJ
ejde-636	36	25	solutions	solution	NOUN
ejde-636	36	26	[	[	X
ejde-636	36	27	6	6	NUM
ejde-636	36	28	,	,	PUNCT
ejde-636	36	29	3	3	NUM
ejde-636	36	30	]	]	PUNCT
ejde-636	36	31	.	.	PUNCT
ejde-636	37	1	alternatively	alternatively	ADV
ejde-636	37	2	,	,	PUNCT
ejde-636	37	3	we	we	PRON
ejde-636	37	4	can	can	AUX
ejde-636	37	5	search	search	VERB
ejde-636	37	6	for	for	ADP
ejde-636	37	7	solutions	solution	NOUN
ejde-636	37	8	to	to	ADP
ejde-636	37	9	(	(	PUNCT
ejde-636	37	10	1.2	1.2	NUM
ejde-636	37	11	)	)	PUNCT
ejde-636	37	12	with	with	ADP
ejde-636	37	13	a	a	DET
ejde-636	37	14	prescribed	prescribed	ADJ
ejde-636	37	15	l2	l2	NOUN
ejde-636	37	16	-	-	PUNCT
ejde-636	37	17	norm	norm	NOUN
ejde-636	37	18	.	.	PUNCT
ejde-636	38	1	define	define	VERB
ejde-636	38	2	the	the	DET
ejde-636	38	3	energy	energy	NOUN
ejde-636	38	4	functional	functional	ADJ
ejde-636	38	5	on	on	ADP
ejde-636	38	6	h2	h2	PROPN
ejde-636	38	7	=	=	SYM
ejde-636	38	8	h2(rn	h2(rn	PROPN
ejde-636	38	9	,	,	PUNCT
ejde-636	38	10	c	c	NOUN
ejde-636	38	11	)	)	PUNCT
ejde-636	38	12	by	by	ADP
ejde-636	38	13	ep	ep	PROPN
ejde-636	38	14	,	,	PUNCT
ejde-636	38	15	q(u	q(u	NOUN
ejde-636	38	16	)	)	PUNCT
ejde-636	38	17	:	:	PUNCT
ejde-636	39	1	=	=	SYM
ejde-636	39	2	ϵ	ϵ	SYM
ejde-636	39	3	2	2	NUM
ejde-636	39	4	∥∆u∥22	∥∆u∥22	X
ejde-636	39	5	+	+	CCONJ
ejde-636	39	6	γ	γ	X
ejde-636	39	7	2	2	NUM
ejde-636	39	8	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	39	9	−	−	PROPN
ejde-636	39	10	µ	µ	X
ejde-636	39	11	q	q	X
ejde-636	39	12	∥u∥qq	∥u∥qq	ADP
ejde-636	39	13	−	−	PROPN
ejde-636	39	14	1	1	NUM
ejde-636	39	15	p	p	NOUN
ejde-636	39	16	∥u∥pp	∥u∥pp	PROPN
ejde-636	39	17	.	.	PUNCT
ejde-636	40	1	it	it	PRON
ejde-636	40	2	is	be	AUX
ejde-636	40	3	standard	standard	ADJ
ejde-636	40	4	to	to	PART
ejde-636	40	5	check	check	VERB
ejde-636	40	6	that	that	SCONJ
ejde-636	40	7	ep	ep	PROPN
ejde-636	40	8	,	,	PUNCT
ejde-636	40	9	q	q	X
ejde-636	40	10	is	be	AUX
ejde-636	40	11	of	of	ADP
ejde-636	40	12	class	class	NOUN
ejde-636	40	13	c1	c1	PROPN
ejde-636	40	14	and	and	CCONJ
ejde-636	40	15	a	a	DET
ejde-636	40	16	critical	critical	ADJ
ejde-636	40	17	point	point	NOUN
ejde-636	40	18	of	of	ADP
ejde-636	40	19	ep	ep	PROPN
ejde-636	40	20	,	,	PUNCT
ejde-636	40	21	q	q	PUNCT
ejde-636	40	22	restricted	restrict	VERB
ejde-636	40	23	to	to	ADP
ejde-636	40	24	the	the	DET
ejde-636	40	25	mass	mass	ADJ
ejde-636	40	26	constraint	constraint	NOUN
ejde-636	40	27	s(c	s(c	NOUN
ejde-636	40	28	)	)	PUNCT
ejde-636	40	29	=	=	PRON
ejde-636	40	30	{	{	PUNCT
ejde-636	40	31	u	u	NOUN
ejde-636	40	32	∈	∈	PROPN
ejde-636	40	33	h2	h2	NOUN
ejde-636	40	34	:	:	PUNCT
ejde-636	41	1	∥u∥22	∥u∥22	PROPN
ejde-636	41	2	=	=	SYM
ejde-636	41	3	c	c	AUX
ejde-636	41	4	}	}	PUNCT
ejde-636	41	5	gives	give	VERB
ejde-636	41	6	rise	rise	NOUN
ejde-636	41	7	to	to	ADP
ejde-636	41	8	a	a	DET
ejde-636	41	9	solution	solution	NOUN
ejde-636	41	10	to	to	ADP
ejde-636	41	11	(	(	PUNCT
ejde-636	41	12	1.2	1.2	NUM
ejde-636	41	13	)	)	PUNCT
ejde-636	41	14	with	with	ADP
ejde-636	41	15	∥u∥22	∥u∥22	PROPN
ejde-636	41	16	=	=	PROPN
ejde-636	41	17	c.	c.	PROPN
ejde-636	41	18	if	if	SCONJ
ejde-636	41	19	µ	µ	X
ejde-636	41	20	=	=	SYM
ejde-636	41	21	0	0	NUM
ejde-636	41	22	,	,	PUNCT
ejde-636	41	23	the	the	DET
ejde-636	41	24	corresponding	corresponding	ADJ
ejde-636	41	25	functional	functional	NOUN
ejde-636	41	26	is	be	AUX
ejde-636	41	27	denoted	denote	VERB
ejde-636	41	28	by	by	ADP
ejde-636	41	29	ep	ep	PROPN
ejde-636	41	30	.	.	PUNCT
ejde-636	42	1	when	when	SCONJ
ejde-636	42	2	ϵ	ϵ	X
ejde-636	42	3	>	>	X
ejde-636	42	4	0	0	NUM
ejde-636	42	5	and	and	CCONJ
ejde-636	42	6	γ	γ	X
ejde-636	42	7	>	>	X
ejde-636	42	8	0	0	NUM
ejde-636	42	9	,	,	PUNCT
ejde-636	42	10	with	with	ADP
ejde-636	42	11	a	a	DET
ejde-636	42	12	pure	pure	ADJ
ejde-636	42	13	mass	mass	ADJ
ejde-636	42	14	subcritical	subcritical	ADJ
ejde-636	42	15	nonlinearity	nonlinearity	NOUN
ejde-636	42	16	,	,	PUNCT
ejde-636	42	17	i.e.	i.e.	X
ejde-636	42	18	,	,	PUNCT
ejde-636	42	19	2	2	NUM
ejde-636	42	20	<	<	X
ejde-636	42	21	p	p	X
ejde-636	42	22	<	<	X
ejde-636	42	23	p	p	NOUN
ejde-636	42	24	as	as	SCONJ
ejde-636	42	25	considered	consider	VERB
ejde-636	42	26	in	in	ADP
ejde-636	42	27	[	[	X
ejde-636	42	28	5	5	NUM
ejde-636	42	29	]	]	PUNCT
ejde-636	42	30	,	,	PUNCT
ejde-636	42	31	the	the	DET
ejde-636	42	32	functional	functional	ADJ
ejde-636	42	33	ep	ep	PROPN
ejde-636	42	34	has	have	AUX
ejde-636	42	35	been	be	AUX
ejde-636	42	36	shown	show	VERB
ejde-636	42	37	to	to	PART
ejde-636	42	38	be	be	AUX
ejde-636	42	39	bounded	bound	VERB
ejde-636	42	40	from	from	ADP
ejde-636	42	41	below	below	ADP
ejde-636	42	42	on	on	ADP
ejde-636	42	43	s(c	s(c	NUM
ejde-636	42	44	)	)	PUNCT
ejde-636	42	45	,	,	PUNCT
ejde-636	42	46	and	and	CCONJ
ejde-636	42	47	critical	critical	ADJ
ejde-636	42	48	points	point	NOUN
ejde-636	42	49	of	of	ADP
ejde-636	42	50	e	e	NOUN
ejde-636	42	51	can	can	AUX
ejde-636	42	52	be	be	AUX
ejde-636	42	53	sought	seek	VERB
ejde-636	42	54	as	as	ADP
ejde-636	42	55	global	global	ADJ
ejde-636	42	56	minimizers	minimizer	NOUN
ejde-636	42	57	for	for	ADP
ejde-636	42	58	any	any	DET
ejde-636	42	59	c	c	PROPN
ejde-636	42	60	>	>	X
ejde-636	43	1	0	0	X
ejde-636	43	2	.	.	PUNCT
ejde-636	43	3	bonheure	bonheure	NOUN
ejde-636	43	4	et	et	PROPN
ejde-636	43	5	al	al	PROPN
ejde-636	44	1	[	[	X
ejde-636	44	2	3	3	NUM
ejde-636	44	3	]	]	PUNCT
ejde-636	44	4	investigated	investigate	VERB
ejde-636	44	5	the	the	DET
ejde-636	44	6	existence	existence	NOUN
ejde-636	44	7	of	of	ADP
ejde-636	44	8	normalized	normalize	VERB
ejde-636	44	9	ground	ground	NOUN
ejde-636	44	10	states	state	NOUN
ejde-636	44	11	of	of	ADP
ejde-636	44	12	(	(	PUNCT
ejde-636	44	13	1.2	1.2	NUM
ejde-636	44	14	)	)	PUNCT
ejde-636	44	15	by	by	ADP
ejde-636	44	16	exploiting	exploit	VERB
ejde-636	44	17	the	the	DET
ejde-636	44	18	constrained	constrain	VERB
ejde-636	44	19	minimization	minimization	NOUN
ejde-636	44	20	method	method	NOUN
ejde-636	44	21	and	and	CCONJ
ejde-636	44	22	explored	explore	VERB
ejde-636	44	23	the	the	DET
ejde-636	44	24	normalized	normalize	VERB
ejde-636	44	25	solutions	solution	NOUN
ejde-636	44	26	of	of	ADP
ejde-636	44	27	equation	equation	NOUN
ejde-636	44	28	(	(	PUNCT
ejde-636	44	29	1.2	1.2	NUM
ejde-636	44	30	)	)	PUNCT
ejde-636	44	31	with	with	ADP
ejde-636	44	32	pure	pure	ADJ
ejde-636	44	33	mass	mass	NOUN
ejde-636	44	34	-	-	PUNCT
ejde-636	44	35	critical	critical	ADJ
ejde-636	44	36	and	and	CCONJ
ejde-636	44	37	mass	mass	ADJ
ejde-636	44	38	-	-	PUNCT
ejde-636	44	39	supcritical	supcritical	ADJ
ejde-636	44	40	nonlinearity	nonlinearity	NOUN
ejde-636	44	41	,	,	PUNCT
ejde-636	44	42	i.e.	i.e.	X
ejde-636	44	43	,	,	PUNCT
ejde-636	45	1	p	p	NOUN
ejde-636	45	2	≤	≤	NOUN
ejde-636	45	3	p	p	NOUN
ejde-636	45	4	<	<	X
ejde-636	45	5	4∗.	4∗.	NUM
ejde-636	45	6	when	when	SCONJ
ejde-636	45	7	ϵ	ϵ	PROPN
ejde-636	45	8	=	=	SYM
ejde-636	45	9	1	1	NUM
ejde-636	45	10	,	,	PUNCT
ejde-636	45	11	γ	γ	X
ejde-636	45	12	<	<	X
ejde-636	45	13	0	0	NUM
ejde-636	45	14	and	and	CCONJ
ejde-636	45	15	µ	µ	X
ejde-636	45	16	=	=	SYM
ejde-636	45	17	0	0	NUM
ejde-636	45	18	,	,	PUNCT
ejde-636	45	19	luo	luo	PROPN
ejde-636	45	20	et	et	PROPN
ejde-636	45	21	al	al	PROPN
ejde-636	46	1	[	[	X
ejde-636	46	2	24	24	NUM
ejde-636	46	3	]	]	PUNCT
ejde-636	46	4	used	use	VERB
ejde-636	46	5	a	a	DET
ejde-636	46	6	profile	profile	NOUN
ejde-636	46	7	decomposition	decomposition	NOUN
ejde-636	46	8	technique	technique	NOUN
ejde-636	46	9	to	to	PART
ejde-636	46	10	study	study	VERB
ejde-636	46	11	the	the	DET
ejde-636	46	12	existence	existence	NOUN
ejde-636	46	13	of	of	ADP
ejde-636	46	14	ground	ground	NOUN
ejde-636	46	15	states	state	NOUN
ejde-636	46	16	for	for	ADP
ejde-636	46	17	(	(	PUNCT
ejde-636	46	18	1.2	1.2	NUM
ejde-636	46	19	)	)	PUNCT
ejde-636	46	20	with	with	ADP
ejde-636	46	21	c	c	NOUN
ejde-636	46	22	=	=	SYM
ejde-636	46	23	1	1	NUM
ejde-636	46	24	and	and	CCONJ
ejde-636	46	25	2	2	NUM
ejde-636	46	26	<	<	X
ejde-636	46	27	p	p	X
ejde-636	46	28	≤	≤	PROPN
ejde-636	46	29	p.	p.	NOUN
ejde-636	46	30	boussaid	boussaid	VERB
ejde-636	46	31	et	et	PROPN
ejde-636	46	32	al	al	PROPN
ejde-636	47	1	[	[	X
ejde-636	47	2	9	9	NUM
ejde-636	47	3	]	]	PUNCT
ejde-636	47	4	obtained	obtain	VERB
ejde-636	47	5	the	the	DET
ejde-636	47	6	existence	existence	NOUN
ejde-636	47	7	of	of	ADP
ejde-636	47	8	normalized	normalize	VERB
ejde-636	47	9	ground	ground	NOUN
ejde-636	47	10	state	state	NOUN
ejde-636	47	11	solutions	solution	NOUN
ejde-636	47	12	for	for	ADP
ejde-636	47	13	all	all	DET
ejde-636	47	14	c	c	PROPN
ejde-636	47	15	>	>	X
ejde-636	47	16	0	0	PROPN
ejde-636	47	17	,	,	PUNCT
ejde-636	47	18	γ	γ	X
ejde-636	47	19	<	<	X
ejde-636	47	20	0	0	NUM
ejde-636	47	21	and	and	CCONJ
ejde-636	47	22	2	2	NUM
ejde-636	47	23	<	<	X
ejde-636	47	24	p	p	X
ejde-636	47	25	≤	≤	ADJ
ejde-636	47	26	p	p	NOUN
ejde-636	47	27	without	without	ADP
ejde-636	47	28	the	the	DET
ejde-636	47	29	restriction	restriction	NOUN
ejde-636	47	30	on	on	ADP
ejde-636	47	31	c	c	PROPN
ejde-636	47	32	and	and	CCONJ
ejde-636	47	33	γ	γ	NOUN
ejde-636	47	34	imposed	impose	VERB
ejde-636	47	35	in	in	ADP
ejde-636	47	36	[	[	X
ejde-636	47	37	24	24	NUM
ejde-636	47	38	]	]	PUNCT
ejde-636	47	39	.	.	PUNCT
ejde-636	48	1	for	for	ADP
ejde-636	48	2	p	p	NOUN
ejde-636	48	3	<	<	X
ejde-636	48	4	p	p	X
ejde-636	48	5	<	<	X
ejde-636	48	6	4∗	4∗	NOUN
ejde-636	48	7	,	,	PUNCT
ejde-636	48	8	luo	luo	PROPN
ejde-636	48	9	-	-	PROPN
ejde-636	48	10	yang	yang	PROPN
ejde-636	48	11	[	[	X
ejde-636	48	12	25	25	NUM
ejde-636	48	13	]	]	PUNCT
ejde-636	48	14	identified	identify	VERB
ejde-636	48	15	at	at	ADP
ejde-636	48	16	least	least	ADV
ejde-636	48	17	two	two	NUM
ejde-636	48	18	radial	radial	ADJ
ejde-636	48	19	normalized	normalize	VERB
ejde-636	48	20	solutions	solution	NOUN
ejde-636	48	21	:	:	PUNCT
ejde-636	48	22	a	a	DET
ejde-636	48	23	ground	ground	NOUN
ejde-636	48	24	state	state	NOUN
ejde-636	48	25	and	and	CCONJ
ejde-636	48	26	an	an	DET
ejde-636	48	27	excited	excited	ADJ
ejde-636	48	28	state	state	NOUN
ejde-636	48	29	,	,	PUNCT
ejde-636	48	30	along	along	ADP
ejde-636	48	31	with	with	ADP
ejde-636	48	32	associated	associated	ADJ
ejde-636	48	33	asymptotic	asymptotic	ADJ
ejde-636	48	34	properties	property	NOUN
ejde-636	48	35	.	.	PUNCT
ejde-636	49	1	recently	recently	ADV
ejde-636	49	2	,	,	PUNCT
ejde-636	49	3	fernández	fernández	PROPN
ejde-636	49	4	et	et	NOUN
ejde-636	49	5	al	al	PROPN
ejde-636	50	1	[	[	X
ejde-636	50	2	14	14	NUM
ejde-636	50	3	]	]	PUNCT
ejde-636	50	4	utilized	utilize	VERB
ejde-636	50	5	the	the	DET
ejde-636	50	6	tomas	tomas	PROPN
ejde-636	50	7	-	-	PUNCT
ejde-636	50	8	stein	stein	PROPN
ejde-636	50	9	inequality	inequality	NOUN
ejde-636	50	10	to	to	PART
ejde-636	50	11	develop	develop	VERB
ejde-636	50	12	a	a	DET
ejde-636	50	13	novel	novel	ADJ
ejde-636	50	14	approach	approach	NOUN
ejde-636	50	15	for	for	ADP
ejde-636	50	16	establishing	establish	VERB
ejde-636	50	17	non	non	ADJ
ejde-636	50	18	-	-	ADJ
ejde-636	50	19	homogeneous	homogeneous	ADJ
ejde-636	50	20	gagliardo	gagliardo	NOUN
ejde-636	50	21	-	-	PUNCT
ejde-636	50	22	nirenberg	nirenberg	NOUN
ejde-636	50	23	-	-	PUNCT
ejde-636	50	24	type	type	NOUN
ejde-636	50	25	inequalities	inequality	NOUN
ejde-636	50	26	in	in	ADP
ejde-636	50	27	rn	rn	PROPN
ejde-636	50	28	.	.	PUNCT
ejde-636	51	1	these	these	DET
ejde-636	51	2	inequalities	inequality	NOUN
ejde-636	51	3	play	play	VERB
ejde-636	51	4	a	a	DET
ejde-636	51	5	crucial	crucial	ADJ
ejde-636	51	6	role	role	NOUN
ejde-636	51	7	in	in	ADP
ejde-636	51	8	proving	prove	VERB
ejde-636	51	9	optimal	optimal	ADJ
ejde-636	51	10	results	result	NOUN
ejde-636	51	11	regarding	regard	VERB
ejde-636	51	12	the	the	DET
ejde-636	51	13	existence	existence	NOUN
ejde-636	51	14	of	of	ADP
ejde-636	51	15	global	global	ADJ
ejde-636	51	16	minimizers	minimizer	NOUN
ejde-636	51	17	for	for	ADP
ejde-636	51	18	2	2	NUM
ejde-636	51	19	<	<	X
ejde-636	51	20	q	q	PROPN
ejde-636	51	21	≤	≤	PROPN
ejde-636	51	22	p.	p.	NOUN
ejde-636	51	23	additionally	additionally	ADV
ejde-636	51	24	,	,	PUNCT
ejde-636	51	25	for	for	ADP
ejde-636	51	26	the	the	DET
ejde-636	51	27	case	case	NOUN
ejde-636	51	28	2	2	NUM
ejde-636	51	29	<	<	X
ejde-636	51	30	q	q	X
ejde-636	51	31	≤	≤	PROPN
ejde-636	51	32	p	p	X
ejde-636	51	33	,	,	PUNCT
ejde-636	51	34	they	they	PRON
ejde-636	51	35	showed	show	VERB
ejde-636	51	36	the	the	DET
ejde-636	51	37	existence	existence	NOUN
ejde-636	51	38	of	of	ADP
ejde-636	51	39	local	local	ADJ
ejde-636	51	40	minimizers	minimizer	NOUN
ejde-636	51	41	in	in	ADP
ejde-636	51	42	h2(rn	h2(rn	PROPN
ejde-636	51	43	)	)	PUNCT
ejde-636	51	44	but	but	CCONJ
ejde-636	51	45	not	not	PART
ejde-636	51	46	h2	h2	NOUN
ejde-636	51	47	r	r	NOUN
ejde-636	51	48	(	(	PUNCT
ejde-636	51	49	rn	rn	PROPN
ejde-636	51	50	)	)	PUNCT
ejde-636	51	51	.	.	PUNCT
ejde-636	52	1	when	when	SCONJ
ejde-636	52	2	ϵ	ϵ	X
ejde-636	52	3	>	>	X
ejde-636	52	4	0	0	NUM
ejde-636	52	5	and	and	CCONJ
ejde-636	52	6	γ	γ	X
ejde-636	52	7	=	=	SYM
ejde-636	52	8	0	0	NUM
ejde-636	52	9	,	,	PUNCT
ejde-636	52	10	equation	equation	NOUN
ejde-636	52	11	(	(	PUNCT
ejde-636	52	12	1.1	1.1	NUM
ejde-636	52	13	)	)	PUNCT
ejde-636	52	14	becomes	become	VERB
ejde-636	52	15	the	the	DET
ejde-636	52	16	biharmonic	biharmonic	NOUN
ejde-636	52	17	nonlinear	nonlinear	PROPN
ejde-636	52	18	schrödinger	schrödinger	NOUN
ejde-636	52	19	equation	equation	NOUN
ejde-636	52	20	,	,	PUNCT
ejde-636	52	21	in	in	ADP
ejde-636	52	22	which	which	PRON
ejde-636	52	23	the	the	DET
ejde-636	52	24	stability	stability	NOUN
ejde-636	52	25	of	of	ADP
ejde-636	52	26	solitons	soliton	NOUN
ejde-636	52	27	in	in	ADP
ejde-636	52	28	magnetic	magnetic	ADJ
ejde-636	52	29	materials	material	NOUN
ejde-636	52	30	was	be	AUX
ejde-636	52	31	investigated	investigate	VERB
ejde-636	52	32	[	[	X
ejde-636	52	33	16	16	NUM
ejde-636	52	34	,	,	PUNCT
ejde-636	52	35	37	37	NUM
ejde-636	52	36	]	]	PUNCT
ejde-636	52	37	.	.	PUNCT
ejde-636	53	1	phan	phan	PROPN
ejde-636	54	1	[	[	X
ejde-636	54	2	33	33	NUM
ejde-636	54	3	]	]	PUNCT
ejde-636	54	4	presented	present	VERB
ejde-636	54	5	the	the	DET
ejde-636	54	6	existence	existence	NOUN
ejde-636	54	7	of	of	ADP
ejde-636	54	8	normalized	normalize	VERB
ejde-636	54	9	ground	ground	NOUN
ejde-636	54	10	state	state	NOUN
ejde-636	54	11	solutions	solution	NOUN
ejde-636	54	12	of	of	ADP
ejde-636	54	13	(	(	PUNCT
ejde-636	54	14	1.1	1.1	NUM
ejde-636	54	15	)	)	PUNCT
ejde-636	54	16	for	for	ADP
ejde-636	54	17	ϵ	ϵ	PROPN
ejde-636	54	18	>	>	SYM
ejde-636	54	19	0	0	PROPN
ejde-636	54	20	and	and	CCONJ
ejde-636	54	21	γ	γ	X
ejde-636	54	22	=	=	SYM
ejde-636	54	23	0	0	NUM
ejde-636	54	24	with	with	ADP
ejde-636	54	25	the	the	DET
ejde-636	54	26	pure	pure	ADJ
ejde-636	54	27	mass	mass	ADJ
ejde-636	54	28	-	-	PUNCT
ejde-636	54	29	critical	critical	ADJ
ejde-636	54	30	nonlinearity	nonlinearity	NOUN
ejde-636	54	31	.	.	PUNCT
ejde-636	55	1	the	the	DET
ejde-636	55	2	case	case	NOUN
ejde-636	55	3	involving	involve	VERB
ejde-636	55	4	mass	mass	ADJ
ejde-636	55	5	supercritical	supercritical	ADJ
ejde-636	55	6	nonlinearities	nonlinearitie	NOUN
ejde-636	55	7	was	be	AUX
ejde-636	55	8	discussed	discuss	VERB
ejde-636	55	9	in	in	ADP
ejde-636	55	10	[	[	X
ejde-636	55	11	27	27	NUM
ejde-636	55	12	]	]	PUNCT
ejde-636	55	13	,	,	PUNCT
ejde-636	55	14	where	where	SCONJ
ejde-636	55	15	normalized	normalize	VERB
ejde-636	55	16	ground	ground	NOUN
ejde-636	55	17	states	state	NOUN
ejde-636	55	18	were	be	AUX
ejde-636	55	19	shown	show	VERB
ejde-636	55	20	to	to	PART
ejde-636	55	21	exist	exist	VERB
ejde-636	55	22	for	for	ADP
ejde-636	55	23	2	2	NUM
ejde-636	55	24	<	<	X
ejde-636	55	25	q	q	X
ejde-636	55	26	<	<	X
ejde-636	55	27	p	p	X
ejde-636	55	28	<	<	X
ejde-636	55	29	p	p	X
ejde-636	55	30	=	=	PROPN
ejde-636	55	31	4∗.	4∗.	NUM
ejde-636	55	32	the	the	DET
ejde-636	55	33	existence	existence	NOUN
ejde-636	55	34	of	of	ADP
ejde-636	55	35	normalized	normalize	VERB
ejde-636	55	36	ground	ground	NOUN
ejde-636	55	37	state	state	NOUN
ejde-636	55	38	solutions	solution	NOUN
ejde-636	55	39	for	for	ADP
ejde-636	55	40	p	p	PROPN
ejde-636	55	41	≤	≤	ADJ
ejde-636	55	42	q	q	NOUN
ejde-636	55	43	<	<	AUX
ejde-636	55	44	p	p	X
ejde-636	55	45	≤	≤	PUNCT
ejde-636	55	46	4∗	4∗	NUM
ejde-636	55	47	was	be	AUX
ejde-636	55	48	shown	show	VERB
ejde-636	55	49	in	in	ADP
ejde-636	55	50	[	[	X
ejde-636	55	51	28	28	NUM
ejde-636	55	52	]	]	PUNCT
ejde-636	55	53	.	.	PUNCT
ejde-636	56	1	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	56	2	dispersion	dispersion	NOUN
ejde-636	56	3	nonlinear	nonlinear	NOUN
ejde-636	56	4	schrödinger	schrödinger	NOUN
ejde-636	56	5	equation	equation	NOUN
ejde-636	56	6	3	3	NUM
ejde-636	56	7	as	as	ADP
ejde-636	56	8	for	for	ADP
ejde-636	56	9	the	the	DET
ejde-636	56	10	case	case	NOUN
ejde-636	56	11	ϵ	ϵ	X
ejde-636	56	12	>	>	X
ejde-636	56	13	0	0	PROPN
ejde-636	56	14	,	,	PUNCT
ejde-636	56	15	γ	γ	X
ejde-636	56	16	>	>	X
ejde-636	56	17	0	0	NUM
ejde-636	56	18	and	and	CCONJ
ejde-636	56	19	µ	µ	X
ejde-636	56	20	>	>	X
ejde-636	56	21	0	0	NUM
ejde-636	56	22	,	,	PUNCT
ejde-636	56	23	however	however	ADV
ejde-636	56	24	,	,	PUNCT
ejde-636	56	25	as	as	ADV
ejde-636	56	26	far	far	ADV
ejde-636	56	27	as	as	SCONJ
ejde-636	56	28	we	we	PRON
ejde-636	56	29	know	know	VERB
ejde-636	56	30	,	,	PUNCT
ejde-636	56	31	very	very	ADV
ejde-636	56	32	little	little	ADJ
ejde-636	56	33	has	have	AUX
ejde-636	56	34	been	be	AUX
ejde-636	56	35	known	know	VERB
ejde-636	56	36	for	for	ADP
ejde-636	56	37	the	the	DET
ejde-636	56	38	mixed	mixed	ADJ
ejde-636	56	39	dispersion	dispersion	NOUN
ejde-636	56	40	fourth	fourth	ADJ
ejde-636	56	41	-	-	PUNCT
ejde-636	56	42	order	order	NOUN
ejde-636	56	43	nonlinear	nonlinear	NOUN
ejde-636	56	44	schrödinger	schrödinger	NOUN
ejde-636	56	45	equation	equation	NOUN
ejde-636	56	46	with	with	ADP
ejde-636	56	47	combined	combined	ADJ
ejde-636	56	48	nonlinearities	nonlinearitie	NOUN
ejde-636	56	49	.	.	PUNCT
ejde-636	57	1	this	this	PRON
ejde-636	57	2	constitutes	constitute	VERB
ejde-636	57	3	one	one	NUM
ejde-636	57	4	of	of	ADP
ejde-636	57	5	our	our	PRON
ejde-636	57	6	primary	primary	ADJ
ejde-636	57	7	motivations	motivation	NOUN
ejde-636	57	8	of	of	ADP
ejde-636	57	9	study	study	NOUN
ejde-636	57	10	in	in	ADP
ejde-636	57	11	the	the	DET
ejde-636	57	12	existence	existence	NOUN
ejde-636	57	13	of	of	ADP
ejde-636	57	14	normalized	normalize	VERB
ejde-636	57	15	ground	ground	NOUN
ejde-636	57	16	state	state	NOUN
ejde-636	57	17	solutions	solution	NOUN
ejde-636	57	18	of	of	ADP
ejde-636	57	19	(	(	PUNCT
ejde-636	57	20	1.1	1.1	NUM
ejde-636	57	21	)	)	PUNCT
ejde-636	57	22	for	for	ADP
ejde-636	57	23	2	2	NUM
ejde-636	57	24	<	<	X
ejde-636	57	25	q	q	X
ejde-636	57	26	<	<	X
ejde-636	57	27	2	2	NUM
ejde-636	57	28	+	+	SYM
ejde-636	57	29	4	4	NUM
ejde-636	57	30	n	n	NOUN
ejde-636	57	31	<	<	X
ejde-636	57	32	p	p	X
ejde-636	57	33	<	<	X
ejde-636	57	34	p	p	X
ejde-636	57	35	≤	≤	NUM
ejde-636	57	36	4∗	4∗	NOUN
ejde-636	57	37	and	and	CCONJ
ejde-636	57	38	p	p	NOUN
ejde-636	57	39	≤	≤	NOUN
ejde-636	57	40	q	q	NOUN
ejde-636	57	41	<	<	X
ejde-636	57	42	p	p	X
ejde-636	57	43	<	<	X
ejde-636	57	44	4∗	4∗	NOUN
ejde-636	57	45	,	,	PUNCT
ejde-636	57	46	respectively	respectively	ADV
ejde-636	57	47	.	.	PUNCT
ejde-636	58	1	for	for	ADP
ejde-636	58	2	simplicity	simplicity	NOUN
ejde-636	58	3	,	,	PUNCT
ejde-636	58	4	we	we	PRON
ejde-636	58	5	set	set	VERB
ejde-636	58	6	ϵ	ϵ	X
ejde-636	58	7	=	=	SYM
ejde-636	58	8	γ	γ	X
ejde-636	58	9	=	=	SYM
ejde-636	58	10	1	1	NUM
ejde-636	58	11	.	.	PUNCT
ejde-636	59	1	definition	definition	NOUN
ejde-636	59	2	1.1	1.1	NUM
ejde-636	59	3	.	.	PUNCT
ejde-636	60	1	we	we	PRON
ejde-636	60	2	say	say	VERB
ejde-636	60	3	that	that	SCONJ
ejde-636	60	4	a	a	DET
ejde-636	60	5	solution	solution	NOUN
ejde-636	60	6	uc	uc	PROPN
ejde-636	60	7	∈	∈	PROPN
ejde-636	60	8	s(c	s(c	PROPN
ejde-636	60	9	)	)	PUNCT
ejde-636	60	10	of	of	ADP
ejde-636	60	11	equation	equation	NOUN
ejde-636	60	12	(	(	PUNCT
ejde-636	60	13	1.2	1.2	NUM
ejde-636	60	14	)	)	PUNCT
ejde-636	60	15	is	be	AUX
ejde-636	60	16	a	a	DET
ejde-636	60	17	ground	ground	NOUN
ejde-636	60	18	state	state	NOUN
ejde-636	60	19	solution	solution	NOUN
ejde-636	60	20	to	to	ADP
ejde-636	60	21	(	(	PUNCT
ejde-636	60	22	1.2	1.2	NUM
ejde-636	60	23	)	)	PUNCT
ejde-636	60	24	if	if	SCONJ
ejde-636	60	25	it	it	PRON
ejde-636	60	26	possesses	possess	VERB
ejde-636	60	27	the	the	DET
ejde-636	60	28	minimal	minimal	ADJ
ejde-636	60	29	energy	energy	NOUN
ejde-636	60	30	among	among	ADP
ejde-636	60	31	all	all	DET
ejde-636	60	32	solutions	solution	NOUN
ejde-636	60	33	in	in	ADP
ejde-636	60	34	s(c	s(c	PROPN
ejde-636	60	35	)	)	PUNCT
ejde-636	60	36	,	,	PUNCT
ejde-636	60	37	i.e.	i.e.	X
ejde-636	60	38	,	,	PUNCT
ejde-636	60	39	if	if	SCONJ
ejde-636	60	40	ep	ep	PROPN
ejde-636	60	41	,	,	PUNCT
ejde-636	60	42	q(uc	q(uc	PROPN
ejde-636	60	43	)	)	PUNCT
ejde-636	60	44	=	=	PUNCT
ejde-636	60	45	inf{ep	inf{ep	NOUN
ejde-636	60	46	,	,	PUNCT
ejde-636	60	47	q(u	q(u	NOUN
ejde-636	60	48	)	)	PUNCT
ejde-636	60	49	,	,	PUNCT
ejde-636	60	50	u	u	PROPN
ejde-636	60	51	∈	∈	PROPN
ejde-636	60	52	s(c	s(c	PROPN
ejde-636	60	53	)	)	PUNCT
ejde-636	60	54	,	,	PUNCT
ejde-636	60	55	(	(	PUNCT
ejde-636	60	56	ep	ep	NOUN
ejde-636	60	57	,	,	PUNCT
ejde-636	60	58	q|s(c	q|s(c	NOUN
ejde-636	60	59	)	)	PUNCT
ejde-636	60	60	)	)	PUNCT
ejde-636	60	61	′(u	′(u	NOUN
ejde-636	60	62	)	)	PUNCT
ejde-636	60	63	=	=	PUNCT
ejde-636	60	64	0	0	NUM
ejde-636	60	65	}	}	PUNCT
ejde-636	60	66	.	.	PUNCT
ejde-636	61	1	we	we	PRON
ejde-636	61	2	start	start	VERB
ejde-636	61	3	with	with	ADP
ejde-636	61	4	the	the	DET
ejde-636	61	5	case	case	NOUN
ejde-636	61	6	2	2	NUM
ejde-636	61	7	<	<	X
ejde-636	61	8	q	q	X
ejde-636	61	9	<	<	X
ejde-636	61	10	2	2	NUM
ejde-636	61	11	+	+	SYM
ejde-636	61	12	4	4	NUM
ejde-636	61	13	n	n	NOUN
ejde-636	61	14	<	<	X
ejde-636	61	15	p	p	X
ejde-636	61	16	<	<	X
ejde-636	61	17	p	p	X
ejde-636	61	18	≤	≤	NUM
ejde-636	61	19	4∗	4∗	NUM
ejde-636	61	20	by	by	ADP
ejde-636	61	21	setting	set	VERB
ejde-636	61	22	v	v	NOUN
ejde-636	61	23	(	(	PUNCT
ejde-636	61	24	c	c	NOUN
ejde-636	61	25	)	)	PUNCT
ejde-636	61	26	:	:	PUNCT
ejde-636	62	1	=	=	SYM
ejde-636	62	2	{	{	PUNCT
ejde-636	62	3	u	u	NOUN
ejde-636	62	4	∈	∈	PROPN
ejde-636	62	5	s(c	s(c	PROPN
ejde-636	62	6	)	)	PUNCT
ejde-636	62	7	:	:	PUNCT
ejde-636	62	8	∥∆u∥22	∥∆u∥22	X
ejde-636	63	1	+	+	CCONJ
ejde-636	63	2	|∇u∥22	|∇u∥22	PROPN
ejde-636	63	3	<	<	X
ejde-636	63	4	ρ0	ρ0	PROPN
ejde-636	63	5	}	}	PUNCT
ejde-636	63	6	,	,	PUNCT
ejde-636	63	7	∂v	∂v	PROPN
ejde-636	63	8	(	(	PUNCT
ejde-636	63	9	c	c	X
ejde-636	63	10	)	)	PUNCT
ejde-636	63	11	=	=	PRON
ejde-636	63	12	{	{	PUNCT
ejde-636	63	13	u	u	NOUN
ejde-636	63	14	∈	∈	PROPN
ejde-636	63	15	s(c	s(c	PROPN
ejde-636	63	16	)	)	PUNCT
ejde-636	63	17	:	:	PUNCT
ejde-636	63	18	∥∆u∥22	∥∆u∥22	X
ejde-636	63	19	+	+	CCONJ
ejde-636	63	20	|∇u∥22	|∇u∥22	PROPN
ejde-636	63	21	=	=	SYM
ejde-636	63	22	ρ0	ρ0	PROPN
ejde-636	63	23	}	}	PUNCT
ejde-636	63	24	,	,	PUNCT
ejde-636	63	25	where	where	SCONJ
ejde-636	63	26	ρ0	ρ0	PROPN
ejde-636	63	27	is	be	AUX
ejde-636	63	28	a	a	DET
ejde-636	63	29	suitable	suitable	ADJ
ejde-636	63	30	positive	positive	ADJ
ejde-636	63	31	constant	constant	NOUN
ejde-636	63	32	.	.	PUNCT
ejde-636	64	1	for	for	ADP
ejde-636	64	2	any	any	DET
ejde-636	64	3	given	give	VERB
ejde-636	64	4	µ	µ	PROPN
ejde-636	64	5	>	>	X
ejde-636	64	6	0	0	NUM
ejde-636	64	7	,	,	PUNCT
ejde-636	64	8	we	we	PRON
ejde-636	64	9	aim	aim	VERB
ejde-636	64	10	to	to	PART
ejde-636	64	11	determine	determine	VERB
ejde-636	64	12	a	a	DET
ejde-636	64	13	specific	specific	ADJ
ejde-636	64	14	value	value	NOUN
ejde-636	64	15	c0	c0	NOUN
ejde-636	64	16	=	=	SYM
ejde-636	64	17	c0(µ	c0(µ	PROPN
ejde-636	64	18	)	)	PUNCT
ejde-636	64	19	>	>	X
ejde-636	64	20	0	0	NUM
ejde-636	65	1	such	such	ADJ
ejde-636	65	2	that	that	PRON
ejde-636	65	3	for	for	ADP
ejde-636	65	4	any	any	DET
ejde-636	65	5	c	c	PROPN
ejde-636	65	6	∈	∈	PROPN
ejde-636	65	7	(	(	PUNCT
ejde-636	65	8	0	0	NUM
ejde-636	65	9	,	,	PUNCT
ejde-636	65	10	c0	c0	NOUN
ejde-636	65	11	)	)	PUNCT
ejde-636	65	12	it	it	PRON
ejde-636	65	13	holds	hold	VERB
ejde-636	65	14	mp	mp	PROPN
ejde-636	65	15	,	,	PUNCT
ejde-636	65	16	q(c	q(c	PROPN
ejde-636	65	17	)	)	PUNCT
ejde-636	65	18	:	:	PUNCT
ejde-636	66	1	=	=	NUM
ejde-636	66	2	inf	inf	NOUN
ejde-636	66	3	u∈v	u∈v	NOUN
ejde-636	66	4	(	(	PUNCT
ejde-636	66	5	c	c	NOUN
ejde-636	66	6	)	)	PUNCT
ejde-636	66	7	ep	ep	NOUN
ejde-636	66	8	,	,	PUNCT
ejde-636	66	9	q(u	q(u	ADJ
ejde-636	66	10	)	)	PUNCT
ejde-636	66	11	<	<	X
ejde-636	66	12	0	0	PUNCT
ejde-636	66	13	<	<	X
ejde-636	66	14	inf	inf	PROPN
ejde-636	66	15	u∈∂v	u∈∂v	X
ejde-636	66	16	(	(	PUNCT
ejde-636	66	17	c	c	NOUN
ejde-636	66	18	)	)	PUNCT
ejde-636	66	19	ep	ep	NOUN
ejde-636	66	20	,	,	PUNCT
ejde-636	66	21	q(u	q(u	NOUN
ejde-636	66	22	)	)	PUNCT
ejde-636	66	23	.	.	PUNCT
ejde-636	67	1	theorem	theorem	VERB
ejde-636	67	2	1.2	1.2	NUM
ejde-636	67	3	.	.	PUNCT
ejde-636	68	1	let	let	VERB
ejde-636	68	2	n	n	PRON
ejde-636	68	3	≥	≥	NUM
ejde-636	68	4	5	5	NUM
ejde-636	68	5	,	,	PUNCT
ejde-636	68	6	µ	µ	X
ejde-636	68	7	>	>	X
ejde-636	68	8	0	0	NUM
ejde-636	68	9	and	and	CCONJ
ejde-636	68	10	2	2	NUM
ejde-636	68	11	<	<	X
ejde-636	68	12	q	q	X
ejde-636	68	13	<	<	X
ejde-636	68	14	2	2	NUM
ejde-636	68	15	+	+	SYM
ejde-636	68	16	4	4	NUM
ejde-636	68	17	n	n	NOUN
ejde-636	68	18	<	<	X
ejde-636	68	19	p	p	X
ejde-636	68	20	<	<	X
ejde-636	68	21	p	p	X
ejde-636	68	22	≤	≤	NUM
ejde-636	68	23	4∗.	4∗.	NUM
ejde-636	68	24	for	for	ADP
ejde-636	68	25	any	any	DET
ejde-636	68	26	µ	µ	X
ejde-636	68	27	>	>	X
ejde-636	68	28	0	0	NUM
ejde-636	68	29	,	,	PUNCT
ejde-636	68	30	there	there	PRON
ejde-636	68	31	exists	exist	VERB
ejde-636	68	32	c0	c0	PROPN
ejde-636	68	33	=	=	SYM
ejde-636	68	34	c0(µ	c0(µ	PROPN
ejde-636	68	35	)	)	PUNCT
ejde-636	68	36	>	>	X
ejde-636	68	37	0	0	NUM
ejde-636	69	1	such	such	ADJ
ejde-636	69	2	that	that	PRON
ejde-636	69	3	for	for	ADP
ejde-636	69	4	any	any	DET
ejde-636	69	5	c	c	PROPN
ejde-636	69	6	∈	∈	PROPN
ejde-636	69	7	(	(	PUNCT
ejde-636	69	8	0	0	NUM
ejde-636	69	9	,	,	PUNCT
ejde-636	69	10	c0	c0	NOUN
ejde-636	69	11	)	)	PUNCT
ejde-636	69	12	,	,	PUNCT
ejde-636	69	13	the	the	DET
ejde-636	69	14	constraint	constraint	ADJ
ejde-636	69	15	functional	functional	ADJ
ejde-636	69	16	ep	ep	PROPN
ejde-636	69	17	,	,	PUNCT
ejde-636	69	18	q|s(c	q|s(c	PROPN
ejde-636	69	19	)	)	PUNCT
ejde-636	69	20	admits	admit	VERB
ejde-636	69	21	a	a	DET
ejde-636	69	22	ground	ground	NOUN
ejde-636	69	23	state	state	NOUN
ejde-636	69	24	,	,	PUNCT
ejde-636	69	25	which	which	PRON
ejde-636	69	26	corresponds	correspond	VERB
ejde-636	69	27	to	to	ADP
ejde-636	69	28	a	a	DET
ejde-636	69	29	local	local	ADJ
ejde-636	69	30	minimizer	minimizer	NOUN
ejde-636	69	31	of	of	ADP
ejde-636	69	32	ep	ep	PROPN
ejde-636	69	33	,	,	PUNCT
ejde-636	69	34	q	q	NOUN
ejde-636	69	35	in	in	ADP
ejde-636	69	36	the	the	DET
ejde-636	69	37	set	set	NOUN
ejde-636	69	38	v	v	NOUN
ejde-636	69	39	(	(	PUNCT
ejde-636	69	40	c	c	NOUN
ejde-636	69	41	)	)	PUNCT
ejde-636	69	42	.	.	PUNCT
ejde-636	70	1	as	as	ADP
ejde-636	70	2	p	p	X
ejde-636	70	3	>	>	X
ejde-636	70	4	p	p	X
ejde-636	70	5	,	,	PUNCT
ejde-636	70	6	it	it	PRON
ejde-636	70	7	is	be	AUX
ejde-636	70	8	evident	evident	ADJ
ejde-636	70	9	that	that	SCONJ
ejde-636	70	10	the	the	DET
ejde-636	70	11	constrained	constrained	ADJ
ejde-636	70	12	functional	functional	ADJ
ejde-636	70	13	ep	ep	NOUN
ejde-636	70	14	,	,	PUNCT
ejde-636	70	15	q|s(c	q|s(c	PROPN
ejde-636	70	16	)	)	PUNCT
ejde-636	70	17	is	be	AUX
ejde-636	70	18	unbounded	unbounde	VERB
ejde-636	70	19	from	from	ADP
ejde-636	70	20	below	below	ADV
ejde-636	70	21	.	.	PUNCT
ejde-636	71	1	however	however	ADV
ejde-636	71	2	,	,	PUNCT
ejde-636	71	3	the	the	DET
ejde-636	71	4	presence	presence	NOUN
ejde-636	71	5	of	of	ADP
ejde-636	71	6	the	the	DET
ejde-636	71	7	lower	low	ADJ
ejde-636	71	8	order	order	NOUN
ejde-636	71	9	term	term	NOUN
ejde-636	71	10	|u|q−2u	|u|q−2u	INTJ
ejde-636	71	11	with	with	ADP
ejde-636	71	12	2	2	NUM
ejde-636	71	13	<	<	X
ejde-636	71	14	q	q	X
ejde-636	71	15	<	<	X
ejde-636	71	16	2	2	NUM
ejde-636	71	17	+	+	SYM
ejde-636	71	18	4	4	NUM
ejde-636	71	19	n	n	PRON
ejde-636	71	20	creates	create	VERB
ejde-636	71	21	a	a	DET
ejde-636	71	22	geometry	geometry	NOUN
ejde-636	71	23	of	of	ADP
ejde-636	71	24	local	local	ADJ
ejde-636	71	25	minima	minima	NOUN
ejde-636	71	26	on	on	ADP
ejde-636	71	27	s(c	s(c	NUM
ejde-636	71	28	)	)	PUNCT
ejde-636	71	29	for	for	ADP
ejde-636	71	30	sufficiently	sufficiently	ADV
ejde-636	71	31	small	small	ADJ
ejde-636	71	32	c	c	NOUN
ejde-636	71	33	>	>	X
ejde-636	71	34	0	0	X
ejde-636	71	35	.	.	PUNCT
ejde-636	72	1	the	the	DET
ejde-636	72	2	challenge	challenge	NOUN
ejde-636	72	3	in	in	ADP
ejde-636	72	4	establishing	establish	VERB
ejde-636	72	5	the	the	DET
ejde-636	72	6	existence	existence	NOUN
ejde-636	72	7	of	of	ADP
ejde-636	72	8	local	local	ADJ
ejde-636	72	9	minimizers	minimizer	NOUN
ejde-636	72	10	arises	arise	VERB
ejde-636	72	11	from	from	ADP
ejde-636	72	12	the	the	DET
ejde-636	72	13	lack	lack	NOUN
ejde-636	72	14	of	of	ADP
ejde-636	72	15	compactness	compactness	NOUN
ejde-636	72	16	of	of	ADP
ejde-636	72	17	the	the	DET
ejde-636	72	18	bounded	bounded	ADJ
ejde-636	72	19	minimizing	minimize	VERB
ejde-636	72	20	sequence	sequence	NOUN
ejde-636	72	21	{	{	PUNCT
ejde-636	72	22	un	un	PROPN
ejde-636	72	23	}	}	PUNCT
ejde-636	72	24	⊂	⊂	ADJ
ejde-636	72	25	v	v	X
ejde-636	72	26	(	(	PUNCT
ejde-636	72	27	c	c	NOUN
ejde-636	72	28	)	)	PUNCT
ejde-636	72	29	due	due	ADP
ejde-636	72	30	to	to	ADP
ejde-636	72	31	the	the	DET
ejde-636	72	32	noncompact	noncompact	NOUN
ejde-636	72	33	embedding	embed	VERB
ejde-636	72	34	h2(rn	h2(rn	X
ejde-636	72	35	)	)	PUNCT
ejde-636	72	36	↪	↪	PROPN
ejde-636	72	37	→	→	SYM
ejde-636	72	38	l2(rn	l2(rn	PROPN
ejde-636	72	39	)	)	PUNCT
ejde-636	72	40	.	.	PUNCT
ejde-636	73	1	by	by	ADP
ejde-636	73	2	employing	employ	VERB
ejde-636	73	3	a	a	DET
ejde-636	73	4	minimization	minimization	NOUN
ejde-636	73	5	approach	approach	NOUN
ejde-636	73	6	and	and	CCONJ
ejde-636	73	7	incorporating	incorporate	VERB
ejde-636	73	8	the	the	DET
ejde-636	73	9	subadditivity	subadditivity	NOUN
ejde-636	73	10	of	of	ADP
ejde-636	73	11	ground	ground	NOUN
ejde-636	73	12	state	state	NOUN
ejde-636	73	13	energy	energy	NOUN
ejde-636	73	14	,	,	PUNCT
ejde-636	73	15	we	we	PRON
ejde-636	73	16	overcome	overcome	VERB
ejde-636	73	17	this	this	DET
ejde-636	73	18	obstacle	obstacle	NOUN
ejde-636	73	19	and	and	CCONJ
ejde-636	73	20	demonstrate	demonstrate	VERB
ejde-636	73	21	the	the	DET
ejde-636	73	22	existence	existence	NOUN
ejde-636	73	23	of	of	ADP
ejde-636	73	24	local	local	ADJ
ejde-636	73	25	minima	minima	NOUN
ejde-636	73	26	.	.	PUNCT
ejde-636	74	1	furthermore	furthermore	ADV
ejde-636	74	2	,	,	PUNCT
ejde-636	74	3	we	we	PRON
ejde-636	74	4	find	find	VERB
ejde-636	74	5	that	that	SCONJ
ejde-636	74	6	any	any	DET
ejde-636	74	7	ground	ground	NOUN
ejde-636	74	8	state	state	NOUN
ejde-636	74	9	serves	serve	VERB
ejde-636	74	10	as	as	ADP
ejde-636	74	11	a	a	DET
ejde-636	74	12	local	local	ADJ
ejde-636	74	13	minimum	minimum	NOUN
ejde-636	74	14	for	for	ADP
ejde-636	74	15	the	the	DET
ejde-636	74	16	associated	associated	ADJ
ejde-636	74	17	energy	energy	NOUN
ejde-636	74	18	functional	functional	ADJ
ejde-636	74	19	.	.	PUNCT
ejde-636	75	1	theorem	theorem	VERB
ejde-636	75	2	1.3	1.3	NUM
ejde-636	75	3	.	.	PUNCT
ejde-636	76	1	let	let	VERB
ejde-636	76	2	n	n	PRON
ejde-636	76	3	≥	≥	NUM
ejde-636	76	4	5	5	NUM
ejde-636	76	5	,	,	PUNCT
ejde-636	76	6	µ	µ	X
ejde-636	76	7	>	>	SYM
ejde-636	76	8	0	0	NUM
ejde-636	77	1	and	and	CCONJ
ejde-636	77	2	p	p	NOUN
ejde-636	77	3	≤	≤	NOUN
ejde-636	77	4	q	q	NOUN
ejde-636	77	5	<	<	X
ejde-636	77	6	p	p	X
ejde-636	77	7	<	<	X
ejde-636	77	8	4∗.	4∗.	NUM
ejde-636	77	9	if	if	SCONJ
ejde-636	77	10	q	q	PROPN
ejde-636	77	11	=	=	SYM
ejde-636	77	12	p	p	NOUN
ejde-636	77	13	,	,	PUNCT
ejde-636	77	14	we	we	PRON
ejde-636	77	15	assume	assume	VERB
ejde-636	77	16	that	that	SCONJ
ejde-636	77	17	µc4	µc4	PROPN
ejde-636	77	18	/	/	SYM
ejde-636	77	19	n	n	NOUN
ejde-636	77	20	<	<	X
ejde-636	77	21	n+4	n+4	NUM
ejde-636	77	22	ncq	ncq	NOUN
ejde-636	77	23	n	n	CCONJ
ejde-636	77	24	,	,	PUNCT
ejde-636	77	25	q	q	PROPN
ejde-636	77	26	.	.	PUNCT
ejde-636	78	1	then	then	ADV
ejde-636	78	2	there	there	PRON
ejde-636	78	3	exists	exist	VERB
ejde-636	78	4	a	a	DET
ejde-636	78	5	sufficiently	sufficiently	ADV
ejde-636	78	6	small	small	ADJ
ejde-636	78	7	c∗	c∗	NOUN
ejde-636	78	8	>	>	X
ejde-636	78	9	0	0	NUM
ejde-636	78	10	such	such	ADJ
ejde-636	78	11	that	that	PRON
ejde-636	78	12	for	for	ADP
ejde-636	78	13	any	any	DET
ejde-636	78	14	c	c	PROPN
ejde-636	78	15	∈	∈	PROPN
ejde-636	78	16	(	(	PUNCT
ejde-636	78	17	0	0	NUM
ejde-636	78	18	,	,	PUNCT
ejde-636	78	19	c∗	c∗	NOUN
ejde-636	78	20	)	)	PUNCT
ejde-636	78	21	,	,	PUNCT
ejde-636	78	22	the	the	DET
ejde-636	78	23	constrained	constrain	VERB
ejde-636	78	24	functional	functional	ADJ
ejde-636	78	25	ep	ep	NOUN
ejde-636	78	26	,	,	PUNCT
ejde-636	78	27	q|s(c	q|s(c	PROPN
ejde-636	78	28	)	)	PUNCT
ejde-636	78	29	possesses	possess	VERB
ejde-636	78	30	a	a	DET
ejde-636	78	31	critical	critical	ADJ
ejde-636	78	32	point	point	NOUN
ejde-636	78	33	u	u	NOUN
ejde-636	78	34	at	at	ADP
ejde-636	78	35	a	a	DET
ejde-636	78	36	positive	positive	ADJ
ejde-636	78	37	level	level	NOUN
ejde-636	78	38	ep	ep	NOUN
ejde-636	78	39	,	,	PUNCT
ejde-636	78	40	q(u	q(u	ADJ
ejde-636	78	41	)	)	PUNCT
ejde-636	78	42	>	>	X
ejde-636	78	43	0	0	PUNCT
ejde-636	78	44	with	with	ADP
ejde-636	78	45	the	the	DET
ejde-636	78	46	following	follow	VERB
ejde-636	78	47	properties	property	NOUN
ejde-636	78	48	:	:	PUNCT
ejde-636	78	49	u	u	NOUN
ejde-636	78	50	satisfies	satisfie	NOUN
ejde-636	78	51	(	(	PUNCT
ejde-636	78	52	1.2	1.2	NUM
ejde-636	78	53	)	)	PUNCT
ejde-636	78	54	for	for	ADP
ejde-636	78	55	some	some	DET
ejde-636	78	56	ω	ω	NOUN
ejde-636	78	57	>	>	X
ejde-636	78	58	0	0	PUNCT
ejde-636	78	59	and	and	CCONJ
ejde-636	78	60	represents	represent	VERB
ejde-636	78	61	a	a	DET
ejde-636	78	62	normalized	normalize	VERB
ejde-636	78	63	ground	ground	NOUN
ejde-636	78	64	state	state	NOUN
ejde-636	78	65	of	of	ADP
ejde-636	78	66	(	(	PUNCT
ejde-636	78	67	1.2	1.2	NUM
ejde-636	78	68	)	)	PUNCT
ejde-636	78	69	on	on	ADP
ejde-636	78	70	s(c	s(c	NUM
ejde-636	78	71	)	)	PUNCT
ejde-636	78	72	.	.	PUNCT
ejde-636	79	1	we	we	PRON
ejde-636	79	2	introduce	introduce	VERB
ejde-636	79	3	the	the	DET
ejde-636	79	4	nehari	nehari	NOUN
ejde-636	79	5	-	-	PUNCT
ejde-636	79	6	pohozaev	pohozaev	NOUN
ejde-636	79	7	set	set	NOUN
ejde-636	79	8	of	of	ADP
ejde-636	79	9	ep	ep	PROPN
ejde-636	79	10	,	,	PUNCT
ejde-636	79	11	q|s(c	q|s(c	PROPN
ejde-636	79	12	)	)	PUNCT
ejde-636	79	13	as	as	SCONJ
ejde-636	79	14	follows	follow	VERB
ejde-636	79	15	qp	qp	ADP
ejde-636	79	16	,	,	PUNCT
ejde-636	79	17	q(c	q(c	PROPN
ejde-636	79	18	)	)	PUNCT
ejde-636	79	19	=	=	SYM
ejde-636	79	20	{	{	PUNCT
ejde-636	79	21	u	u	NOUN
ejde-636	79	22	∈	∈	PROPN
ejde-636	79	23	s(c	s(c	PROPN
ejde-636	79	24	)	)	PUNCT
ejde-636	79	25	:	:	PUNCT
ejde-636	79	26	qp	qp	ADP
ejde-636	79	27	,	,	PUNCT
ejde-636	79	28	q(u	q(u	ADJ
ejde-636	79	29	)	)	PUNCT
ejde-636	79	30	=	=	SYM
ejde-636	79	31	0	0	NUM
ejde-636	79	32	}	}	PUNCT
ejde-636	79	33	,	,	PUNCT
ejde-636	79	34	where	where	SCONJ
ejde-636	79	35	qp	qp	NOUN
ejde-636	79	36	,	,	PUNCT
ejde-636	79	37	q(u	q(u	ADJ
ejde-636	79	38	)	)	PUNCT
ejde-636	79	39	=	=	PRON
ejde-636	80	1	∥∆u∥22	∥∆u∥22	X
ejde-636	80	2	+	+	CCONJ
ejde-636	80	3	1	1	NUM
ejde-636	80	4	2	2	NUM
ejde-636	80	5	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	80	6	−	−	ADP
ejde-636	80	7	µγq∥u∥qq	µγq∥u∥qq	ADP
ejde-636	80	8	−	−	NOUN
ejde-636	80	9	γp∥u∥pp	γp∥u∥pp	PROPN
ejde-636	80	10	,	,	PUNCT
ejde-636	80	11	γr	γr	X
ejde-636	80	12	:	:	PUNCT
ejde-636	80	13	=	=	SYM
ejde-636	80	14	n(r	n(r	NOUN
ejde-636	80	15	−	−	NOUN
ejde-636	80	16	2	2	X
ejde-636	80	17	)	)	PUNCT
ejde-636	80	18	4r	4r	NOUN
ejde-636	80	19	=	=	SYM
ejde-636	80	20	n	n	PRON
ejde-636	80	21	2	2	NUM
ejde-636	80	22	(	(	PUNCT
ejde-636	80	23	1	1	NUM
ejde-636	80	24	2	2	NUM
ejde-636	80	25	−	−	NUM
ejde-636	80	26	1	1	NUM
ejde-636	80	27	r	r	NOUN
ejde-636	80	28	)	)	PUNCT
ejde-636	80	29	,	,	PUNCT
ejde-636	80	30	∀r	∀r	X
ejde-636	80	31	∈	∈	NOUN
ejde-636	80	32	(	(	PUNCT
ejde-636	80	33	2	2	NUM
ejde-636	80	34	,	,	PUNCT
ejde-636	80	35	4∗	4∗	NOUN
ejde-636	80	36	]	]	X
ejde-636	80	37	.	.	PUNCT
ejde-636	80	38	4	4	NUM
ejde-636	80	39	z.	z.	PROPN
ejde-636	80	40	ma	ma	PROPN
ejde-636	80	41	,	,	PUNCT
ejde-636	80	42	x.	x.	PROPN
ejde-636	80	43	chang	chang	PROPN
ejde-636	80	44	,	,	PUNCT
ejde-636	80	45	z.	z.	PROPN
ejde-636	80	46	feng	feng	PROPN
ejde-636	80	47	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	80	48	it	it	PRON
ejde-636	80	49	is	be	AUX
ejde-636	80	50	easily	easily	ADV
ejde-636	80	51	seen	see	VERB
ejde-636	80	52	that	that	SCONJ
ejde-636	80	53	all	all	DET
ejde-636	80	54	critical	critical	ADJ
ejde-636	80	55	points	point	NOUN
ejde-636	80	56	of	of	ADP
ejde-636	80	57	ep	ep	PROPN
ejde-636	80	58	,	,	PUNCT
ejde-636	80	59	q|s(c	q|s(c	PROPN
ejde-636	80	60	)	)	PUNCT
ejde-636	80	61	lie	lie	NOUN
ejde-636	80	62	in	in	ADP
ejde-636	80	63	qp	qp	NOUN
ejde-636	80	64	,	,	PUNCT
ejde-636	80	65	q(c	q(c	PROPN
ejde-636	80	66	)	)	PUNCT
ejde-636	80	67	.	.	PUNCT
ejde-636	81	1	to	to	PART
ejde-636	81	2	prove	prove	VERB
ejde-636	81	3	theorem	theorem	VERB
ejde-636	81	4	1.3	1.3	NUM
ejde-636	81	5	,	,	PUNCT
ejde-636	81	6	we	we	PRON
ejde-636	81	7	shall	shall	AUX
ejde-636	81	8	employ	employ	VERB
ejde-636	81	9	a	a	DET
ejde-636	81	10	direct	direct	ADJ
ejde-636	81	11	minimization	minimization	NOUN
ejde-636	81	12	method	method	NOUN
ejde-636	81	13	for	for	ADP
ejde-636	81	14	ep	ep	PROPN
ejde-636	81	15	,	,	PUNCT
ejde-636	81	16	q	q	NOUN
ejde-636	81	17	on	on	ADP
ejde-636	81	18	qp	qp	NOUN
ejde-636	81	19	,	,	PUNCT
ejde-636	81	20	q(c	q(c	PROPN
ejde-636	81	21	)	)	PUNCT
ejde-636	81	22	.	.	PUNCT
ejde-636	82	1	a	a	DET
ejde-636	82	2	crucial	crucial	ADJ
ejde-636	82	3	step	step	NOUN
ejde-636	82	4	is	be	AUX
ejde-636	82	5	to	to	PART
ejde-636	82	6	show	show	VERB
ejde-636	82	7	the	the	DET
ejde-636	82	8	convergence	convergence	NOUN
ejde-636	82	9	of	of	ADP
ejde-636	82	10	a	a	DET
ejde-636	82	11	minimizing	minimize	VERB
ejde-636	82	12	sequence	sequence	NOUN
ejde-636	82	13	{	{	PUNCT
ejde-636	82	14	un	un	PROPN
ejde-636	82	15	}	}	PUNCT
ejde-636	82	16	⊂	⊂	PROPN
ejde-636	82	17	qp	qp	NOUN
ejde-636	82	18	,	,	PUNCT
ejde-636	82	19	q(c	q(c	PROPN
ejde-636	82	20	)	)	PUNCT
ejde-636	82	21	of	of	ADP
ejde-636	82	22	ep	ep	PROPN
ejde-636	82	23	,	,	PUNCT
ejde-636	82	24	q	q	NOUN
ejde-636	82	25	at	at	ADP
ejde-636	82	26	mp	mp	PROPN
ejde-636	82	27	,	,	PUNCT
ejde-636	82	28	q(c	q(c	PROPN
ejde-636	82	29	)	)	PUNCT
ejde-636	82	30	.	.	PUNCT
ejde-636	83	1	the	the	DET
ejde-636	83	2	sign	sign	NOUN
ejde-636	83	3	of	of	ADP
ejde-636	83	4	the	the	DET
ejde-636	83	5	lagrange	lagrange	NOUN
ejde-636	83	6	multiplier	multiplier	ADV
ejde-636	83	7	ω	ω	PROPN
ejde-636	83	8	∈	∈	NOUN
ejde-636	83	9	r	r	NOUN
ejde-636	83	10	plays	play	VERB
ejde-636	83	11	a	a	DET
ejde-636	83	12	pivotal	pivotal	ADJ
ejde-636	83	13	role	role	NOUN
ejde-636	83	14	in	in	ADP
ejde-636	83	15	the	the	DET
ejde-636	83	16	analysis	analysis	NOUN
ejde-636	83	17	.	.	PUNCT
ejde-636	84	1	however	however	ADV
ejde-636	84	2	,	,	PUNCT
ejde-636	84	3	tackling	tackle	VERB
ejde-636	84	4	this	this	DET
ejde-636	84	5	issue	issue	NOUN
ejde-636	84	6	is	be	AUX
ejde-636	84	7	challenging	challenge	VERB
ejde-636	84	8	because	because	SCONJ
ejde-636	84	9	of	of	ADP
ejde-636	84	10	the	the	DET
ejde-636	84	11	presence	presence	NOUN
ejde-636	84	12	of	of	ADP
ejde-636	84	13	the	the	DET
ejde-636	84	14	term	term	NOUN
ejde-636	84	15	∥∇u∥2	∥∇u∥2	NOUN
ejde-636	84	16	.	.	PROPN
ejde-636	84	17	as	as	SCONJ
ejde-636	84	18	demonstrated	demonstrate	VERB
ejde-636	84	19	in	in	ADP
ejde-636	84	20	lemma	lemma	PROPN
ejde-636	84	21	4.5	4.5	NUM
ejde-636	84	22	,	,	PUNCT
ejde-636	84	23	we	we	PRON
ejde-636	84	24	identify	identify	VERB
ejde-636	84	25	a	a	DET
ejde-636	84	26	sufficiently	sufficiently	ADV
ejde-636	84	27	small	small	ADJ
ejde-636	84	28	c∗	c∗	NOUN
ejde-636	84	29	>	>	X
ejde-636	84	30	0	0	NUM
ejde-636	85	1	such	such	ADJ
ejde-636	85	2	that	that	PRON
ejde-636	85	3	for	for	ADP
ejde-636	85	4	any	any	DET
ejde-636	85	5	c	c	PROPN
ejde-636	85	6	∈	∈	PROPN
ejde-636	85	7	(	(	PUNCT
ejde-636	85	8	0	0	NUM
ejde-636	85	9	,	,	PUNCT
ejde-636	85	10	c∗	c∗	NOUN
ejde-636	85	11	)	)	PUNCT
ejde-636	85	12	,	,	PUNCT
ejde-636	85	13	the	the	DET
ejde-636	85	14	corresponding	correspond	VERB
ejde-636	85	15	ωc	ωc	ADP
ejde-636	85	16	remains	remain	VERB
ejde-636	85	17	positive	positive	ADJ
ejde-636	85	18	.	.	PUNCT
ejde-636	86	1	another	another	DET
ejde-636	86	2	difficulty	difficulty	NOUN
ejde-636	86	3	comes	come	VERB
ejde-636	86	4	from	from	ADP
ejde-636	86	5	weak	weak	ADJ
ejde-636	86	6	limits	limit	NOUN
ejde-636	86	7	of	of	ADP
ejde-636	86	8	the	the	DET
ejde-636	86	9	minimizing	minimize	VERB
ejde-636	86	10	sequence	sequence	NOUN
ejde-636	86	11	,	,	PUNCT
ejde-636	86	12	which	which	PRON
ejde-636	86	13	may	may	AUX
ejde-636	86	14	violate	violate	VERB
ejde-636	86	15	the	the	DET
ejde-636	86	16	constraint	constraint	NOUN
ejde-636	86	17	due	due	ADP
ejde-636	86	18	to	to	ADP
ejde-636	86	19	the	the	DET
ejde-636	86	20	non	non	ADJ
ejde-636	86	21	-	-	NOUN
ejde-636	86	22	compactness	compactness	NOUN
ejde-636	86	23	of	of	ADP
ejde-636	86	24	the	the	DET
ejde-636	86	25	embedding	embed	VERB
ejde-636	86	26	h2(rn	h2(rn	X
ejde-636	86	27	)	)	PUNCT
ejde-636	86	28	↪	↪	PROPN
ejde-636	86	29	→	→	SYM
ejde-636	86	30	l2(rn	l2(rn	PROPN
ejde-636	86	31	)	)	PUNCT
ejde-636	86	32	.	.	PUNCT
ejde-636	87	1	overcoming	overcome	VERB
ejde-636	87	2	this	this	DET
ejde-636	87	3	obstacle	obstacle	NOUN
ejde-636	87	4	,	,	PUNCT
ejde-636	87	5	we	we	PRON
ejde-636	87	6	need	need	VERB
ejde-636	87	7	to	to	PART
ejde-636	87	8	show	show	VERB
ejde-636	87	9	that	that	SCONJ
ejde-636	87	10	the	the	DET
ejde-636	87	11	mapping	mapping	NOUN
ejde-636	87	12	c	c	PROPN
ejde-636	87	13	7→	7→	NUM
ejde-636	87	14	mp	mp	PROPN
ejde-636	87	15	,	,	PUNCT
ejde-636	87	16	q(c	q(c	PROPN
ejde-636	87	17	)	)	PUNCT
ejde-636	87	18	is	be	AUX
ejde-636	87	19	strictly	strictly	ADV
ejde-636	87	20	decreasing	decrease	VERB
ejde-636	87	21	.	.	PUNCT
ejde-636	88	1	this	this	PRON
ejde-636	88	2	,	,	PUNCT
ejde-636	88	3	together	together	ADV
ejde-636	88	4	with	with	ADP
ejde-636	88	5	the	the	DET
ejde-636	88	6	relationship	relationship	NOUN
ejde-636	88	7	between	between	ADP
ejde-636	88	8	the	the	DET
ejde-636	88	9	energy	energy	NOUN
ejde-636	88	10	functional	functional	ADJ
ejde-636	88	11	ep	ep	PROPN
ejde-636	88	12	,	,	PUNCT
ejde-636	88	13	q	q	PROPN
ejde-636	88	14	and	and	CCONJ
ejde-636	88	15	the	the	DET
ejde-636	88	16	nehari	nehari	NOUN
ejde-636	88	17	-	-	PUNCT
ejde-636	88	18	pohozaev	pohozaev	NOUN
ejde-636	88	19	functional	functional	ADJ
ejde-636	88	20	qp	qp	ADP
ejde-636	88	21	,	,	PUNCT
ejde-636	88	22	q	q	NOUN
ejde-636	88	23	,	,	PUNCT
ejde-636	88	24	leads	lead	VERB
ejde-636	88	25	to	to	ADP
ejde-636	88	26	strong	strong	ADJ
ejde-636	88	27	convergence	convergence	NOUN
ejde-636	88	28	of	of	ADP
ejde-636	88	29	the	the	DET
ejde-636	88	30	minimizing	minimize	VERB
ejde-636	88	31	sequence	sequence	NOUN
ejde-636	88	32	in	in	ADP
ejde-636	88	33	h2(rn	h2(rn	PROPN
ejde-636	88	34	)	)	PUNCT
ejde-636	88	35	.	.	PUNCT
ejde-636	89	1	subsequently	subsequently	ADV
ejde-636	89	2	,	,	PUNCT
ejde-636	89	3	by	by	ADP
ejde-636	89	4	showing	show	VERB
ejde-636	89	5	that	that	PRON
ejde-636	89	6	qp	qp	NOUN
ejde-636	89	7	,	,	PUNCT
ejde-636	89	8	q(c	q(c	PROPN
ejde-636	89	9	)	)	PUNCT
ejde-636	89	10	is	be	AUX
ejde-636	89	11	a	a	DET
ejde-636	89	12	natural	natural	ADJ
ejde-636	89	13	constraint	constraint	NOUN
ejde-636	89	14	,	,	PUNCT
ejde-636	89	15	we	we	PRON
ejde-636	89	16	observe	observe	VERB
ejde-636	89	17	that	that	SCONJ
ejde-636	89	18	the	the	DET
ejde-636	89	19	minimizer	minimizer	NOUN
ejde-636	89	20	of	of	ADP
ejde-636	89	21	ep	ep	PROPN
ejde-636	89	22	,	,	PUNCT
ejde-636	89	23	q	q	NOUN
ejde-636	89	24	on	on	ADP
ejde-636	89	25	qp	qp	NOUN
ejde-636	89	26	,	,	PUNCT
ejde-636	89	27	q(c	q(c	PROPN
ejde-636	89	28	)	)	PUNCT
ejde-636	89	29	constitutes	constitute	VERB
ejde-636	89	30	a	a	DET
ejde-636	89	31	normalized	normalize	VERB
ejde-636	89	32	ground	ground	NOUN
ejde-636	89	33	state	state	NOUN
ejde-636	89	34	solution	solution	NOUN
ejde-636	89	35	of	of	ADP
ejde-636	89	36	(	(	PUNCT
ejde-636	89	37	1.2	1.2	NUM
ejde-636	89	38	)	)	PUNCT
ejde-636	89	39	.	.	PUNCT
ejde-636	90	1	the	the	DET
ejde-636	90	2	paper	paper	NOUN
ejde-636	90	3	is	be	AUX
ejde-636	90	4	organized	organize	VERB
ejde-636	90	5	as	as	SCONJ
ejde-636	90	6	follows	follow	VERB
ejde-636	90	7	.	.	PUNCT
ejde-636	91	1	in	in	ADP
ejde-636	91	2	section	section	NOUN
ejde-636	91	3	2	2	NUM
ejde-636	91	4	,	,	PUNCT
ejde-636	91	5	we	we	PRON
ejde-636	91	6	provide	provide	VERB
ejde-636	91	7	some	some	DET
ejde-636	91	8	preliminary	preliminary	ADJ
ejde-636	91	9	concepts	concept	NOUN
ejde-636	91	10	and	and	CCONJ
ejde-636	91	11	lemmas	lemma	NOUN
ejde-636	91	12	that	that	PRON
ejde-636	91	13	will	will	AUX
ejde-636	91	14	be	be	AUX
ejde-636	91	15	utilized	utilize	VERB
ejde-636	91	16	throughout	throughout	ADP
ejde-636	91	17	the	the	DET
ejde-636	91	18	paper	paper	NOUN
ejde-636	91	19	.	.	PUNCT
ejde-636	92	1	we	we	PRON
ejde-636	92	2	prove	prove	VERB
ejde-636	92	3	theorem	theorem	VERB
ejde-636	92	4	1.2	1.2	NUM
ejde-636	92	5	in	in	ADP
ejde-636	92	6	section	section	NOUN
ejde-636	92	7	3	3	NUM
ejde-636	92	8	and	and	CCONJ
ejde-636	92	9	prove	prove	VERB
ejde-636	92	10	theorem	theorem	VERB
ejde-636	92	11	1.3	1.3	NUM
ejde-636	92	12	in	in	ADP
ejde-636	92	13	section	section	NOUN
ejde-636	92	14	4	4	NUM
ejde-636	92	15	,	,	PUNCT
ejde-636	92	16	respectively	respectively	ADV
ejde-636	92	17	.	.	PUNCT
ejde-636	93	1	2	2	X
ejde-636	93	2	.	.	X
ejde-636	93	3	preliminary	preliminary	ADJ
ejde-636	93	4	results	result	NOUN
ejde-636	93	5	throughout	throughout	ADP
ejde-636	93	6	this	this	DET
ejde-636	93	7	article	article	NOUN
ejde-636	93	8	,	,	PUNCT
ejde-636	93	9	for	for	ADP
ejde-636	93	10	1	1	NUM
ejde-636	93	11	≤	≤	NOUN
ejde-636	93	12	r	r	NOUN
ejde-636	93	13	<	<	X
ejde-636	93	14	∞	∞	PROPN
ejde-636	93	15	,	,	PUNCT
ejde-636	93	16	lr(rn	lr(rn	PROPN
ejde-636	93	17	)	)	PUNCT
ejde-636	93	18	denotes	denote	VERB
ejde-636	93	19	the	the	DET
ejde-636	93	20	standard	standard	ADJ
ejde-636	93	21	lebesgue	lebesgue	NOUN
ejde-636	93	22	space	space	NOUN
ejde-636	93	23	with	with	ADP
ejde-636	93	24	norm	norm	NOUN
ejde-636	93	25	∥u∥rr	∥u∥rr	X
ejde-636	93	26	:	:	PUNCT
ejde-636	94	1	=	=	SYM
ejde-636	94	2	∫	∫	PROPN
ejde-636	94	3	rn	rn	PROPN
ejde-636	94	4	|u|rdx	|u|rdx	PROPN
ejde-636	94	5	.	.	PUNCT
ejde-636	95	1	additionally	additionally	ADV
ejde-636	95	2	,	,	PUNCT
ejde-636	95	3	the	the	DET
ejde-636	95	4	positive	positive	ADJ
ejde-636	95	5	constants	constant	NOUN
ejde-636	95	6	are	be	AUX
ejde-636	95	7	denote	denote	VERB
ejde-636	95	8	by	by	ADP
ejde-636	95	9	c	c	PROPN
ejde-636	95	10	,	,	PUNCT
ejde-636	95	11	c1	c1	PROPN
ejde-636	95	12	,	,	PUNCT
ejde-636	95	13	c2	c2	PROPN
ejde-636	95	14	,	,	PUNCT
ejde-636	95	15	.	.	PUNCT
ejde-636	95	16	.	.	PUNCT
ejde-636	96	1	.	.	PUNCT
ejde-636	97	1	,	,	PUNCT
ejde-636	97	2	with	with	ADP
ejde-636	97	3	values	value	NOUN
ejde-636	97	4	that	that	PRON
ejde-636	97	5	may	may	AUX
ejde-636	97	6	vary	vary	VERB
ejde-636	97	7	from	from	ADP
ejde-636	97	8	line	line	NOUN
ejde-636	97	9	to	to	ADP
ejde-636	97	10	line	line	NOUN
ejde-636	97	11	.	.	PUNCT
ejde-636	98	1	the	the	DET
ejde-636	98	2	open	open	ADJ
ejde-636	98	3	ball	ball	PROPN
ejde-636	98	4	in	in	ADP
ejde-636	98	5	rn	rn	PROPN
ejde-636	98	6	is	be	AUX
ejde-636	98	7	denoted	denote	VERB
ejde-636	98	8	as	as	ADP
ejde-636	98	9	br(x	br(x	NOUN
ejde-636	98	10	)	)	PUNCT
ejde-636	98	11	with	with	ADP
ejde-636	98	12	center	center	NOUN
ejde-636	98	13	at	at	ADP
ejde-636	98	14	x	x	PUNCT
ejde-636	98	15	and	and	CCONJ
ejde-636	98	16	radius	radius	PROPN
ejde-636	98	17	r.	r.	PROPN
ejde-636	98	18	in	in	ADP
ejde-636	98	19	this	this	DET
ejde-636	98	20	section	section	NOUN
ejde-636	98	21	,	,	PUNCT
ejde-636	98	22	we	we	PRON
ejde-636	98	23	present	present	VERB
ejde-636	98	24	some	some	DET
ejde-636	98	25	preliminary	preliminary	ADJ
ejde-636	98	26	results	result	NOUN
ejde-636	98	27	which	which	PRON
ejde-636	98	28	will	will	AUX
ejde-636	98	29	be	be	AUX
ejde-636	98	30	used	use	VERB
ejde-636	98	31	in	in	ADP
ejde-636	98	32	the	the	DET
ejde-636	98	33	next	next	ADJ
ejde-636	98	34	two	two	NUM
ejde-636	98	35	sections	section	NOUN
ejde-636	98	36	.	.	PUNCT
ejde-636	99	1	we	we	PRON
ejde-636	99	2	start	start	VERB
ejde-636	99	3	with	with	ADP
ejde-636	99	4	recalling	recall	VERB
ejde-636	99	5	the	the	DET
ejde-636	99	6	well	well	ADV
ejde-636	99	7	-	-	PUNCT
ejde-636	99	8	known	know	VERB
ejde-636	99	9	gagliardo	gagliardo	NOUN
ejde-636	99	10	-	-	PUNCT
ejde-636	99	11	nirenberg	nirenberg	NOUN
ejde-636	99	12	inequality	inequality	NOUN
ejde-636	99	13	and	and	CCONJ
ejde-636	99	14	sobolev	sobolev	NOUN
ejde-636	99	15	inequality	inequality	NOUN
ejde-636	99	16	.	.	PUNCT
ejde-636	100	1	lemma	lemma	PROPN
ejde-636	100	2	2.1	2.1	NUM
ejde-636	100	3	(	(	PUNCT
ejde-636	100	4	[	[	X
ejde-636	100	5	31	31	NUM
ejde-636	100	6	]	]	PUNCT
ejde-636	100	7	)	)	PUNCT
ejde-636	100	8	.	.	PUNCT
ejde-636	101	1	if	if	SCONJ
ejde-636	101	2	n	n	NUM
ejde-636	101	3	≥	≥	NOUN
ejde-636	101	4	5	5	NUM
ejde-636	101	5	and	and	CCONJ
ejde-636	101	6	2	2	NUM
ejde-636	101	7	<	<	X
ejde-636	101	8	r	r	NOUN
ejde-636	101	9	<	<	X
ejde-636	101	10	4∗	4∗	NOUN
ejde-636	101	11	,	,	PUNCT
ejde-636	101	12	then	then	ADV
ejde-636	101	13	the	the	DET
ejde-636	101	14	gagliardo	gagliardo	NOUN
ejde-636	101	15	-	-	PUNCT
ejde-636	101	16	nirenberg	nirenberg	PROPN
ejde-636	101	17	inequality	inequality	NOUN
ejde-636	101	18	∥u∥rr	∥u∥rr	PROPN
ejde-636	101	19	≤	≤	PROPN
ejde-636	101	20	cr	cr	PROPN
ejde-636	101	21	n	n	CCONJ
ejde-636	101	22	,	,	PUNCT
ejde-636	101	23	r∥∆u∥	r∥∆u∥	ADJ
ejde-636	101	24	rγr	rγr	NOUN
ejde-636	101	25	2	2	NUM
ejde-636	101	26	∥u∥r(1−γr	∥u∥r(1−γr	PROPN
ejde-636	101	27	)	)	PUNCT
ejde-636	101	28	2	2	NUM
ejde-636	101	29	holds	hold	VERB
ejde-636	101	30	for	for	ADP
ejde-636	101	31	u	u	NOUN
ejde-636	101	32	∈	∈	PROPN
ejde-636	101	33	h2(rn	h2(rn	PROPN
ejde-636	101	34	)	)	PUNCT
ejde-636	101	35	,	,	PUNCT
ejde-636	101	36	where	where	SCONJ
ejde-636	101	37	cn	cn	PROPN
ejde-636	101	38	,	,	PUNCT
ejde-636	101	39	r	r	NOUN
ejde-636	101	40	denotes	denote	VERB
ejde-636	101	41	the	the	DET
ejde-636	101	42	sharp	sharp	ADJ
ejde-636	101	43	constant	constant	ADJ
ejde-636	101	44	.	.	PUNCT
ejde-636	102	1	lemma	lemma	PROPN
ejde-636	102	2	2.2	2.2	NUM
ejde-636	102	3	(	(	PUNCT
ejde-636	102	4	[	[	X
ejde-636	102	5	36	36	NUM
ejde-636	102	6	]	]	NUM
ejde-636	102	7	)	)	PUNCT
ejde-636	102	8	.	.	PUNCT
ejde-636	103	1	when	when	SCONJ
ejde-636	103	2	n	n	X
ejde-636	103	3	≥	≥	NOUN
ejde-636	103	4	5	5	NUM
ejde-636	103	5	,	,	PUNCT
ejde-636	103	6	we	we	PRON
ejde-636	103	7	have	have	VERB
ejde-636	103	8	s∥u∥24∗	s∥u∥24∗	VERB
ejde-636	103	9	≤	≤	NUM
ejde-636	103	10	∥∆u∥22	∥∆u∥22	NOUN
ejde-636	103	11	,	,	PUNCT
ejde-636	103	12	∀u	∀u	NOUN
ejde-636	103	13	∈	∈	NOUN
ejde-636	103	14	h2(rn	h2(rn	PROPN
ejde-636	103	15	)	)	PUNCT
ejde-636	103	16	,	,	PUNCT
ejde-636	103	17	where	where	SCONJ
ejde-636	103	18	s	s	VERB
ejde-636	103	19	>	>	X
ejde-636	103	20	0	0	PUNCT
ejde-636	103	21	depending	depend	VERB
ejde-636	103	22	only	only	ADV
ejde-636	103	23	on	on	ADP
ejde-636	103	24	n	n	PRON
ejde-636	103	25	denotes	denote	VERB
ejde-636	103	26	an	an	DET
ejde-636	103	27	optimal	optimal	ADJ
ejde-636	103	28	constant	constant	ADJ
ejde-636	103	29	.	.	PUNCT
ejde-636	104	1	note	note	VERB
ejde-636	104	2	that	that	SCONJ
ejde-636	104	3	the	the	DET
ejde-636	104	4	following	follow	VERB
ejde-636	104	5	interpolation	interpolation	NOUN
ejde-636	104	6	inequality	inequality	NOUN
ejde-636	104	7	holds:∫	holds:∫	PROPN
ejde-636	104	8	rn	rn	PROPN
ejde-636	104	9	|∇u|2dx	|∇u|2dx	VERB
ejde-636	104	10	≤	≤	PROPN
ejde-636	104	11	(	(	PUNCT
ejde-636	104	12	∫	∫	PROPN
ejde-636	104	13	rn	rn	PROPN
ejde-636	104	14	|∆u|2dx	|∆u|2dx	PROPN
ejde-636	104	15	)	)	PUNCT
ejde-636	104	16	1/2(∫	1/2(∫	PROPN
ejde-636	104	17	rn	rn	PROPN
ejde-636	104	18	|u|2dx	|u|2dx	VERB
ejde-636	104	19	)	)	PUNCT
ejde-636	104	20	1/2	1/2	NUM
ejde-636	104	21	,	,	PUNCT
ejde-636	104	22	∀u	∀u	NOUN
ejde-636	104	23	∈	∈	NOUN
ejde-636	104	24	h2(rn	h2(rn	PROPN
ejde-636	104	25	)	)	PUNCT
ejde-636	104	26	.	.	PUNCT
ejde-636	105	1	(	(	PUNCT
ejde-636	105	2	2.1	2.1	NUM
ejde-636	105	3	)	)	PUNCT
ejde-636	105	4	by	by	ADP
ejde-636	105	5	similar	similar	ADJ
ejde-636	105	6	arguments	argument	NOUN
ejde-636	105	7	as	as	ADP
ejde-636	105	8	those	those	PRON
ejde-636	105	9	in	in	ADP
ejde-636	105	10	[	[	X
ejde-636	105	11	39	39	NUM
ejde-636	105	12	]	]	PUNCT
ejde-636	105	13	,	,	PUNCT
ejde-636	105	14	we	we	PRON
ejde-636	105	15	can	can	AUX
ejde-636	105	16	obtain	obtain	VERB
ejde-636	105	17	the	the	DET
ejde-636	105	18	lions	lion	NOUN
ejde-636	105	19	’	'	PUNCT
ejde-636	105	20	type	type	NOUN
ejde-636	105	21	lemma	lemma	PROPN
ejde-636	105	22	in	in	ADP
ejde-636	105	23	h2(rn	h2(rn	PROPN
ejde-636	105	24	)	)	PUNCT
ejde-636	105	25	.	.	PUNCT
ejde-636	106	1	lemma	lemma	PROPN
ejde-636	106	2	2.3	2.3	NUM
ejde-636	106	3	.	.	PUNCT
ejde-636	107	1	assume	assume	VERB
ejde-636	107	2	that	that	SCONJ
ejde-636	107	3	{	{	PUNCT
ejde-636	107	4	un	un	PROPN
ejde-636	107	5	}	}	PUNCT
ejde-636	107	6	is	be	AUX
ejde-636	107	7	bounded	bound	VERB
ejde-636	107	8	in	in	ADP
ejde-636	107	9	h2(rn	h2(rn	PROPN
ejde-636	107	10	)	)	PUNCT
ejde-636	107	11	.	.	PUNCT
ejde-636	108	1	for	for	ADP
ejde-636	108	2	any	any	DET
ejde-636	108	3	r	r	NOUN
ejde-636	108	4	>	>	X
ejde-636	108	5	0	0	NUM
ejde-636	108	6	,	,	PUNCT
ejde-636	108	7	if	if	SCONJ
ejde-636	108	8	sup	sup	PROPN
ejde-636	108	9	y∈rn	y∈rn	PROPN
ejde-636	108	10	∫	∫	PROPN
ejde-636	108	11	br(y	br(y	NUM
ejde-636	108	12	)	)	PUNCT
ejde-636	108	13	|un|2dx→	|un|2dx→	NOUN
ejde-636	108	14	0	0	NUM
ejde-636	108	15	as	as	ADP
ejde-636	108	16	n→	n→	PROPN
ejde-636	108	17	∞	∞	PROPN
ejde-636	108	18	,	,	PUNCT
ejde-636	108	19	then	then	ADV
ejde-636	108	20	un	un	PROPN
ejde-636	108	21	→	→	SYM
ejde-636	108	22	0	0	NUM
ejde-636	108	23	in	in	ADP
ejde-636	108	24	lr(rn	lr(rn	PROPN
ejde-636	108	25	)	)	PUNCT
ejde-636	108	26	for	for	ADP
ejde-636	108	27	r	r	PROPN
ejde-636	108	28	∈	∈	PROPN
ejde-636	108	29	(	(	PUNCT
ejde-636	108	30	2	2	NUM
ejde-636	108	31	,	,	PUNCT
ejde-636	108	32	4∗	4∗	NOUN
ejde-636	108	33	)	)	PUNCT
ejde-636	108	34	.	.	PUNCT
ejde-636	109	1	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	109	2	dispersion	dispersion	NOUN
ejde-636	109	3	nonlinear	nonlinear	NOUN
ejde-636	109	4	schrödinger	schrödinger	NOUN
ejde-636	109	5	equation	equation	NOUN
ejde-636	109	6	5	5	NUM
ejde-636	109	7	to	to	PART
ejde-636	109	8	understand	understand	VERB
ejde-636	109	9	the	the	DET
ejde-636	109	10	geometry	geometry	NOUN
ejde-636	109	11	of	of	ADP
ejde-636	109	12	the	the	DET
ejde-636	109	13	constrained	constrain	VERB
ejde-636	109	14	functional	functional	ADJ
ejde-636	109	15	,	,	PUNCT
ejde-636	109	16	we	we	PRON
ejde-636	109	17	consider	consider	VERB
ejde-636	109	18	the	the	DET
ejde-636	109	19	function	function	NOUN
ejde-636	109	20	f(c	f(c	PROPN
ejde-636	109	21	,	,	PUNCT
ejde-636	109	22	ρ	ρ	NOUN
ejde-636	109	23	)	)	PUNCT
ejde-636	109	24	defined	define	VERB
ejde-636	109	25	on	on	ADP
ejde-636	109	26	r+	r+	X
ejde-636	109	27	×	×	NOUN
ejde-636	109	28	r+	r+	NOUN
ejde-636	109	29	by	by	ADP
ejde-636	109	30	f(c	f(c	PROPN
ejde-636	109	31	,	,	PUNCT
ejde-636	109	32	ρ	ρ	NOUN
ejde-636	109	33	)	)	PUNCT
ejde-636	109	34	=	=	SYM
ejde-636	109	35	1	1	NUM
ejde-636	109	36	2	2	NUM
ejde-636	109	37	−	−	PROPN
ejde-636	109	38	µ	µ	PROPN
ejde-636	109	39	q	q	X
ejde-636	109	40	cq	cq	PROPN
ejde-636	109	41	n	n	CCONJ
ejde-636	109	42	,	,	PUNCT
ejde-636	109	43	qρ	qρ	VERB
ejde-636	109	44	α0cα1	α0cα1	NOUN
ejde-636	109	45	−	−	PUNCT
ejde-636	109	46	cp	cp	NUM
ejde-636	109	47	n	n	NOUN
ejde-636	109	48	,	,	PUNCT
ejde-636	109	49	p	p	X
ejde-636	109	50	p	p	X
ejde-636	109	51	ρα2cα3	ρα2cα3	NOUN
ejde-636	109	52	,	,	PUNCT
ejde-636	109	53	and	and	CCONJ
ejde-636	109	54	its	its	PRON
ejde-636	109	55	restriction	restriction	NOUN
ejde-636	109	56	gc(ρ	gc(ρ	NOUN
ejde-636	109	57	)	)	PUNCT
ejde-636	109	58	is	be	AUX
ejde-636	109	59	defined	define	VERB
ejde-636	109	60	on	on	ADP
ejde-636	109	61	(	(	PUNCT
ejde-636	109	62	0,∞	0,∞	NOUN
ejde-636	109	63	)	)	PUNCT
ejde-636	109	64	by	by	ADP
ejde-636	109	65	ρ	ρ	PROPN
ejde-636	109	66	7→	7→	NUM
ejde-636	109	67	gc(ρ	gc(ρ	NOUN
ejde-636	109	68	)	)	PUNCT
ejde-636	109	69	:	:	PUNCT
ejde-636	110	1	=	=	SYM
ejde-636	110	2	f(c	f(c	PROPN
ejde-636	110	3	,	,	PUNCT
ejde-636	110	4	ρ	ρ	PROPN
ejde-636	110	5	)	)	PUNCT
ejde-636	110	6	for	for	ADP
ejde-636	110	7	each	each	DET
ejde-636	110	8	c	c	PROPN
ejde-636	110	9	∈	∈	PROPN
ejde-636	110	10	(	(	PUNCT
ejde-636	110	11	0,∞	0,∞	NOUN
ejde-636	110	12	)	)	PUNCT
ejde-636	110	13	,	,	PUNCT
ejde-636	110	14	where	where	SCONJ
ejde-636	110	15	α0	α0	ADJ
ejde-636	110	16	=	=	SYM
ejde-636	110	17	n(q	n(q	PROPN
ejde-636	110	18	−	−	PROPN
ejde-636	110	19	2	2	NUM
ejde-636	110	20	)	)	PUNCT
ejde-636	110	21	8	8	NUM
ejde-636	110	22	−	−	PROPN
ejde-636	110	23	1	1	NUM
ejde-636	110	24	,	,	PUNCT
ejde-636	110	25	α1	α1	PROPN
ejde-636	110	26	=	=	SYM
ejde-636	110	27	2n	2n	NUM
ejde-636	110	28	−	−	X
ejde-636	110	29	q(n	q(n	NOUN
ejde-636	110	30	−	−	PROPN
ejde-636	110	31	4	4	NUM
ejde-636	110	32	)	)	PUNCT
ejde-636	110	33	8	8	NUM
ejde-636	110	34	,	,	PUNCT
ejde-636	110	35	α2	α2	NOUN
ejde-636	110	36	=	=	SYM
ejde-636	110	37	n(p−	n(p−	X
ejde-636	110	38	2	2	NUM
ejde-636	110	39	)	)	PUNCT
ejde-636	110	40	8	8	NUM
ejde-636	110	41	−	−	NUM
ejde-636	110	42	1	1	NUM
ejde-636	110	43	,	,	PUNCT
ejde-636	110	44	α3	α3	NOUN
ejde-636	110	45	=	=	SYM
ejde-636	110	46	2n	2n	NUM
ejde-636	111	1	−	−	ADP
ejde-636	111	2	p(n	p(n	PROPN
ejde-636	111	3	−	−	PROPN
ejde-636	111	4	4	4	NUM
ejde-636	111	5	)	)	PUNCT
ejde-636	111	6	8	8	NUM
ejde-636	111	7	.	.	PUNCT
ejde-636	112	1	note	note	VERB
ejde-636	112	2	that	that	SCONJ
ejde-636	112	3	for	for	ADP
ejde-636	112	4	any	any	DET
ejde-636	112	5	n	n	PRON
ejde-636	112	6	≥	≥	NOUN
ejde-636	112	7	5	5	NUM
ejde-636	112	8	and	and	CCONJ
ejde-636	112	9	2	2	NUM
ejde-636	112	10	<	<	X
ejde-636	112	11	q	q	X
ejde-636	112	12	<	<	X
ejde-636	112	13	2	2	NUM
ejde-636	112	14	+	+	SYM
ejde-636	112	15	4	4	NUM
ejde-636	112	16	n	n	NOUN
ejde-636	112	17	<	<	X
ejde-636	112	18	p	p	X
ejde-636	112	19	<	<	X
ejde-636	112	20	p	p	X
ejde-636	112	21	≤	≤	PROPN
ejde-636	112	22	4∗	4∗	NOUN
ejde-636	112	23	,	,	PUNCT
ejde-636	112	24	we	we	PRON
ejde-636	112	25	have	have	VERB
ejde-636	112	26	α0	α0	ADJ
ejde-636	112	27	∈	∈	PROPN
ejde-636	112	28	(	(	PUNCT
ejde-636	112	29	−1,−	−1,−	PROPN
ejde-636	112	30	1	1	NUM
ejde-636	112	31	2	2	NUM
ejde-636	112	32	)	)	PUNCT
ejde-636	112	33	,	,	PUNCT
ejde-636	112	34	α1	α1	PROPN
ejde-636	112	35	∈	∈	PROPN
ejde-636	112	36	(	(	PUNCT
ejde-636	112	37	n+4	n+4	NUM
ejde-636	112	38	2n	2n	NUM
ejde-636	112	39	,	,	PUNCT
ejde-636	112	40	1	1	NUM
ejde-636	112	41	)	)	PUNCT
ejde-636	112	42	,	,	PUNCT
ejde-636	112	43	α2	α2	PROPN
ejde-636	112	44	∈	∈	PROPN
ejde-636	112	45	(	(	PUNCT
ejde-636	112	46	0	0	NUM
ejde-636	112	47	,	,	PUNCT
ejde-636	112	48	4	4	NUM
ejde-636	112	49	n−4	n−4	PROPN
ejde-636	112	50	]	]	PUNCT
ejde-636	112	51	,	,	PUNCT
ejde-636	112	52	and	and	CCONJ
ejde-636	112	53	α3	α3	PROPN
ejde-636	112	54	∈	∈	PROPN
ejde-636	113	1	[	[	X
ejde-636	113	2	0	0	NUM
ejde-636	113	3	,	,	PUNCT
ejde-636	113	4	4	4	NUM
ejde-636	113	5	n	n	NUM
ejde-636	113	6	)	)	PUNCT
ejde-636	113	7	.	.	PUNCT
ejde-636	114	1	lemma	lemma	PROPN
ejde-636	114	2	2.4	2.4	NUM
ejde-636	114	3	.	.	PUNCT
ejde-636	115	1	for	for	ADP
ejde-636	115	2	each	each	DET
ejde-636	115	3	c	c	PROPN
ejde-636	115	4	>	>	X
ejde-636	115	5	0	0	PROPN
ejde-636	115	6	,	,	PUNCT
ejde-636	115	7	the	the	DET
ejde-636	115	8	function	function	NOUN
ejde-636	115	9	gc(ρ	gc(ρ	NOUN
ejde-636	115	10	)	)	PUNCT
ejde-636	115	11	has	have	VERB
ejde-636	115	12	a	a	DET
ejde-636	115	13	unique	unique	ADJ
ejde-636	115	14	global	global	ADJ
ejde-636	115	15	maximum	maximum	NOUN
ejde-636	115	16	and	and	CCONJ
ejde-636	115	17	the	the	DET
ejde-636	115	18	maximum	maximum	ADJ
ejde-636	115	19	value	value	NOUN
ejde-636	115	20	satisfies	satisfy	VERB
ejde-636	115	21	max	max	PROPN
ejde-636	115	22	ρ>0	ρ>0	PROPN
ejde-636	115	23	gc(ρ	gc(ρ	NOUN
ejde-636	115	24	)	)	PUNCT
ejde-636	116	1			NOUN
ejde-636	116	2	>	>	X
ejde-636	116	3	0	0	PUNCT
ejde-636	117	1	if	if	SCONJ
ejde-636	117	2	c	c	PROPN
ejde-636	117	3	<	<	X
ejde-636	117	4	c0	c0	X
ejde-636	117	5	,	,	PUNCT
ejde-636	117	6	=	=	NOUN
ejde-636	117	7	0	0	PUNCT
ejde-636	118	1	if	if	SCONJ
ejde-636	118	2	c	c	NOUN
ejde-636	118	3	=	=	SYM
ejde-636	118	4	c0	c0	X
ejde-636	118	5	,	,	PUNCT
ejde-636	118	6	maxρ>0	maxρ>0	X
ejde-636	118	7	gc(ρ	gc(ρ	NOUN
ejde-636	118	8	)	)	PUNCT
ejde-636	118	9	<	<	X
ejde-636	118	10	0	0	PUNCT
ejde-636	119	1	if	if	SCONJ
ejde-636	119	2	c	c	PROPN
ejde-636	119	3	>	>	X
ejde-636	119	4	c0	c0	PROPN
ejde-636	119	5	,	,	PUNCT
ejde-636	119	6	where	where	SCONJ
ejde-636	119	7	c0	c0	PROPN
ejde-636	119	8	=	=	PUNCT
ejde-636	119	9	(	(	PUNCT
ejde-636	119	10	1	1	NUM
ejde-636	119	11	2k	2k	NUM
ejde-636	119	12	)	)	PUNCT
ejde-636	119	13	n/4	n/4	X
ejde-636	119	14	>	>	X
ejde-636	119	15	0	0	PUNCT
ejde-636	119	16	(	(	PUNCT
ejde-636	119	17	2.2	2.2	NUM
ejde-636	119	18	)	)	PUNCT
ejde-636	119	19	with	with	ADP
ejde-636	119	20	k	k	PROPN
ejde-636	119	21	=	=	PUNCT
ejde-636	119	22	µ	µ	PROPN
ejde-636	119	23	q	q	X
ejde-636	119	24	cq	cq	PROPN
ejde-636	119	25	n	n	CCONJ
ejde-636	119	26	,	,	PUNCT
ejde-636	119	27	q	q	X
ejde-636	119	28	[	[	PUNCT
ejde-636	119	29	−	−	PROPN
ejde-636	119	30	α0	α0	ADJ
ejde-636	119	31	α2	α2	NOUN
ejde-636	119	32	µp	µp	PROPN
ejde-636	119	33	q	q	PROPN
ejde-636	119	34	cq	cq	PROPN
ejde-636	119	35	n	n	CCONJ
ejde-636	119	36	,	,	PUNCT
ejde-636	119	37	q	q	PROPN
ejde-636	119	38	cp	cp	INTJ
ejde-636	119	39	n	n	PROPN
ejde-636	119	40	,	,	PUNCT
ejde-636	119	41	p	p	X
ejde-636	119	42	]	]	X
ejde-636	119	43	α0	α0	ADJ
ejde-636	119	44	α2−α0	α2−α0	NUM
ejde-636	119	45	+	+	CCONJ
ejde-636	119	46	cp	cp	NUM
ejde-636	119	47	n	n	NOUN
ejde-636	119	48	,	,	PUNCT
ejde-636	119	49	p	p	X
ejde-636	119	50	p	p	X
ejde-636	119	51	[	[	PUNCT
ejde-636	119	52	−	−	PROPN
ejde-636	119	53	α0	α0	ADJ
ejde-636	119	54	α2	α2	NOUN
ejde-636	119	55	µp	µp	PROPN
ejde-636	119	56	q	q	PROPN
ejde-636	119	57	cq	cq	PROPN
ejde-636	119	58	n	n	CCONJ
ejde-636	119	59	,	,	PUNCT
ejde-636	119	60	q	q	PROPN
ejde-636	119	61	cp	cp	INTJ
ejde-636	119	62	n	n	PROPN
ejde-636	119	63	,	,	PUNCT
ejde-636	119	64	p	p	X
ejde-636	119	65	]	]	X
ejde-636	119	66	α2	α2	PROPN
ejde-636	119	67	α2−α0	α2−α0	NUM
ejde-636	119	68	>	>	X
ejde-636	119	69	0	0	X
ejde-636	119	70	.	.	PUNCT
ejde-636	119	71	proof	proof	NOUN
ejde-636	119	72	.	.	PUNCT
ejde-636	120	1	from	from	ADP
ejde-636	120	2	the	the	DET
ejde-636	120	3	definition	definition	NOUN
ejde-636	120	4	of	of	ADP
ejde-636	120	5	gc(ρ	gc(ρ	NOUN
ejde-636	120	6	)	)	PUNCT
ejde-636	120	7	it	it	PRON
ejde-636	120	8	follows	follow	VERB
ejde-636	120	9	that	that	SCONJ
ejde-636	120	10	g′c(ρ	g′c(ρ	NOUN
ejde-636	120	11	)	)	PUNCT
ejde-636	121	1	=	=	PUNCT
ejde-636	121	2	−α0	−α0	NOUN
ejde-636	121	3	µ	µ	X
ejde-636	121	4	q	q	PROPN
ejde-636	121	5	cq	cq	PROPN
ejde-636	121	6	n	n	CCONJ
ejde-636	121	7	,	,	PUNCT
ejde-636	121	8	qρ	qρ	ADP
ejde-636	121	9	α0−1cα1	α0−1cα1	PROPN
ejde-636	121	10	−	−	PROPN
ejde-636	121	11	α2	α2	ADJ
ejde-636	121	12	1	1	NUM
ejde-636	121	13	p	p	NOUN
ejde-636	121	14	cp	cp	NUM
ejde-636	121	15	n	n	CCONJ
ejde-636	121	16	,	,	PUNCT
ejde-636	121	17	pρ	pρ	ADP
ejde-636	121	18	α2−1cα3	α2−1cα3	PROPN
ejde-636	121	19	.	.	PUNCT
ejde-636	122	1	hence	hence	ADV
ejde-636	122	2	,	,	PUNCT
ejde-636	122	3	the	the	DET
ejde-636	122	4	equation	equation	NOUN
ejde-636	122	5	g′c(ρ	g′c(ρ	PROPN
ejde-636	122	6	)	)	PUNCT
ejde-636	123	1	=	=	SYM
ejde-636	123	2	0	0	PUNCT
ejde-636	123	3	has	have	VERB
ejde-636	123	4	a	a	DET
ejde-636	123	5	unique	unique	ADJ
ejde-636	123	6	solution	solution	NOUN
ejde-636	123	7	:	:	PUNCT
ejde-636	123	8	ρc	ρc	NOUN
ejde-636	123	9	=	=	PUNCT
ejde-636	123	10	[	[	PUNCT
ejde-636	123	11	−	−	PROPN
ejde-636	123	12	α0	α0	ADJ
ejde-636	123	13	α2	α2	NOUN
ejde-636	123	14	µp	µp	PROPN
ejde-636	123	15	q	q	PROPN
ejde-636	123	16	cq	cq	PROPN
ejde-636	123	17	n	n	CCONJ
ejde-636	123	18	,	,	PUNCT
ejde-636	123	19	q	q	PROPN
ejde-636	123	20	cp	cp	INTJ
ejde-636	123	21	n	n	PROPN
ejde-636	123	22	,	,	PUNCT
ejde-636	123	23	p	p	X
ejde-636	123	24	]	]	X
ejde-636	123	25	1	1	NUM
ejde-636	123	26	α2−α0	α2−α0	NUM
ejde-636	123	27	c	c	NOUN
ejde-636	123	28	α1−α3	α1−α3	NUM
ejde-636	123	29	α2−α0	α2−α0	NUM
ejde-636	123	30	.	.	PUNCT
ejde-636	124	1	(	(	PUNCT
ejde-636	124	2	2.3	2.3	NUM
ejde-636	124	3	)	)	PUNCT
ejde-636	124	4	taking	take	VERB
ejde-636	124	5	into	into	ADP
ejde-636	124	6	account	account	NOUN
ejde-636	124	7	that	that	DET
ejde-636	124	8	gc(ρ	gc(ρ	NOUN
ejde-636	124	9	)	)	PUNCT
ejde-636	124	10	→	→	PUNCT
ejde-636	125	1	−∞	−∞	X
ejde-636	125	2	as	as	ADP
ejde-636	125	3	ρ	ρ	PROPN
ejde-636	125	4	→	→	SYM
ejde-636	125	5	0	0	NUM
ejde-636	125	6	and	and	CCONJ
ejde-636	125	7	gc(ρ	gc(ρ	NOUN
ejde-636	125	8	)	)	PUNCT
ejde-636	125	9	→	→	PUNCT
ejde-636	125	10	−∞	−∞	X
ejde-636	125	11	as	as	ADP
ejde-636	125	12	ρ	ρ	PROPN
ejde-636	125	13	→	→	SYM
ejde-636	125	14	∞	∞	PROPN
ejde-636	125	15	,	,	PUNCT
ejde-636	125	16	we	we	PRON
ejde-636	125	17	obtain	obtain	VERB
ejde-636	125	18	that	that	PRON
ejde-636	125	19	ρc	ρc	VERB
ejde-636	125	20	is	be	AUX
ejde-636	125	21	the	the	DET
ejde-636	125	22	unique	unique	ADJ
ejde-636	125	23	global	global	ADJ
ejde-636	125	24	maximum	maximum	ADJ
ejde-636	125	25	point	point	NOUN
ejde-636	125	26	of	of	ADP
ejde-636	125	27	gc(ρ	gc(ρ	NOUN
ejde-636	125	28	)	)	PUNCT
ejde-636	125	29	and	and	CCONJ
ejde-636	125	30	the	the	DET
ejde-636	125	31	maximum	maximum	ADJ
ejde-636	125	32	value	value	NOUN
ejde-636	125	33	is	be	AUX
ejde-636	125	34	max	max	PROPN
ejde-636	125	35	ρ>0	ρ>0	PROPN
ejde-636	125	36	gc(ρ	gc(ρ	NOUN
ejde-636	125	37	)	)	PUNCT
ejde-636	125	38	=	=	SYM
ejde-636	125	39	1	1	NUM
ejde-636	125	40	2	2	NUM
ejde-636	125	41	−	−	PROPN
ejde-636	125	42	µ	µ	PROPN
ejde-636	125	43	q	q	X
ejde-636	125	44	cq	cq	PROPN
ejde-636	125	45	n	n	CCONJ
ejde-636	125	46	,	,	PUNCT
ejde-636	125	47	q	q	X
ejde-636	126	1	[	[	PUNCT
ejde-636	126	2	−	−	PROPN
ejde-636	126	3	α0	α0	ADJ
ejde-636	126	4	α2	α2	NOUN
ejde-636	126	5	µp	µp	PROPN
ejde-636	126	6	q	q	PROPN
ejde-636	126	7	cq	cq	PROPN
ejde-636	126	8	n	n	CCONJ
ejde-636	126	9	,	,	PUNCT
ejde-636	126	10	q	q	PROPN
ejde-636	126	11	cp	cp	INTJ
ejde-636	126	12	n	n	PROPN
ejde-636	126	13	,	,	PUNCT
ejde-636	126	14	p	p	X
ejde-636	126	15	]	]	X
ejde-636	126	16	α0	α0	ADJ
ejde-636	126	17	α2−α0	α2−α0	NUM
ejde-636	126	18	c	c	PROPN
ejde-636	126	19	α0(α1−α3	α0(α1−α3	NUM
ejde-636	126	20	)	)	PUNCT
ejde-636	126	21	α2−α0	α2−α0	NUM
ejde-636	126	22	cα1	cα1	NOUN
ejde-636	126	23	−	−	NOUN
ejde-636	126	24	cp	cp	INTJ
ejde-636	126	25	n	n	NOUN
ejde-636	126	26	,	,	PUNCT
ejde-636	126	27	p	p	X
ejde-636	126	28	p	p	X
ejde-636	126	29	[	[	PUNCT
ejde-636	126	30	−	−	PROPN
ejde-636	126	31	α0	α0	ADJ
ejde-636	126	32	α2	α2	NOUN
ejde-636	126	33	µp	µp	PROPN
ejde-636	126	34	q	q	PROPN
ejde-636	126	35	cq	cq	PROPN
ejde-636	126	36	n	n	CCONJ
ejde-636	126	37	,	,	PUNCT
ejde-636	126	38	q	q	PROPN
ejde-636	126	39	cp	cp	INTJ
ejde-636	126	40	n	n	PROPN
ejde-636	126	41	,	,	PUNCT
ejde-636	126	42	p	p	X
ejde-636	126	43	]	]	X
ejde-636	126	44	α2	α2	ADJ
ejde-636	126	45	α2−α0	α2−α0	NUM
ejde-636	126	46	c	c	NOUN
ejde-636	126	47	α2(α1−α3	α2(α1−α3	NUM
ejde-636	126	48	)	)	PUNCT
ejde-636	126	49	α2−α0	α2−α0	NUM
ejde-636	126	50	cα3	cα3	X
ejde-636	126	51	=	=	SYM
ejde-636	126	52	1	1	NUM
ejde-636	126	53	2	2	NUM
ejde-636	126	54	−	−	PROPN
ejde-636	126	55	µ	µ	PROPN
ejde-636	126	56	q	q	X
ejde-636	126	57	cq	cq	PROPN
ejde-636	126	58	n	n	CCONJ
ejde-636	126	59	,	,	PUNCT
ejde-636	126	60	q	q	X
ejde-636	127	1	[	[	PUNCT
ejde-636	127	2	−	−	PROPN
ejde-636	127	3	α0	α0	ADJ
ejde-636	127	4	α2	α2	NOUN
ejde-636	127	5	µp	µp	PROPN
ejde-636	127	6	q	q	PROPN
ejde-636	127	7	cq	cq	PROPN
ejde-636	127	8	n	n	CCONJ
ejde-636	127	9	,	,	PUNCT
ejde-636	127	10	q	q	PROPN
ejde-636	127	11	cp	cp	INTJ
ejde-636	127	12	n	n	PROPN
ejde-636	127	13	,	,	PUNCT
ejde-636	127	14	p	p	X
ejde-636	127	15	]	]	X
ejde-636	127	16	α0	α0	ADJ
ejde-636	127	17	α2−α0	α2−α0	NUM
ejde-636	127	18	c	c	NOUN
ejde-636	127	19	α1α2−α0α3	α1α2−α0α3	INTJ
ejde-636	127	20	α2−α0	α2−α0	NUM
ejde-636	127	21	−	−	PROPN
ejde-636	127	22	cp	cp	INTJ
ejde-636	127	23	n	n	PROPN
ejde-636	127	24	,	,	PUNCT
ejde-636	127	25	p	p	X
ejde-636	127	26	p	p	X
ejde-636	127	27	[	[	PUNCT
ejde-636	127	28	−	−	PROPN
ejde-636	127	29	α0	α0	ADJ
ejde-636	127	30	α2	α2	NOUN
ejde-636	127	31	µp	µp	PROPN
ejde-636	127	32	q	q	PROPN
ejde-636	127	33	cq	cq	PROPN
ejde-636	127	34	n	n	CCONJ
ejde-636	127	35	,	,	PUNCT
ejde-636	127	36	q	q	PROPN
ejde-636	127	37	cp	cp	INTJ
ejde-636	127	38	n	n	PROPN
ejde-636	127	39	,	,	PUNCT
ejde-636	127	40	p	p	X
ejde-636	127	41	]	]	X
ejde-636	127	42	α2	α2	ADJ
ejde-636	127	43	α2−α0	α2−α0	NUM
ejde-636	127	44	c	c	NOUN
ejde-636	127	45	α1α2−α0α3	α1α2−α0α3	PRON
ejde-636	127	46	α2−α0	α2−α0	NUM
ejde-636	127	47	6	6	NUM
ejde-636	127	48	z.	z.	PROPN
ejde-636	127	49	ma	ma	PROPN
ejde-636	127	50	,	,	PUNCT
ejde-636	127	51	x.	x.	PROPN
ejde-636	127	52	chang	chang	PROPN
ejde-636	127	53	,	,	PUNCT
ejde-636	127	54	z.	z.	PROPN
ejde-636	127	55	feng	feng	PROPN
ejde-636	127	56	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	127	57	=	=	SYM
ejde-636	127	58	1	1	NUM
ejde-636	127	59	2	2	NUM
ejde-636	127	60	−kcn/4	−kcn/4	PROPN
ejde-636	127	61	.	.	PUNCT
ejde-636	128	1	by	by	ADP
ejde-636	128	2	the	the	DET
ejde-636	128	3	definition	definition	NOUN
ejde-636	128	4	of	of	ADP
ejde-636	128	5	c0	c0	NOUN
ejde-636	128	6	,	,	PUNCT
ejde-636	128	7	we	we	PRON
ejde-636	128	8	obtain	obtain	AUX
ejde-636	128	9	maxρ>0	maxρ>0	X
ejde-636	128	10	gc0(ρ	gc0(ρ	PROPN
ejde-636	128	11	)	)	PUNCT
ejde-636	128	12	=	=	SYM
ejde-636	128	13	0	0	X
ejde-636	128	14	.	.	PUNCT
ejde-636	128	15	□	□	PUNCT
ejde-636	128	16	remark	remark	NOUN
ejde-636	128	17	2.5	2.5	NUM
ejde-636	128	18	.	.	PUNCT
ejde-636	129	1	when	when	SCONJ
ejde-636	129	2	p	p	PROPN
ejde-636	129	3	=	=	SYM
ejde-636	129	4	4∗	4∗	PROPN
ejde-636	129	5	,	,	PUNCT
ejde-636	129	6	we	we	PRON
ejde-636	129	7	use	use	VERB
ejde-636	129	8	s−4∗/2	s−4∗/2	NOUN
ejde-636	129	9	instead	instead	ADV
ejde-636	129	10	of	of	ADP
ejde-636	129	11	cp	cp	NUM
ejde-636	129	12	n	n	CCONJ
ejde-636	129	13	,	,	PUNCT
ejde-636	129	14	p	p	X
ejde-636	129	15	,	,	PUNCT
ejde-636	129	16	where	where	SCONJ
ejde-636	129	17	s	s	NOUN
ejde-636	129	18	is	be	AUX
ejde-636	129	19	the	the	DET
ejde-636	129	20	optimal	optimal	ADJ
ejde-636	129	21	constant	constant	NOUN
ejde-636	129	22	given	give	VERB
ejde-636	129	23	in	in	ADP
ejde-636	129	24	lemma	lemma	PROPN
ejde-636	129	25	2.2	2.2	NUM
ejde-636	129	26	.	.	PUNCT
ejde-636	130	1	lemma	lemma	PROPN
ejde-636	130	2	2.6	2.6	NUM
ejde-636	130	3	.	.	PUNCT
ejde-636	131	1	let	let	AUX
ejde-636	131	2	(	(	PUNCT
ejde-636	131	3	c1	c1	NOUN
ejde-636	131	4	,	,	PUNCT
ejde-636	131	5	ρ1	ρ1	PROPN
ejde-636	131	6	)	)	PUNCT
ejde-636	131	7	∈	∈	PROPN
ejde-636	131	8	r+	r+	NOUN
ejde-636	131	9	×	×	NOUN
ejde-636	131	10	r+	r+	NOUN
ejde-636	131	11	be	be	AUX
ejde-636	131	12	such	such	ADJ
ejde-636	131	13	that	that	SCONJ
ejde-636	131	14	f(c1	f(c1	NOUN
ejde-636	131	15	,	,	PUNCT
ejde-636	131	16	ρ1	ρ1	PROPN
ejde-636	131	17	)	)	PUNCT
ejde-636	131	18	≥	≥	NOUN
ejde-636	131	19	0	0	NUM
ejde-636	131	20	.	.	PUNCT
ejde-636	132	1	then	then	ADV
ejde-636	132	2	for	for	ADP
ejde-636	132	3	any	any	DET
ejde-636	132	4	c2	c2	PROPN
ejde-636	132	5	∈	∈	PROPN
ejde-636	132	6	(	(	PUNCT
ejde-636	132	7	0	0	NUM
ejde-636	132	8	,	,	PUNCT
ejde-636	132	9	c1	c1	NOUN
ejde-636	132	10	]	]	PUNCT
ejde-636	132	11	we	we	PRON
ejde-636	132	12	have	have	VERB
ejde-636	132	13	f(c2	f(c2	ADJ
ejde-636	132	14	,	,	PUNCT
ejde-636	132	15	ρ2	ρ2	PROPN
ejde-636	132	16	)	)	PUNCT
ejde-636	132	17	≥	≥	NOUN
ejde-636	132	18	0	0	NUM
ejde-636	132	19	,	,	PUNCT
ejde-636	132	20	if	if	SCONJ
ejde-636	132	21	ρ2	ρ2	PROPN
ejde-636	132	22	∈	∈	PROPN
ejde-636	132	23	[	[	PUNCT
ejde-636	132	24	c2	c2	PROPN
ejde-636	132	25	c1	c1	PROPN
ejde-636	132	26	ρ1	ρ1	PROPN
ejde-636	132	27	,	,	PUNCT
ejde-636	132	28	ρ1	ρ1	PROPN
ejde-636	132	29	]	]	PUNCT
ejde-636	132	30	.	.	PUNCT
ejde-636	133	1	proof	proof	NOUN
ejde-636	133	2	.	.	PUNCT
ejde-636	134	1	since	since	SCONJ
ejde-636	134	2	c→	c→	PROPN
ejde-636	134	3	f	f	X
ejde-636	134	4	(	(	PUNCT
ejde-636	134	5	·	·	PUNCT
ejde-636	134	6	,	,	PUNCT
ejde-636	134	7	ρ	ρ	PROPN
ejde-636	134	8	)	)	PUNCT
ejde-636	134	9	is	be	AUX
ejde-636	134	10	a	a	DET
ejde-636	134	11	non	non	ADJ
ejde-636	134	12	-	-	ADJ
ejde-636	134	13	increasing	increasing	ADJ
ejde-636	134	14	function	function	NOUN
ejde-636	134	15	,	,	PUNCT
ejde-636	134	16	we	we	PRON
ejde-636	134	17	have	have	VERB
ejde-636	134	18	f(c2	f(c2	ADJ
ejde-636	134	19	,	,	PUNCT
ejde-636	134	20	ρ1	ρ1	PROPN
ejde-636	134	21	)	)	PUNCT
ejde-636	134	22	≥	≥	NOUN
ejde-636	134	23	f(c1	f(c1	NOUN
ejde-636	134	24	,	,	PUNCT
ejde-636	134	25	ρ1	ρ1	PROPN
ejde-636	134	26	)	)	PUNCT
ejde-636	134	27	≥	≥	NOUN
ejde-636	134	28	0	0	NUM
ejde-636	134	29	.	.	PUNCT
ejde-636	135	1	taking	take	VERB
ejde-636	135	2	into	into	ADP
ejde-636	135	3	account	account	NOUN
ejde-636	135	4	α0	α0	ADJ
ejde-636	135	5	+	+	CCONJ
ejde-636	135	6	α1	α1	PROPN
ejde-636	135	7	=	=	SYM
ejde-636	135	8	q−2	q−2	PROPN
ejde-636	135	9	2	2	NUM
ejde-636	135	10	and	and	CCONJ
ejde-636	135	11	α2	α2	ADJ
ejde-636	135	12	+	+	CCONJ
ejde-636	135	13	α3	α3	NOUN
ejde-636	135	14	=	=	SYM
ejde-636	135	15	p−2	p−2	PROPN
ejde-636	135	16	2	2	NUM
ejde-636	135	17	,	,	PUNCT
ejde-636	135	18	we	we	PRON
ejde-636	135	19	obtain	obtain	VERB
ejde-636	135	20	f(c2	f(c2	NOUN
ejde-636	135	21	,	,	PUNCT
ejde-636	135	22	c2	c2	PROPN
ejde-636	135	23	c1	c1	PROPN
ejde-636	135	24	ρ1)−	ρ1)−	PROPN
ejde-636	135	25	f(c1	f(c1	PROPN
ejde-636	135	26	,	,	PUNCT
ejde-636	135	27	ρ1	ρ1	NOUN
ejde-636	135	28	)	)	PUNCT
ejde-636	135	29	=	=	SYM
ejde-636	135	30	µ	µ	X
ejde-636	135	31	q	q	X
ejde-636	135	32	cq	cq	PROPN
ejde-636	135	33	n	n	CCONJ
ejde-636	135	34	,	,	PUNCT
ejde-636	135	35	qρ	qρ	PROPN
ejde-636	135	36	α1	α1	PROPN
ejde-636	135	37	1	1	NUM
ejde-636	135	38	cα1	cα1	NOUN
ejde-636	135	39	1	1	NUM
ejde-636	135	40	(	(	PUNCT
ejde-636	135	41	1−	1−	NUM
ejde-636	135	42	(	(	PUNCT
ejde-636	135	43	c2	c2	PROPN
ejde-636	135	44	c1	c1	PROPN
ejde-636	135	45	)	)	PUNCT
ejde-636	135	46	α0+α1	α0+α1	NUM
ejde-636	135	47	)	)	PUNCT
ejde-636	136	1	+	+	CCONJ
ejde-636	137	1	1	1	NUM
ejde-636	137	2	p	p	NOUN
ejde-636	137	3	cp	cp	NUM
ejde-636	137	4	n	n	CCONJ
ejde-636	137	5	,	,	PUNCT
ejde-636	137	6	pρ	pρ	ADP
ejde-636	137	7	α1	α1	PROPN
ejde-636	137	8	1	1	NUM
ejde-636	137	9	cα3	cα3	NOUN
ejde-636	137	10	1	1	NUM
ejde-636	137	11	(	(	PUNCT
ejde-636	137	12	1−	1−	NUM
ejde-636	137	13	(	(	PUNCT
ejde-636	137	14	c2	c2	PROPN
ejde-636	137	15	c1	c1	PROPN
ejde-636	137	16	)	)	PUNCT
ejde-636	137	17	α2+α3	α2+α3	X
ejde-636	137	18	)	)	PUNCT
ejde-636	137	19	=	=	SYM
ejde-636	137	20	µ	µ	X
ejde-636	137	21	q	q	X
ejde-636	137	22	cq	cq	PROPN
ejde-636	137	23	n	n	CCONJ
ejde-636	137	24	,	,	PUNCT
ejde-636	137	25	qρ	qρ	PROPN
ejde-636	137	26	α1	α1	PROPN
ejde-636	137	27	1	1	NUM
ejde-636	137	28	cα1	cα1	NOUN
ejde-636	137	29	1	1	NUM
ejde-636	137	30	(	(	PUNCT
ejde-636	137	31	1−	1−	NUM
ejde-636	137	32	(	(	PUNCT
ejde-636	137	33	c2	c2	PROPN
ejde-636	137	34	c1	c1	PROPN
ejde-636	137	35	)	)	PUNCT
ejde-636	138	1	q−2	q−2	PROPN
ejde-636	138	2	2	2	NUM
ejde-636	138	3	)	)	PUNCT
ejde-636	138	4	+	+	CCONJ
ejde-636	138	5	1	1	NUM
ejde-636	138	6	p	p	NOUN
ejde-636	138	7	cp	cp	NUM
ejde-636	138	8	n	n	CCONJ
ejde-636	138	9	,	,	PUNCT
ejde-636	138	10	pρ	pρ	ADP
ejde-636	138	11	α1	α1	PROPN
ejde-636	138	12	1	1	NUM
ejde-636	138	13	cα3	cα3	NOUN
ejde-636	138	14	1	1	NUM
ejde-636	138	15	(	(	PUNCT
ejde-636	138	16	1−	1−	NUM
ejde-636	138	17	(	(	PUNCT
ejde-636	138	18	c2	c2	PROPN
ejde-636	138	19	c1	c1	PROPN
ejde-636	138	20	)	)	PUNCT
ejde-636	139	1	p−2	p−2	NOUN
ejde-636	139	2	2	2	NUM
ejde-636	139	3	)	)	PUNCT
ejde-636	139	4	.	.	PUNCT
ejde-636	140	1	since	since	SCONJ
ejde-636	140	2	c2	c2	PROPN
ejde-636	140	3	<	<	X
ejde-636	140	4	c1	c1	PROPN
ejde-636	140	5	,	,	PUNCT
ejde-636	140	6	2	2	NUM
ejde-636	140	7	<	<	X
ejde-636	140	8	q	q	X
ejde-636	140	9	<	<	X
ejde-636	140	10	2	2	NUM
ejde-636	140	11	+	+	SYM
ejde-636	140	12	4	4	NUM
ejde-636	140	13	n	n	NOUN
ejde-636	140	14	and	and	CCONJ
ejde-636	140	15	p	p	X
ejde-636	140	16	<	<	X
ejde-636	140	17	p	p	X
ejde-636	140	18	≤	≤	PROPN
ejde-636	140	19	4∗	4∗	NOUN
ejde-636	140	20	,	,	PUNCT
ejde-636	140	21	we	we	PRON
ejde-636	140	22	derive	derive	VERB
ejde-636	140	23	f(c2	f(c2	NOUN
ejde-636	140	24	,	,	PUNCT
ejde-636	140	25	c2	c2	PROPN
ejde-636	140	26	c1	c1	PROPN
ejde-636	140	27	ρ1	ρ1	PROPN
ejde-636	140	28	)	)	PUNCT
ejde-636	140	29	≥	≥	PROPN
ejde-636	140	30	f(c1	f(c1	NOUN
ejde-636	140	31	,	,	PUNCT
ejde-636	140	32	ρ1	ρ1	PROPN
ejde-636	140	33	)	)	PUNCT
ejde-636	140	34	≥	≥	NOUN
ejde-636	140	35	0	0	NUM
ejde-636	140	36	.	.	PUNCT
ejde-636	141	1	we	we	PRON
ejde-636	141	2	claim	claim	VERB
ejde-636	141	3	that	that	SCONJ
ejde-636	141	4	if	if	SCONJ
ejde-636	141	5	gc2	gc2	PROPN
ejde-636	141	6	(	(	PUNCT
ejde-636	141	7	c2	c2	PROPN
ejde-636	141	8	c1	c1	PROPN
ejde-636	141	9	ρ	ρ	PROPN
ejde-636	141	10	)	)	PUNCT
ejde-636	141	11	≥	≥	NOUN
ejde-636	141	12	0	0	NUM
ejde-636	141	13	and	and	CCONJ
ejde-636	141	14	gc2(ρ1	gc2(ρ1	PROPN
ejde-636	141	15	)	)	PUNCT
ejde-636	141	16	≥	≥	NOUN
ejde-636	141	17	0	0	NUM
ejde-636	141	18	,	,	PUNCT
ejde-636	141	19	then	then	ADV
ejde-636	141	20	f(c2	f(c2	PROPN
ejde-636	141	21	,	,	PUNCT
ejde-636	141	22	ρ	ρ	NOUN
ejde-636	141	23	)	)	PUNCT
ejde-636	141	24	=	=	SYM
ejde-636	141	25	gc2(ρ	gc2(ρ	NOUN
ejde-636	141	26	)	)	PUNCT
ejde-636	141	27	≥	≥	NOUN
ejde-636	141	28	0	0	NUM
ejde-636	141	29	,	,	PUNCT
ejde-636	141	30	for	for	ADP
ejde-636	141	31	ρ	ρ	PROPN
ejde-636	141	32	∈	∈	PROPN
ejde-636	141	33	[	[	PUNCT
ejde-636	141	34	c2	c2	PROPN
ejde-636	141	35	c1	c1	PROPN
ejde-636	141	36	ρ	ρ	PROPN
ejde-636	141	37	,	,	PUNCT
ejde-636	141	38	ρ1	ρ1	PROPN
ejde-636	141	39	]	]	PUNCT
ejde-636	141	40	.	.	PUNCT
ejde-636	142	1	indeed	indeed	ADV
ejde-636	142	2	,	,	PUNCT
ejde-636	142	3	if	if	SCONJ
ejde-636	142	4	gc2(ρ	gc2(ρ	PROPN
ejde-636	142	5	)	)	PUNCT
ejde-636	143	1	<	<	X
ejde-636	143	2	0	0	NUM
ejde-636	144	1	for	for	ADP
ejde-636	144	2	some	some	DET
ejde-636	144	3	ρ	ρ	NUM
ejde-636	144	4	∈	∈	PROPN
ejde-636	144	5	[	[	PUNCT
ejde-636	144	6	c2c1	c2c1	PROPN
ejde-636	144	7	ρ	ρ	PROPN
ejde-636	144	8	,	,	PUNCT
ejde-636	144	9	ρ1	ρ1	PROPN
ejde-636	144	10	]	]	PUNCT
ejde-636	144	11	,	,	PUNCT
ejde-636	144	12	then	then	ADV
ejde-636	144	13	there	there	PRON
ejde-636	144	14	exists	exist	VERB
ejde-636	144	15	a	a	DET
ejde-636	144	16	local	local	ADJ
ejde-636	144	17	minimum	minimum	NOUN
ejde-636	144	18	point	point	NOUN
ejde-636	144	19	on	on	ADP
ejde-636	144	20	(	(	PUNCT
ejde-636	144	21	c2c1	c2c1	PROPN
ejde-636	144	22	ρ	ρ	PROPN
ejde-636	144	23	,	,	PUNCT
ejde-636	144	24	ρ1	ρ1	PROPN
ejde-636	144	25	)	)	PUNCT
ejde-636	144	26	.	.	PUNCT
ejde-636	145	1	this	this	PRON
ejde-636	145	2	contradicts	contradict	VERB
ejde-636	145	3	the	the	DET
ejde-636	145	4	fact	fact	NOUN
ejde-636	145	5	in	in	ADP
ejde-636	145	6	lemma	lemma	PROPN
ejde-636	145	7	2.4	2.4	NUM
ejde-636	145	8	that	that	SCONJ
ejde-636	145	9	the	the	DET
ejde-636	145	10	function	function	NOUN
ejde-636	145	11	gc2(ρ	gc2(ρ	PROPN
ejde-636	145	12	)	)	PUNCT
ejde-636	145	13	has	have	AUX
ejde-636	145	14	a	a	DET
ejde-636	145	15	unique	unique	ADJ
ejde-636	145	16	critical	critical	ADJ
ejde-636	145	17	point	point	NOUN
ejde-636	145	18	which	which	PRON
ejde-636	145	19	has	have	VERB
ejde-636	145	20	to	to	PART
ejde-636	145	21	be	be	AUX
ejde-636	145	22	its	its	PRON
ejde-636	145	23	unique	unique	ADJ
ejde-636	145	24	global	global	ADJ
ejde-636	145	25	maximum	maximum	NOUN
ejde-636	145	26	.	.	PUNCT
ejde-636	146	1	□	□	PUNCT
ejde-636	146	2	lemma	lemma	PROPN
ejde-636	146	3	2.7	2.7	NUM
ejde-636	146	4	.	.	PUNCT
ejde-636	147	1	for	for	ADP
ejde-636	147	2	p	p	NOUN
ejde-636	147	3	<	<	X
ejde-636	147	4	q	q	X
ejde-636	147	5	<	<	X
ejde-636	147	6	p	p	X
ejde-636	147	7	<	<	X
ejde-636	147	8	4∗	4∗	NOUN
ejde-636	147	9	,	,	PUNCT
ejde-636	147	10	a	a	DET
ejde-636	147	11	>	>	X
ejde-636	147	12	0	0	NUM
ejde-636	147	13	,	,	PUNCT
ejde-636	147	14	b	b	PROPN
ejde-636	147	15	≥	≥	NOUN
ejde-636	147	16	0	0	NUM
ejde-636	147	17	,	,	PUNCT
ejde-636	147	18	c	c	X
ejde-636	147	19	≥	≥	NUM
ejde-636	147	20	0	0	NUM
ejde-636	147	21	and	and	CCONJ
ejde-636	147	22	d	d	X
ejde-636	147	23	≥	≥	NOUN
ejde-636	147	24	0	0	NUM
ejde-636	147	25	with	with	ADP
ejde-636	147	26	c	c	PROPN
ejde-636	147	27	+	+	CCONJ
ejde-636	147	28	d	d	X
ejde-636	147	29	>	>	X
ejde-636	147	30	0	0	PROPN
ejde-636	147	31	,	,	PUNCT
ejde-636	147	32	which	which	PRON
ejde-636	147	33	are	be	AUX
ejde-636	147	34	independent	independent	ADJ
ejde-636	147	35	of	of	ADP
ejde-636	147	36	t	t	PROPN
ejde-636	147	37	,	,	PUNCT
ejde-636	147	38	we	we	PRON
ejde-636	147	39	denote	denote	VERB
ejde-636	147	40	h(a	h(a	PROPN
ejde-636	147	41	,	,	PUNCT
ejde-636	147	42	b	b	PROPN
ejde-636	147	43	,	,	PUNCT
ejde-636	147	44	c	c	NOUN
ejde-636	147	45	,	,	PUNCT
ejde-636	147	46	d	d	NOUN
ejde-636	147	47	)	)	PUNCT
ejde-636	147	48	=	=	SYM
ejde-636	147	49	max	max	PROPN
ejde-636	147	50	t>0	t>0	NOUN
ejde-636	147	51	{	{	PUNCT
ejde-636	147	52	a	a	DET
ejde-636	147	53	·	·	PUNCT
ejde-636	147	54	t2	t2	NOUN
ejde-636	147	55	+	+	CCONJ
ejde-636	147	56	b	b	PROPN
ejde-636	147	57	·	·	PUNCT
ejde-636	147	58	t−	t−	PROPN
ejde-636	147	59	c	c	NOUN
ejde-636	147	60	·	·	PUNCT
ejde-636	147	61	t	t	PROPN
ejde-636	147	62	n(q−2	n(q−2	PUNCT
ejde-636	147	63	)	)	PUNCT
ejde-636	147	64	4	4	NUM
ejde-636	147	65	−	−	PROPN
ejde-636	147	66	d	d	PROPN
ejde-636	147	67	·	·	PUNCT
ejde-636	147	68	t	t	PROPN
ejde-636	147	69	n(p−2	n(p−2	PROPN
ejde-636	147	70	)	)	PUNCT
ejde-636	147	71	4	4	NUM
ejde-636	147	72	}	}	PUNCT
ejde-636	147	73	.	.	PUNCT
ejde-636	148	1	then	then	ADV
ejde-636	148	2	the	the	DET
ejde-636	148	3	function	function	NOUN
ejde-636	148	4	(	(	PUNCT
ejde-636	148	5	a	a	PRON
ejde-636	148	6	,	,	PUNCT
ejde-636	148	7	b	b	NOUN
ejde-636	148	8	,	,	PUNCT
ejde-636	148	9	c	c	NOUN
ejde-636	148	10	,	,	PUNCT
ejde-636	148	11	d	d	NOUN
ejde-636	148	12	)	)	PUNCT
ejde-636	148	13	7→	7→	PROPN
ejde-636	148	14	h(a	h(a	PROPN
ejde-636	148	15	,	,	PUNCT
ejde-636	148	16	b	b	PROPN
ejde-636	148	17	,	,	PUNCT
ejde-636	148	18	c	c	NOUN
ejde-636	148	19	,	,	PUNCT
ejde-636	148	20	d	d	NOUN
ejde-636	148	21	)	)	PUNCT
ejde-636	148	22	is	be	AUX
ejde-636	148	23	continuous	continuous	ADJ
ejde-636	148	24	.	.	PUNCT
ejde-636	149	1	proof	proof	NOUN
ejde-636	149	2	.	.	PUNCT
ejde-636	150	1	by	by	ADP
ejde-636	150	2	making	make	VERB
ejde-636	150	3	slight	slight	ADJ
ejde-636	150	4	modifications	modification	NOUN
ejde-636	150	5	to	to	ADP
ejde-636	150	6	the	the	DET
ejde-636	150	7	proof	proof	NOUN
ejde-636	150	8	of	of	ADP
ejde-636	150	9	[	[	X
ejde-636	150	10	2	2	NUM
ejde-636	150	11	,	,	PUNCT
ejde-636	150	12	lemma	lemma	PROPN
ejde-636	150	13	5.2	5.2	NUM
ejde-636	150	14	]	]	PUNCT
ejde-636	150	15	,	,	PUNCT
ejde-636	150	16	we	we	PRON
ejde-636	150	17	can	can	AUX
ejde-636	150	18	arrive	arrive	VERB
ejde-636	150	19	at	at	ADP
ejde-636	150	20	the	the	DET
ejde-636	150	21	desired	desire	VERB
ejde-636	150	22	result	result	NOUN
ejde-636	150	23	.	.	PUNCT
ejde-636	151	1	so	so	ADV
ejde-636	151	2	,	,	PUNCT
ejde-636	151	3	we	we	PRON
ejde-636	151	4	omit	omit	VERB
ejde-636	151	5	the	the	DET
ejde-636	151	6	details	detail	NOUN
ejde-636	151	7	here	here	ADV
ejde-636	151	8	.	.	PUNCT
ejde-636	152	1	□	□	PUNCT
ejde-636	152	2	3	3	X
ejde-636	152	3	.	.	X
ejde-636	152	4	case	case	NOUN
ejde-636	152	5	2	2	NUM
ejde-636	152	6	<	<	X
ejde-636	152	7	q	q	X
ejde-636	152	8	<	<	X
ejde-636	152	9	2	2	NUM
ejde-636	152	10	+	+	SYM
ejde-636	152	11	4	4	NUM
ejde-636	152	12	n	n	NOUN
ejde-636	152	13	<	<	X
ejde-636	152	14	p	p	X
ejde-636	152	15	<	<	X
ejde-636	152	16	p	p	X
ejde-636	152	17	≤	≤	NUM
ejde-636	152	18	4∗	4∗	NOUN
ejde-636	152	19	in	in	ADP
ejde-636	152	20	this	this	DET
ejde-636	152	21	section	section	NOUN
ejde-636	152	22	,	,	PUNCT
ejde-636	152	23	we	we	PRON
ejde-636	152	24	show	show	VERB
ejde-636	152	25	that	that	SCONJ
ejde-636	152	26	ground	ground	NOUN
ejde-636	152	27	states	state	NOUN
ejde-636	152	28	of	of	ADP
ejde-636	152	29	equation	equation	NOUN
ejde-636	152	30	(	(	PUNCT
ejde-636	152	31	1.2	1.2	NUM
ejde-636	152	32	)	)	PUNCT
ejde-636	152	33	exist	exist	VERB
ejde-636	152	34	which	which	PRON
ejde-636	152	35	correspond	correspond	VERB
ejde-636	152	36	to	to	ADP
ejde-636	152	37	the	the	DET
ejde-636	152	38	local	local	ADJ
ejde-636	152	39	minima	minima	NOUN
ejde-636	152	40	of	of	ADP
ejde-636	152	41	the	the	DET
ejde-636	152	42	associated	associated	ADJ
ejde-636	152	43	functional	functional	ADJ
ejde-636	152	44	.	.	PUNCT
ejde-636	153	1	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	153	2	dispersion	dispersion	NOUN
ejde-636	153	3	nonlinear	nonlinear	NOUN
ejde-636	153	4	schrödinger	schrödinger	NOUN
ejde-636	153	5	equation	equation	NOUN
ejde-636	153	6	7	7	NUM
ejde-636	153	7	3.1	3.1	NUM
ejde-636	153	8	.	.	PUNCT
ejde-636	154	1	properties	property	NOUN
ejde-636	154	2	of	of	ADP
ejde-636	154	3	mapping	mapping	NOUN
ejde-636	154	4	c	c	PROPN
ejde-636	154	5	7→	7→	NUM
ejde-636	154	6	mp	mp	PROPN
ejde-636	154	7	,	,	PUNCT
ejde-636	154	8	q(c	q(c	PROPN
ejde-636	154	9	)	)	PUNCT
ejde-636	154	10	.	.	PUNCT
ejde-636	155	1	let	let	VERB
ejde-636	155	2	c0	c0	PROPN
ejde-636	155	3	>	>	X
ejde-636	155	4	0	0	PUNCT
ejde-636	155	5	be	be	AUX
ejde-636	155	6	determined	determine	VERB
ejde-636	155	7	by	by	ADP
ejde-636	155	8	equation	equation	NOUN
ejde-636	155	9	(	(	PUNCT
ejde-636	155	10	2.2	2.2	NUM
ejde-636	155	11	)	)	PUNCT
ejde-636	155	12	and	and	CCONJ
ejde-636	155	13	let	let	VERB
ejde-636	155	14	ρ0	ρ0	VERB
ejde-636	155	15	:	:	PUNCT
ejde-636	155	16	=	=	SYM
ejde-636	155	17	ρc0	ρc0	X
ejde-636	155	18	>	>	X
ejde-636	155	19	0	0	PUNCT
ejde-636	155	20	be	be	AUX
ejde-636	155	21	defined	define	VERB
ejde-636	155	22	by	by	ADP
ejde-636	155	23	equation	equation	NOUN
ejde-636	155	24	(	(	PUNCT
ejde-636	155	25	2.3	2.3	NUM
ejde-636	155	26	)	)	PUNCT
ejde-636	155	27	.	.	PUNCT
ejde-636	156	1	according	accord	VERB
ejde-636	156	2	to	to	ADP
ejde-636	156	3	lemmas	lemmas	PROPN
ejde-636	156	4	2.4	2.4	NUM
ejde-636	156	5	and	and	CCONJ
ejde-636	156	6	2.6	2.6	NUM
ejde-636	156	7	,	,	PUNCT
ejde-636	156	8	it	it	PRON
ejde-636	156	9	follows	follow	VERB
ejde-636	156	10	that	that	SCONJ
ejde-636	156	11	f(c0	f(c0	NOUN
ejde-636	156	12	,	,	PUNCT
ejde-636	156	13	ρ0	ρ0	PROPN
ejde-636	156	14	)	)	PUNCT
ejde-636	156	15	=	=	SYM
ejde-636	156	16	0	0	NUM
ejde-636	156	17	,	,	PUNCT
ejde-636	156	18	and	and	CCONJ
ejde-636	156	19	f(c	f(c	PROPN
ejde-636	156	20	,	,	PUNCT
ejde-636	156	21	ρ0	ρ0	PROPN
ejde-636	156	22	)	)	PUNCT
ejde-636	156	23	>	>	X
ejde-636	156	24	0	0	PUNCT
ejde-636	157	1	for	for	ADP
ejde-636	157	2	all	all	DET
ejde-636	157	3	c	c	NOUN
ejde-636	157	4	∈	∈	PROPN
ejde-636	157	5	(	(	PUNCT
ejde-636	157	6	0	0	NUM
ejde-636	157	7	,	,	PUNCT
ejde-636	157	8	c0	c0	NOUN
ejde-636	157	9	)	)	PUNCT
ejde-636	157	10	.	.	PUNCT
ejde-636	158	1	set	set	VERB
ejde-636	158	2	bρ0	bρ0	PROPN
ejde-636	158	3	=	=	PUNCT
ejde-636	158	4	{	{	PUNCT
ejde-636	158	5	u	u	NOUN
ejde-636	158	6	∈	∈	PROPN
ejde-636	158	7	h2(rn	h2(rn	PROPN
ejde-636	158	8	)	)	PUNCT
ejde-636	158	9	:	:	PUNCT
ejde-636	159	1	∥∆u∥22	∥∆u∥22	X
ejde-636	159	2	+	+	CCONJ
ejde-636	159	3	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	159	4	<	<	X
ejde-636	159	5	ρ0	ρ0	PROPN
ejde-636	159	6	}	}	PUNCT
ejde-636	159	7	and	and	CCONJ
ejde-636	159	8	v	v	X
ejde-636	159	9	(	(	PUNCT
ejde-636	159	10	c	c	NOUN
ejde-636	159	11	)	)	PUNCT
ejde-636	159	12	:	:	PUNCT
ejde-636	159	13	=	=	SYM
ejde-636	159	14	s(c	s(c	ADJ
ejde-636	159	15	)	)	PUNCT
ejde-636	159	16	∩bρ0	∩bρ0	NOUN
ejde-636	159	17	.	.	PUNCT
ejde-636	160	1	for	for	ADP
ejde-636	160	2	c	c	PROPN
ejde-636	160	3	∈	∈	PROPN
ejde-636	160	4	(	(	PUNCT
ejde-636	160	5	0	0	NUM
ejde-636	160	6	,	,	PUNCT
ejde-636	160	7	c0	c0	NOUN
ejde-636	160	8	)	)	PUNCT
ejde-636	160	9	,	,	PUNCT
ejde-636	160	10	we	we	PRON
ejde-636	160	11	consider	consider	VERB
ejde-636	160	12	the	the	DET
ejde-636	160	13	local	local	ADJ
ejde-636	160	14	minimization	minimization	NOUN
ejde-636	160	15	problem	problem	NOUN
ejde-636	160	16	:	:	PUNCT
ejde-636	160	17	mp	mp	NOUN
ejde-636	160	18	,	,	PUNCT
ejde-636	160	19	q(c	q(c	PROPN
ejde-636	160	20	)	)	PUNCT
ejde-636	160	21	=	=	SYM
ejde-636	160	22	inf	inf	NOUN
ejde-636	160	23	u∈v	u∈v	NOUN
ejde-636	160	24	(	(	PUNCT
ejde-636	160	25	c	c	NOUN
ejde-636	160	26	)	)	PUNCT
ejde-636	160	27	ep	ep	NOUN
ejde-636	160	28	,	,	PUNCT
ejde-636	160	29	q(u	q(u	NOUN
ejde-636	160	30	)	)	PUNCT
ejde-636	160	31	.	.	PUNCT
ejde-636	161	1	lemma	lemma	PROPN
ejde-636	161	2	3.1	3.1	NUM
ejde-636	161	3	.	.	PUNCT
ejde-636	162	1	let	let	VERB
ejde-636	162	2	c	c	NOUN
ejde-636	162	3	∈	∈	PROPN
ejde-636	162	4	(	(	PUNCT
ejde-636	162	5	0	0	NUM
ejde-636	162	6	,	,	PUNCT
ejde-636	162	7	c0	c0	NOUN
ejde-636	162	8	)	)	PUNCT
ejde-636	162	9	and	and	CCONJ
ejde-636	162	10	2	2	NUM
ejde-636	162	11	<	<	X
ejde-636	162	12	q	q	X
ejde-636	162	13	<	<	X
ejde-636	162	14	2	2	NUM
ejde-636	162	15	+	+	SYM
ejde-636	162	16	4	4	NUM
ejde-636	162	17	n	n	NOUN
ejde-636	162	18	<	<	X
ejde-636	162	19	p	p	X
ejde-636	162	20	<	<	X
ejde-636	162	21	p	p	X
ejde-636	162	22	≤	≤	NUM
ejde-636	162	23	4∗.	4∗.	NUM
ejde-636	162	24	then	then	ADV
ejde-636	162	25	the	the	DET
ejde-636	162	26	following	follow	VERB
ejde-636	162	27	three	three	NUM
ejde-636	162	28	assertions	assertion	NOUN
ejde-636	162	29	hold	hold	VERB
ejde-636	162	30	.	.	PUNCT
ejde-636	163	1	(	(	PUNCT
ejde-636	163	2	1	1	X
ejde-636	163	3	)	)	PUNCT
ejde-636	163	4	mp	mp	NOUN
ejde-636	163	5	,	,	PUNCT
ejde-636	163	6	q(c	q(c	X
ejde-636	163	7	)	)	PUNCT
ejde-636	163	8	=	=	SYM
ejde-636	163	9	infu∈v	infu∈v	NOUN
ejde-636	163	10	(	(	PUNCT
ejde-636	163	11	c)ep	c)ep	PROPN
ejde-636	163	12	,	,	PUNCT
ejde-636	163	13	q(u	q(u	ADJ
ejde-636	163	14	)	)	PUNCT
ejde-636	163	15	<	<	X
ejde-636	163	16	0	0	PUNCT
ejde-636	163	17	<	<	X
ejde-636	163	18	infu∈∂v	infu∈∂v	PROPN
ejde-636	163	19	(	(	PUNCT
ejde-636	163	20	c)ep	c)ep	PROPN
ejde-636	163	21	,	,	PUNCT
ejde-636	163	22	q(u	q(u	NOUN
ejde-636	163	23	)	)	PUNCT
ejde-636	163	24	;	;	PUNCT
ejde-636	163	25	.	.	PUNCT
ejde-636	164	1	(	(	PUNCT
ejde-636	164	2	2	2	X
ejde-636	164	3	)	)	PUNCT
ejde-636	164	4	the	the	DET
ejde-636	164	5	function	function	NOUN
ejde-636	164	6	c	c	PROPN
ejde-636	164	7	7→	7→	PROPN
ejde-636	164	8	mp	mp	PROPN
ejde-636	164	9	,	,	PUNCT
ejde-636	164	10	q(c	q(c	PROPN
ejde-636	164	11	)	)	PUNCT
ejde-636	164	12	is	be	AUX
ejde-636	164	13	a	a	DET
ejde-636	164	14	continuous	continuous	ADJ
ejde-636	164	15	mapping	mapping	NOUN
ejde-636	164	16	.	.	PUNCT
ejde-636	165	1	(	(	PUNCT
ejde-636	165	2	3	3	X
ejde-636	165	3	)	)	PUNCT
ejde-636	165	4	for	for	ADP
ejde-636	165	5	all	all	PRON
ejde-636	165	6	α	α	PRON
ejde-636	165	7	∈	∈	NOUN
ejde-636	165	8	(	(	PUNCT
ejde-636	165	9	0	0	NUM
ejde-636	165	10	,	,	PUNCT
ejde-636	165	11	c	c	NOUN
ejde-636	165	12	)	)	PUNCT
ejde-636	165	13	,	,	PUNCT
ejde-636	165	14	we	we	PRON
ejde-636	165	15	have	have	VERB
ejde-636	165	16	mp	mp	NOUN
ejde-636	165	17	,	,	PUNCT
ejde-636	165	18	q(c	q(c	PROPN
ejde-636	165	19	)	)	PUNCT
ejde-636	165	20	≤	≤	PROPN
ejde-636	165	21	mp	mp	PROPN
ejde-636	165	22	,	,	PUNCT
ejde-636	165	23	q(α	q(α	PROPN
ejde-636	165	24	)	)	PUNCT
ejde-636	166	1	+	+	NOUN
ejde-636	166	2	mp	mp	PROPN
ejde-636	166	3	,	,	PUNCT
ejde-636	166	4	q(c−	q(c−	PROPN
ejde-636	166	5	α	α	NOUN
ejde-636	166	6	)	)	PUNCT
ejde-636	166	7	.	.	PUNCT
ejde-636	167	1	if	if	SCONJ
ejde-636	167	2	mp	mp	PROPN
ejde-636	167	3	,	,	PUNCT
ejde-636	167	4	q(α	q(α	PROPN
ejde-636	167	5	)	)	PUNCT
ejde-636	167	6	or	or	CCONJ
ejde-636	167	7	mp	mp	PROPN
ejde-636	167	8	,	,	PUNCT
ejde-636	167	9	q(c−	q(c−	PROPN
ejde-636	167	10	α	α	NOUN
ejde-636	167	11	)	)	PUNCT
ejde-636	167	12	is	be	AUX
ejde-636	167	13	attained	attain	VERB
ejde-636	167	14	,	,	PUNCT
ejde-636	167	15	then	then	ADV
ejde-636	167	16	the	the	DET
ejde-636	167	17	inequality	inequality	NOUN
ejde-636	167	18	is	be	AUX
ejde-636	167	19	strict	strict	ADJ
ejde-636	167	20	.	.	PUNCT
ejde-636	168	1	proof	proof	NOUN
ejde-636	168	2	.	.	PUNCT
ejde-636	169	1	(	(	PUNCT
ejde-636	169	2	1	1	X
ejde-636	169	3	)	)	PUNCT
ejde-636	169	4	for	for	ADP
ejde-636	169	5	any	any	DET
ejde-636	169	6	u	u	PROPN
ejde-636	169	7	∈	∈	PROPN
ejde-636	169	8	∂v	∂v	PROPN
ejde-636	169	9	(	(	PUNCT
ejde-636	169	10	c	c	NOUN
ejde-636	169	11	)	)	PUNCT
ejde-636	169	12	,	,	PUNCT
ejde-636	169	13	we	we	PRON
ejde-636	169	14	have	have	AUX
ejde-636	169	15	∥∆u∥22	∥∆u∥22	VERB
ejde-636	169	16	+	+	PUNCT
ejde-636	169	17	∥∇u∥22	∥∇u∥22	X
ejde-636	169	18	=	=	SYM
ejde-636	169	19	ρ0	ρ0	PROPN
ejde-636	169	20	.	.	PUNCT
ejde-636	170	1	applying	apply	VERB
ejde-636	170	2	the	the	DET
ejde-636	170	3	gagliardo	gagliardo	NOUN
ejde-636	170	4	-	-	PUNCT
ejde-636	170	5	nirenberg	nirenberg	PROPN
ejde-636	170	6	inequality	inequality	NOUN
ejde-636	170	7	leads	lead	VERB
ejde-636	170	8	to	to	ADP
ejde-636	170	9	ep	ep	PROPN
ejde-636	170	10	,	,	PUNCT
ejde-636	170	11	q(u	q(u	ADJ
ejde-636	170	12	)	)	PUNCT
ejde-636	170	13	≥	≥	NOUN
ejde-636	170	14	1	1	NUM
ejde-636	170	15	2	2	NUM
ejde-636	170	16	(	(	PUNCT
ejde-636	170	17	∥∆u∥22	∥∆u∥22	X
ejde-636	170	18	+	+	NUM
ejde-636	170	19	∥∇u∥22)−	∥∇u∥22)−	NUM
ejde-636	170	20	µ	µ	PRON
ejde-636	170	21	q	q	X
ejde-636	170	22	cq	cq	PROPN
ejde-636	170	23	n	n	CCONJ
ejde-636	170	24	,	,	PUNCT
ejde-636	170	25	q(∥∆u∥	q(∥∆u∥	PROPN
ejde-636	170	26	2	2	NUM
ejde-636	170	27	2	2	NUM
ejde-636	170	28	+	+	CCONJ
ejde-636	170	29	∥∇u∥22)α0	∥∇u∥22)α0	NUM
ejde-636	170	30	+	+	NOUN
ejde-636	170	31	1(∥u∥22)α1	1(∥u∥22)α1	PROPN
ejde-636	170	32	−	−	PROPN
ejde-636	170	33	cp	cp	NUM
ejde-636	170	34	n	n	PROPN
ejde-636	170	35	,	,	PUNCT
ejde-636	170	36	p	p	PROPN
ejde-636	170	37	p	p	X
ejde-636	170	38	(	(	PUNCT
ejde-636	170	39	∥∆u∥22	∥∆u∥22	X
ejde-636	170	40	+	+	X
ejde-636	171	1	∥∇u∥22)α2	∥∇u∥22)α2	NUM
ejde-636	171	2	+	+	NUM
ejde-636	171	3	1(∥u∥22)α3	1(∥u∥22)α3	NUM
ejde-636	171	4	=	=	SYM
ejde-636	171	5	(	(	PUNCT
ejde-636	171	6	∥∆u∥22	∥∆u∥22	X
ejde-636	171	7	+	+	CCONJ
ejde-636	171	8	∥∇u∥22)f(∥u∥22	∥∇u∥22)f(∥u∥22	ADV
ejde-636	171	9	,	,	PUNCT
ejde-636	171	10	∥∆u∥22	∥∆u∥22	X
ejde-636	171	11	+	+	CCONJ
ejde-636	171	12	∥∇u∥22	∥∇u∥22	NUM
ejde-636	171	13	)	)	PUNCT
ejde-636	171	14	=	=	SYM
ejde-636	171	15	ρ0f(c	ρ0f(c	PROPN
ejde-636	171	16	,	,	PUNCT
ejde-636	171	17	ρ0	ρ0	PROPN
ejde-636	171	18	)	)	PUNCT
ejde-636	171	19	>	>	X
ejde-636	172	1	ρ0f(c0	ρ0f(c0	PROPN
ejde-636	172	2	,	,	PUNCT
ejde-636	172	3	ρ0	ρ0	PROPN
ejde-636	172	4	)	)	PUNCT
ejde-636	172	5	=	=	SYM
ejde-636	172	6	0	0	X
ejde-636	172	7	.	.	PUNCT
ejde-636	172	8	(	(	PUNCT
ejde-636	172	9	3.1	3.1	NUM
ejde-636	172	10	)	)	PUNCT
ejde-636	172	11	let	let	VERB
ejde-636	172	12	u	u	PRON
ejde-636	172	13	∈	∈	PROPN
ejde-636	172	14	s(c	s(c	PROPN
ejde-636	172	15	)	)	PUNCT
ejde-636	172	16	be	be	AUX
ejde-636	172	17	arbitrary	arbitrary	ADJ
ejde-636	172	18	but	but	CCONJ
ejde-636	172	19	fixed	fix	VERB
ejde-636	172	20	.	.	PUNCT
ejde-636	173	1	for	for	ADP
ejde-636	173	2	s	s	PROPN
ejde-636	173	3	∈	∈	PROPN
ejde-636	173	4	r+	r+	X
ejde-636	173	5	,	,	PUNCT
ejde-636	173	6	set	set	VERB
ejde-636	173	7	us(x	us(x	NOUN
ejde-636	173	8	)	)	PUNCT
ejde-636	173	9	=	=	SYM
ejde-636	173	10	sn/2u(sx	sn/2u(sx	PROPN
ejde-636	173	11	)	)	PUNCT
ejde-636	173	12	.	.	PUNCT
ejde-636	174	1	clearly	clearly	ADV
ejde-636	174	2	,	,	PUNCT
ejde-636	174	3	us	us	PROPN
ejde-636	174	4	∈	∈	PROPN
ejde-636	174	5	s(c	s(c	PROPN
ejde-636	174	6	)	)	PUNCT
ejde-636	174	7	for	for	ADP
ejde-636	174	8	any	any	DET
ejde-636	174	9	s	s	PROPN
ejde-636	174	10	∈	∈	NOUN
ejde-636	174	11	r+	r+	NOUN
ejde-636	174	12	.	.	PUNCT
ejde-636	175	1	we	we	PRON
ejde-636	175	2	define	define	VERB
ejde-636	175	3	ψu(s	ψu(s	PUNCT
ejde-636	175	4	)	)	PUNCT
ejde-636	175	5	=	=	SYM
ejde-636	175	6	ep	ep	PROPN
ejde-636	175	7	,	,	PUNCT
ejde-636	175	8	q(us	q(us	PROPN
ejde-636	175	9	)	)	PUNCT
ejde-636	175	10	=	=	SYM
ejde-636	175	11	s4	s4	PROPN
ejde-636	175	12	2	2	NUM
ejde-636	175	13	∥∆u∥22	∥∆u∥22	PUNCT
ejde-636	175	14	+	+	CCONJ
ejde-636	175	15	s2	s2	VERB
ejde-636	175	16	2	2	NUM
ejde-636	175	17	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	175	18	−	−	PROPN
ejde-636	175	19	µ	µ	PRON
ejde-636	175	20	q	q	NOUN
ejde-636	175	21	sn(q−2)/2∥u∥qq	sn(q−2)/2∥u∥qq	NOUN
ejde-636	175	22	−	−	NOUN
ejde-636	175	23	1	1	NUM
ejde-636	175	24	p	p	NOUN
ejde-636	175	25	sn(p−2)/2∥u∥pp	sn(p−2)/2∥u∥pp	NOUN
ejde-636	175	26	,	,	PUNCT
ejde-636	175	27	for	for	ADP
ejde-636	175	28	all	all	PRON
ejde-636	175	29	s	s	VERB
ejde-636	175	30	>	>	X
ejde-636	175	31	0	0	NUM
ejde-636	175	32	.	.	PUNCT
ejde-636	176	1	it	it	PRON
ejde-636	176	2	is	be	AUX
ejde-636	176	3	easily	easily	ADV
ejde-636	176	4	seen	see	VERB
ejde-636	176	5	that	that	PRON
ejde-636	176	6	ψu(s	ψu(s	PUNCT
ejde-636	176	7	)	)	PUNCT
ejde-636	176	8	→	→	SYM
ejde-636	176	9	0−	0−	NUM
ejde-636	176	10	as	as	ADP
ejde-636	176	11	s	s	NOUN
ejde-636	176	12	→	→	SYM
ejde-636	176	13	0	0	NUM
ejde-636	176	14	.	.	PUNCT
ejde-636	177	1	hence	hence	ADV
ejde-636	177	2	,	,	PUNCT
ejde-636	177	3	there	there	PRON
ejde-636	177	4	exists	exist	VERB
ejde-636	177	5	sufficiently	sufficiently	ADV
ejde-636	177	6	small	small	ADJ
ejde-636	177	7	s0	s0	NOUN
ejde-636	177	8	>	>	X
ejde-636	177	9	0	0	NUM
ejde-636	178	1	such	such	ADJ
ejde-636	178	2	that	that	SCONJ
ejde-636	178	3	∥∆us0∥22	∥∆us0∥22	PROPN
ejde-636	178	4	+	+	CCONJ
ejde-636	178	5	∥∇us0∥22	∥∇us0∥22	PROPN
ejde-636	178	6	<	<	X
ejde-636	178	7	ρ0	ρ0	PROPN
ejde-636	178	8	and	and	CCONJ
ejde-636	178	9	ep	ep	PROPN
ejde-636	178	10	,	,	PUNCT
ejde-636	178	11	q(us0	q(us0	PROPN
ejde-636	178	12	)	)	PUNCT
ejde-636	178	13	=	=	SYM
ejde-636	178	14	ψu(s0	ψu(s0	X
ejde-636	178	15	)	)	PUNCT
ejde-636	178	16	<	<	X
ejde-636	178	17	0	0	X
ejde-636	178	18	.	.	PUNCT
ejde-636	179	1	consequently	consequently	ADV
ejde-636	179	2	,	,	PUNCT
ejde-636	179	3	we	we	PRON
ejde-636	179	4	have	have	VERB
ejde-636	179	5	mp	mp	NOUN
ejde-636	179	6	,	,	PUNCT
ejde-636	179	7	q(c	q(c	PROPN
ejde-636	179	8	)	)	PUNCT
ejde-636	179	9	<	<	X
ejde-636	179	10	0	0	X
ejde-636	179	11	.	.	PUNCT
ejde-636	180	1	(	(	PUNCT
ejde-636	180	2	2	2	X
ejde-636	180	3	)	)	PUNCT
ejde-636	180	4	let	let	VERB
ejde-636	180	5	c	c	NOUN
ejde-636	180	6	∈	∈	PROPN
ejde-636	180	7	(	(	PUNCT
ejde-636	180	8	0	0	NUM
ejde-636	180	9	,	,	PUNCT
ejde-636	180	10	c0	c0	NOUN
ejde-636	180	11	)	)	PUNCT
ejde-636	180	12	be	be	AUX
ejde-636	180	13	arbitrary	arbitrary	ADJ
ejde-636	180	14	and	and	CCONJ
ejde-636	180	15	{	{	PUNCT
ejde-636	180	16	cn	cn	PROPN
ejde-636	180	17	}	}	PUNCT
ejde-636	180	18	⊂	⊂	PROPN
ejde-636	180	19	(	(	PUNCT
ejde-636	180	20	0	0	NUM
ejde-636	180	21	,	,	PUNCT
ejde-636	180	22	c0	c0	NOUN
ejde-636	180	23	)	)	PUNCT
ejde-636	180	24	be	be	VERB
ejde-636	180	25	such	such	ADJ
ejde-636	180	26	that	that	SCONJ
ejde-636	180	27	cn	cn	PROPN
ejde-636	180	28	→	→	PROPN
ejde-636	180	29	c.	c.	PROPN
ejde-636	180	30	by	by	ADP
ejde-636	180	31	the	the	DET
ejde-636	180	32	definition	definition	NOUN
ejde-636	180	33	of	of	ADP
ejde-636	180	34	mp	mp	PROPN
ejde-636	180	35	,	,	PUNCT
ejde-636	180	36	q(cn	q(cn	PROPN
ejde-636	180	37	)	)	PUNCT
ejde-636	180	38	with	with	ADP
ejde-636	180	39	mp	mp	PROPN
ejde-636	180	40	,	,	PUNCT
ejde-636	180	41	q(cn	q(cn	PROPN
ejde-636	180	42	)	)	PUNCT
ejde-636	180	43	<	<	X
ejde-636	180	44	0	0	NUM
ejde-636	180	45	,	,	PUNCT
ejde-636	180	46	for	for	ADP
ejde-636	180	47	any	any	DET
ejde-636	180	48	ϵ	ϵ	PROPN
ejde-636	180	49	>	>	X
ejde-636	180	50	0	0	PUNCT
ejde-636	180	51	small	small	ADJ
ejde-636	180	52	enough	enough	ADV
ejde-636	180	53	,	,	PUNCT
ejde-636	180	54	there	there	PRON
ejde-636	180	55	exists	exist	VERB
ejde-636	180	56	un	un	PROPN
ejde-636	180	57	∈	∈	PROPN
ejde-636	180	58	v	v	PROPN
ejde-636	180	59	(	(	PUNCT
ejde-636	180	60	c	c	NOUN
ejde-636	180	61	)	)	PUNCT
ejde-636	180	62	such	such	ADJ
ejde-636	180	63	that	that	SCONJ
ejde-636	180	64	ep	ep	PROPN
ejde-636	180	65	,	,	PUNCT
ejde-636	180	66	q(un	q(un	PROPN
ejde-636	180	67	)	)	PUNCT
ejde-636	180	68	≤	≤	PROPN
ejde-636	180	69	mp	mp	PROPN
ejde-636	180	70	,	,	PUNCT
ejde-636	180	71	q(cn	q(cn	PROPN
ejde-636	180	72	)	)	PUNCT
ejde-636	180	73	+	+	CCONJ
ejde-636	180	74	ϵ	ϵ	PROPN
ejde-636	180	75	and	and	CCONJ
ejde-636	180	76	ep	ep	PROPN
ejde-636	180	77	,	,	PUNCT
ejde-636	180	78	q(un	q(un	PROPN
ejde-636	180	79	)	)	PUNCT
ejde-636	180	80	<	<	X
ejde-636	180	81	0	0	NUM
ejde-636	180	82	.	.	PUNCT
ejde-636	181	1	(	(	PUNCT
ejde-636	181	2	3.2	3.2	NUM
ejde-636	181	3	)	)	PUNCT
ejde-636	181	4	let	let	VERB
ejde-636	181	5	zn	zn	NOUN
ejde-636	181	6	=	=	PUNCT
ejde-636	181	7	√	√	PROPN
ejde-636	181	8	c	c	PROPN
ejde-636	181	9	cn	cn	PROPN
ejde-636	181	10	un	un	PROPN
ejde-636	181	11	.	.	PROPN
ejde-636	182	1	clearly	clearly	ADV
ejde-636	182	2	,	,	PUNCT
ejde-636	182	3	zn	zn	PROPN
ejde-636	182	4	∈	∈	PROPN
ejde-636	182	5	s(c	s(c	PROPN
ejde-636	182	6	)	)	PUNCT
ejde-636	182	7	.	.	PUNCT
ejde-636	183	1	on	on	ADP
ejde-636	183	2	the	the	DET
ejde-636	183	3	one	one	NUM
ejde-636	183	4	hand	hand	NOUN
ejde-636	183	5	,	,	PUNCT
ejde-636	183	6	if	if	SCONJ
ejde-636	183	7	cn	cn	PROPN
ejde-636	183	8	≥	≥	AUX
ejde-636	183	9	c	c	NOUN
ejde-636	183	10	,	,	PUNCT
ejde-636	183	11	then	then	ADV
ejde-636	183	12	∥∆zn∥22	∥∆zn∥22	PROPN
ejde-636	183	13	+	+	CCONJ
ejde-636	183	14	∥∇zn∥22	∥∇zn∥22	PROPN
ejde-636	183	15	=	=	SYM
ejde-636	183	16	c	c	X
ejde-636	183	17	cn	cn	X
ejde-636	183	18	(	(	PUNCT
ejde-636	183	19	∥∆un∥22	∥∆un∥22	PROPN
ejde-636	183	20	+	+	CCONJ
ejde-636	183	21	∥∇un∥22	∥∇un∥22	NUM
ejde-636	183	22	)	)	PUNCT
ejde-636	183	23	<	<	X
ejde-636	183	24	ρ0	ρ0	PROPN
ejde-636	183	25	.	.	PUNCT
ejde-636	183	26	on	on	ADP
ejde-636	183	27	the	the	DET
ejde-636	183	28	other	other	ADJ
ejde-636	183	29	hand	hand	NOUN
ejde-636	183	30	,	,	PUNCT
ejde-636	183	31	if	if	SCONJ
ejde-636	183	32	cn	cn	PROPN
ejde-636	183	33	<	<	X
ejde-636	183	34	c	c	X
ejde-636	183	35	,	,	PUNCT
ejde-636	183	36	by	by	ADP
ejde-636	183	37	lemma	lemma	PROPN
ejde-636	183	38	2.6	2.6	NUM
ejde-636	183	39	and	and	CCONJ
ejde-636	183	40	f(cn	f(cn	PROPN
ejde-636	183	41	,	,	PUNCT
ejde-636	183	42	ρ0	ρ0	PROPN
ejde-636	183	43	)	)	PUNCT
ejde-636	183	44	≥	≥	NOUN
ejde-636	183	45	f(c0	f(c0	NOUN
ejde-636	183	46	,	,	PUNCT
ejde-636	183	47	ρ0	ρ0	PROPN
ejde-636	183	48	)	)	PUNCT
ejde-636	183	49	=	=	SYM
ejde-636	183	50	0	0	NUM
ejde-636	183	51	,	,	PUNCT
ejde-636	183	52	we	we	PRON
ejde-636	183	53	have	have	VERB
ejde-636	183	54	f(cn	f(cn	PROPN
ejde-636	183	55	,	,	PUNCT
ejde-636	183	56	ρ	ρ	PROPN
ejde-636	183	57	)	)	PUNCT
ejde-636	183	58	≥	≥	NOUN
ejde-636	183	59	0	0	NUM
ejde-636	183	60	for	for	ADP
ejde-636	183	61	any	any	DET
ejde-636	183	62	ρ	ρ	PROPN
ejde-636	183	63	∈	∈	PROPN
ejde-636	183	64	[	[	PUNCT
ejde-636	183	65	cnc	cnc	PROPN
ejde-636	183	66	ρ0	ρ0	PROPN
ejde-636	183	67	,	,	PUNCT
ejde-636	183	68	ρ0	ρ0	PROPN
ejde-636	183	69	]	]	PUNCT
ejde-636	183	70	.	.	PUNCT
ejde-636	184	1	however	however	ADV
ejde-636	184	2	,	,	PUNCT
ejde-636	184	3	from	from	ADP
ejde-636	184	4	(	(	PUNCT
ejde-636	184	5	3.1	3.1	NUM
ejde-636	184	6	)	)	PUNCT
ejde-636	184	7	and	and	CCONJ
ejde-636	184	8	(	(	PUNCT
ejde-636	184	9	3.2	3.2	NUM
ejde-636	184	10	)	)	PUNCT
ejde-636	184	11	it	it	PRON
ejde-636	184	12	follows	follow	VERB
ejde-636	184	13	that	that	SCONJ
ejde-636	184	14	f(∥un∥22	f(∥un∥22	NOUN
ejde-636	184	15	,	,	PUNCT
ejde-636	184	16	∥∆un∥22	∥∆un∥22	PUNCT
ejde-636	184	17	+	+	CCONJ
ejde-636	184	18	∥∇un∥22	∥∇un∥22	NUM
ejde-636	184	19	)	)	PUNCT
ejde-636	184	20	<	<	X
ejde-636	184	21	0	0	X
ejde-636	184	22	.	.	PUNCT
ejde-636	185	1	hence	hence	ADV
ejde-636	185	2	,	,	PUNCT
ejde-636	185	3	∥∆un∥22	∥∆un∥22	PUNCT
ejde-636	186	1	+	+	CCONJ
ejde-636	186	2	∥∇un∥22	∥∇un∥22	PROPN
ejde-636	186	3	<	<	X
ejde-636	186	4	cn	cn	X
ejde-636	186	5	c	c	PROPN
ejde-636	186	6	ρ0	ρ0	PROPN
ejde-636	186	7	and	and	CCONJ
ejde-636	186	8	∥∆zn∥22	∥∆zn∥22	PROPN
ejde-636	186	9	+	+	CCONJ
ejde-636	186	10	∥∇zn∥22	∥∇zn∥22	X
ejde-636	186	11	<	<	X
ejde-636	186	12	c	c	X
ejde-636	186	13	cn	cn	PROPN
ejde-636	186	14	·	·	PUNCT
ejde-636	186	15	cn	cn	PROPN
ejde-636	187	1	c	c	NOUN
ejde-636	187	2	ρ0	ρ0	PROPN
ejde-636	187	3	=	=	SYM
ejde-636	187	4	ρ0	ρ0	PROPN
ejde-636	187	5	.	.	PUNCT
ejde-636	188	1	since	since	SCONJ
ejde-636	188	2	zn	zn	PROPN
ejde-636	188	3	∈	∈	PROPN
ejde-636	188	4	v	v	PROPN
ejde-636	188	5	(	(	PUNCT
ejde-636	188	6	c	c	NOUN
ejde-636	188	7	)	)	PUNCT
ejde-636	188	8	,	,	PUNCT
ejde-636	188	9	we	we	PRON
ejde-636	188	10	have	have	VERB
ejde-636	188	11	mp	mp	NOUN
ejde-636	188	12	,	,	PUNCT
ejde-636	188	13	q(c	q(c	PROPN
ejde-636	188	14	)	)	PUNCT
ejde-636	188	15	≤	≤	NOUN
ejde-636	188	16	ep	ep	PROPN
ejde-636	188	17	,	,	PUNCT
ejde-636	188	18	q(zn	q(zn	PROPN
ejde-636	188	19	)	)	PUNCT
ejde-636	188	20	8	8	NUM
ejde-636	188	21	z.	z.	PROPN
ejde-636	188	22	ma	ma	PROPN
ejde-636	188	23	,	,	PUNCT
ejde-636	188	24	x.	x.	PROPN
ejde-636	188	25	chang	chang	PROPN
ejde-636	188	26	,	,	PUNCT
ejde-636	188	27	z.	z.	PROPN
ejde-636	188	28	feng	feng	PROPN
ejde-636	188	29	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	188	30	=	=	SYM
ejde-636	188	31	ep	ep	PROPN
ejde-636	188	32	,	,	PUNCT
ejde-636	188	33	q(un	q(un	PROPN
ejde-636	188	34	)	)	PUNCT
ejde-636	189	1	+	+	CCONJ
ejde-636	189	2	(	(	PUNCT
ejde-636	189	3	ep	ep	PROPN
ejde-636	189	4	,	,	PUNCT
ejde-636	189	5	q(zn)−	q(zn)−	PRON
ejde-636	189	6	ep	ep	PROPN
ejde-636	189	7	,	,	PUNCT
ejde-636	189	8	q(un	q(un	PROPN
ejde-636	189	9	)	)	PUNCT
ejde-636	189	10	)	)	PUNCT
ejde-636	190	1	=	=	SYM
ejde-636	190	2	ep	ep	PROPN
ejde-636	190	3	,	,	PUNCT
ejde-636	190	4	q(un	q(un	PROPN
ejde-636	190	5	)	)	PUNCT
ejde-636	190	6	+	+	CCONJ
ejde-636	190	7	1	1	NUM
ejde-636	190	8	2	2	NUM
ejde-636	190	9	(	(	PUNCT
ejde-636	190	10	c	c	PROPN
ejde-636	190	11	cn	cn	PROPN
ejde-636	190	12	−	−	PROPN
ejde-636	190	13	1)∥∆un∥22	1)∥∆un∥22	NUM
ejde-636	191	1	+	+	CCONJ
ejde-636	191	2	1	1	NUM
ejde-636	191	3	2	2	NUM
ejde-636	191	4	(	(	PUNCT
ejde-636	191	5	c	c	AUX
ejde-636	191	6	cn	cn	PROPN
ejde-636	191	7	−	−	PROPN
ejde-636	192	1	1)∥∇un∥22	1)∥∇un∥22	NUM
ejde-636	192	2	−	−	PROPN
ejde-636	192	3	µ	µ	X
ejde-636	192	4	q	q	X
ejde-636	193	1	[	[	X
ejde-636	193	2	(	(	PUNCT
ejde-636	193	3	c	c	X
ejde-636	193	4	cn	cn	PROPN
ejde-636	193	5	)	)	PUNCT
ejde-636	193	6	q	q	PROPN
ejde-636	193	7	2	2	NUM
ejde-636	193	8	−	−	NOUN
ejde-636	193	9	1]∥un∥qq	1]∥un∥qq	NUM
ejde-636	193	10	−	−	PROPN
ejde-636	193	11	1	1	NUM
ejde-636	193	12	p	p	X
ejde-636	194	1	[	[	X
ejde-636	194	2	(	(	PUNCT
ejde-636	194	3	c	c	PROPN
ejde-636	194	4	cn	cn	PROPN
ejde-636	194	5	)	)	PUNCT
ejde-636	194	6	p/2	p/2	NOUN
ejde-636	194	7	−	−	PROPN
ejde-636	194	8	1]∥un∥pp	1]∥un∥pp	NUM
ejde-636	194	9	.	.	PUNCT
ejde-636	195	1	that	that	PRON
ejde-636	195	2	is	be	AUX
ejde-636	195	3	,	,	PUNCT
ejde-636	195	4	mp	mp	PROPN
ejde-636	195	5	,	,	PUNCT
ejde-636	195	6	q(c	q(c	PROPN
ejde-636	195	7	)	)	PUNCT
ejde-636	195	8	≤	≤	NOUN
ejde-636	195	9	ep	ep	PROPN
ejde-636	195	10	,	,	PUNCT
ejde-636	195	11	q(zn	q(zn	PRON
ejde-636	195	12	)	)	PUNCT
ejde-636	196	1	=	=	SYM
ejde-636	196	2	ep	ep	PROPN
ejde-636	196	3	,	,	PUNCT
ejde-636	196	4	q(un	q(un	PROPN
ejde-636	196	5	)	)	PUNCT
ejde-636	196	6	+	+	CCONJ
ejde-636	196	7	on(1	on(1	NOUN
ejde-636	196	8	)	)	PUNCT
ejde-636	196	9	as	as	ADP
ejde-636	196	10	n→	n→	PROPN
ejde-636	196	11	∞.	∞.	PROPN
ejde-636	196	12	(	(	PUNCT
ejde-636	196	13	3.3	3.3	NUM
ejde-636	196	14	)	)	PUNCT
ejde-636	196	15	using	use	VERB
ejde-636	196	16	(	(	PUNCT
ejde-636	196	17	3.2	3.2	NUM
ejde-636	196	18	)	)	PUNCT
ejde-636	196	19	and	and	CCONJ
ejde-636	196	20	(	(	PUNCT
ejde-636	196	21	3.3	3.3	NUM
ejde-636	196	22	)	)	PUNCT
ejde-636	196	23	yields	yield	NOUN
ejde-636	196	24	mp	mp	PROPN
ejde-636	196	25	,	,	PUNCT
ejde-636	196	26	q(c	q(c	PROPN
ejde-636	196	27	)	)	PUNCT
ejde-636	196	28	≤	≤	PROPN
ejde-636	196	29	mp	mp	PROPN
ejde-636	196	30	,	,	PUNCT
ejde-636	196	31	q(cn	q(cn	PROPN
ejde-636	196	32	)	)	PUNCT
ejde-636	197	1	+	+	CCONJ
ejde-636	197	2	ϵ+	ϵ+	PUNCT
ejde-636	197	3	on(1	on(1	NOUN
ejde-636	197	4	)	)	PUNCT
ejde-636	197	5	.	.	PUNCT
ejde-636	198	1	now	now	ADV
ejde-636	198	2	,	,	PUNCT
ejde-636	198	3	we	we	PRON
ejde-636	198	4	let	let	VERB
ejde-636	198	5	u	u	PRON
ejde-636	198	6	∈	∈	PROPN
ejde-636	198	7	v	v	X
ejde-636	198	8	(	(	PUNCT
ejde-636	198	9	c	c	NOUN
ejde-636	198	10	)	)	PUNCT
ejde-636	198	11	be	be	AUX
ejde-636	198	12	such	such	ADJ
ejde-636	198	13	that	that	SCONJ
ejde-636	198	14	ep	ep	NOUN
ejde-636	198	15	,	,	PUNCT
ejde-636	198	16	q(u	q(u	ADJ
ejde-636	198	17	)	)	PUNCT
ejde-636	198	18	≤	≤	NOUN
ejde-636	198	19	mp	mp	PROPN
ejde-636	198	20	,	,	PUNCT
ejde-636	198	21	q(c	q(c	PROPN
ejde-636	198	22	)	)	PUNCT
ejde-636	198	23	+	+	CCONJ
ejde-636	198	24	ϵ	ϵ	X
ejde-636	198	25	and	and	CCONJ
ejde-636	198	26	ep	ep	PROPN
ejde-636	198	27	,	,	PUNCT
ejde-636	198	28	q(u	q(u	ADJ
ejde-636	198	29	)	)	PUNCT
ejde-636	198	30	<	<	X
ejde-636	198	31	0	0	X
ejde-636	198	32	.	.	PUNCT
ejde-636	199	1	set	set	PROPN
ejde-636	199	2	un	un	PROPN
ejde-636	200	1	:	:	PUNCT
ejde-636	200	2	=	=	SYM
ejde-636	200	3	√	√	ADP
ejde-636	201	1	cn	cn	INTJ
ejde-636	201	2	c	c	AUX
ejde-636	201	3	u.	u.	PROPN
ejde-636	201	4	then	then	ADV
ejde-636	201	5	un	un	PROPN
ejde-636	201	6	∈	∈	PROPN
ejde-636	201	7	s(cn	s(cn	PROPN
ejde-636	201	8	)	)	PUNCT
ejde-636	201	9	,	,	PUNCT
ejde-636	201	10	and	and	CCONJ
ejde-636	201	11	cn	cn	PROPN
ejde-636	201	12	→	→	SYM
ejde-636	201	13	c	c	PROPN
ejde-636	201	14	implies	imply	VERB
ejde-636	201	15	that	that	SCONJ
ejde-636	201	16	∥∆un∥22+∥∇un∥22	∥∆un∥22+∥∇un∥22	PROPN
ejde-636	201	17	<	<	X
ejde-636	201	18	ρ0	ρ0	PROPN
ejde-636	201	19	for	for	ADP
ejde-636	201	20	n	n	X
ejde-636	201	21	large	large	ADJ
ejde-636	201	22	enough	enough	ADV
ejde-636	201	23	.	.	PUNCT
ejde-636	202	1	so	so	ADV
ejde-636	202	2	un	un	PROPN
ejde-636	202	3	∈	∈	PROPN
ejde-636	202	4	v	v	PROPN
ejde-636	202	5	(	(	PUNCT
ejde-636	202	6	cn	cn	PROPN
ejde-636	202	7	)	)	PUNCT
ejde-636	202	8	.	.	PUNCT
ejde-636	203	1	note	note	VERB
ejde-636	203	2	that	that	SCONJ
ejde-636	203	3	ep	ep	PROPN
ejde-636	203	4	,	,	PUNCT
ejde-636	203	5	q(un	q(un	PROPN
ejde-636	203	6	)	)	PUNCT
ejde-636	203	7	→	→	SYM
ejde-636	203	8	ep	ep	PROPN
ejde-636	203	9	,	,	PUNCT
ejde-636	203	10	q(u	q(u	NOUN
ejde-636	203	11	)	)	PUNCT
ejde-636	203	12	.	.	PUNCT
ejde-636	204	1	thus	thus	ADV
ejde-636	204	2	,	,	PUNCT
ejde-636	204	3	we	we	PRON
ejde-636	204	4	obtain	obtain	VERB
ejde-636	204	5	mp	mp	PROPN
ejde-636	204	6	,	,	PUNCT
ejde-636	204	7	q(cn	q(cn	PROPN
ejde-636	204	8	)	)	PUNCT
ejde-636	204	9	≤	≤	NOUN
ejde-636	204	10	ep	ep	PROPN
ejde-636	204	11	,	,	PUNCT
ejde-636	204	12	q(u	q(u	PROPN
ejde-636	204	13	)	)	PUNCT
ejde-636	205	1	+	+	CCONJ
ejde-636	205	2	(	(	PUNCT
ejde-636	205	3	ep	ep	PROPN
ejde-636	205	4	,	,	PUNCT
ejde-636	205	5	q(un)−	q(un)−	PROPN
ejde-636	205	6	ep	ep	PROPN
ejde-636	205	7	,	,	PUNCT
ejde-636	205	8	q(u	q(u	ADJ
ejde-636	205	9	)	)	PUNCT
ejde-636	205	10	)	)	PUNCT
ejde-636	205	11	≤	≤	PROPN
ejde-636	205	12	mp	mp	PROPN
ejde-636	205	13	,	,	PUNCT
ejde-636	205	14	q(c	q(c	PROPN
ejde-636	205	15	)	)	PUNCT
ejde-636	205	16	+	+	CCONJ
ejde-636	205	17	ϵ+	ϵ+	PUNCT
ejde-636	205	18	on(1	on(1	NOUN
ejde-636	205	19	)	)	PUNCT
ejde-636	205	20	.	.	PUNCT
ejde-636	206	1	because	because	SCONJ
ejde-636	206	2	of	of	ADP
ejde-636	206	3	the	the	DET
ejde-636	206	4	arbitrariness	arbitrariness	NOUN
ejde-636	206	5	of	of	ADP
ejde-636	206	6	ϵ	ϵ	PROPN
ejde-636	206	7	>	>	X
ejde-636	206	8	0	0	NUM
ejde-636	206	9	,	,	PUNCT
ejde-636	206	10	we	we	PRON
ejde-636	206	11	infer	infer	VERB
ejde-636	206	12	that	that	SCONJ
ejde-636	206	13	mp	mp	PROPN
ejde-636	206	14	,	,	PUNCT
ejde-636	206	15	q(cn	q(cn	PROPN
ejde-636	206	16	)	)	PUNCT
ejde-636	206	17	→	→	SYM
ejde-636	206	18	mp	mp	PROPN
ejde-636	206	19	,	,	PUNCT
ejde-636	206	20	q(c	q(c	PROPN
ejde-636	206	21	)	)	PUNCT
ejde-636	206	22	.	.	PUNCT
ejde-636	207	1	(	(	PUNCT
ejde-636	207	2	3	3	X
ejde-636	207	3	)	)	PUNCT
ejde-636	207	4	given	give	VERB
ejde-636	207	5	α	α	PRON
ejde-636	207	6	∈	∈	PROPN
ejde-636	207	7	(	(	PUNCT
ejde-636	207	8	0	0	NUM
ejde-636	207	9	,	,	PUNCT
ejde-636	207	10	c	c	NOUN
ejde-636	207	11	)	)	PUNCT
ejde-636	207	12	,	,	PUNCT
ejde-636	207	13	it	it	PRON
ejde-636	207	14	suffices	suffice	VERB
ejde-636	207	15	to	to	PART
ejde-636	207	16	prove	prove	VERB
ejde-636	207	17	that	that	SCONJ
ejde-636	207	18	∀θ	∀θ	PROPN
ejde-636	207	19	∈	∈	PROPN
ejde-636	207	20	(	(	PUNCT
ejde-636	207	21	1	1	NUM
ejde-636	207	22	,	,	PUNCT
ejde-636	207	23	c	c	NOUN
ejde-636	207	24	α	α	NOUN
ejde-636	207	25	]	]	X
ejde-636	207	26	:	:	PUNCT
ejde-636	207	27	mp	mp	NOUN
ejde-636	207	28	,	,	PUNCT
ejde-636	207	29	q(θα	q(θα	NOUN
ejde-636	207	30	)	)	PUNCT
ejde-636	207	31	≤	≤	NOUN
ejde-636	207	32	θmp	θmp	PROPN
ejde-636	207	33	,	,	PUNCT
ejde-636	207	34	q(α	q(α	PROPN
ejde-636	207	35	)	)	PUNCT
ejde-636	207	36	and	and	CCONJ
ejde-636	207	37	that	that	SCONJ
ejde-636	207	38	,	,	PUNCT
ejde-636	207	39	if	if	SCONJ
ejde-636	207	40	mp	mp	PROPN
ejde-636	207	41	,	,	PUNCT
ejde-636	207	42	q(α	q(α	PROPN
ejde-636	207	43	)	)	PUNCT
ejde-636	207	44	is	be	AUX
ejde-636	207	45	attained	attain	VERB
ejde-636	207	46	,	,	PUNCT
ejde-636	207	47	the	the	DET
ejde-636	207	48	inequality	inequality	NOUN
ejde-636	207	49	is	be	AUX
ejde-636	207	50	strict	strict	ADJ
ejde-636	207	51	.	.	PUNCT
ejde-636	208	1	using	use	VERB
ejde-636	208	2	(	(	PUNCT
ejde-636	208	3	i	i	NOUN
ejde-636	208	4	)	)	PUNCT
ejde-636	208	5	,	,	PUNCT
ejde-636	208	6	for	for	ADP
ejde-636	208	7	any	any	DET
ejde-636	208	8	ϵ	ϵ	PROPN
ejde-636	208	9	>	>	X
ejde-636	208	10	0	0	PUNCT
ejde-636	208	11	small	small	ADJ
ejde-636	208	12	enough	enough	ADV
ejde-636	208	13	,	,	PUNCT
ejde-636	208	14	there	there	PRON
ejde-636	208	15	exists	exist	VERB
ejde-636	208	16	u	u	PROPN
ejde-636	208	17	∈	∈	PROPN
ejde-636	208	18	v	v	X
ejde-636	208	19	(	(	PUNCT
ejde-636	208	20	α	α	NOUN
ejde-636	208	21	)	)	PUNCT
ejde-636	208	22	such	such	ADJ
ejde-636	208	23	that	that	SCONJ
ejde-636	208	24	ep	ep	PROPN
ejde-636	208	25	,	,	PUNCT
ejde-636	208	26	q(u	q(u	ADJ
ejde-636	208	27	)	)	PUNCT
ejde-636	208	28	≤	≤	NOUN
ejde-636	208	29	mp	mp	PROPN
ejde-636	208	30	,	,	PUNCT
ejde-636	208	31	q(α	q(α	PROPN
ejde-636	208	32	)	)	PUNCT
ejde-636	209	1	+	+	CCONJ
ejde-636	209	2	ϵ	ϵ	PROPN
ejde-636	209	3	and	and	CCONJ
ejde-636	209	4	ep	ep	PROPN
ejde-636	209	5	,	,	PUNCT
ejde-636	209	6	q(u	q(u	ADJ
ejde-636	209	7	)	)	PUNCT
ejde-636	209	8	<	<	X
ejde-636	209	9	0	0	X
ejde-636	209	10	.	.	PUNCT
ejde-636	210	1	from	from	ADP
ejde-636	210	2	lemma	lemma	PROPN
ejde-636	210	3	2.6	2.6	NUM
ejde-636	210	4	and	and	CCONJ
ejde-636	210	5	f(α	f(α	NOUN
ejde-636	210	6	,	,	PUNCT
ejde-636	210	7	ρ0	ρ0	PROPN
ejde-636	210	8	)	)	PUNCT
ejde-636	210	9	≥	≥	NOUN
ejde-636	210	10	f(c0	f(c0	NOUN
ejde-636	210	11	,	,	PUNCT
ejde-636	210	12	ρ0	ρ0	PROPN
ejde-636	210	13	)	)	PUNCT
ejde-636	210	14	=	=	SYM
ejde-636	210	15	0	0	NUM
ejde-636	210	16	,	,	PUNCT
ejde-636	210	17	it	it	PRON
ejde-636	210	18	follows	follow	VERB
ejde-636	210	19	that	that	SCONJ
ejde-636	210	20	f(α	f(α	NOUN
ejde-636	210	21	,	,	PUNCT
ejde-636	210	22	ρ	ρ	PROPN
ejde-636	210	23	)	)	PUNCT
ejde-636	210	24	≥	≥	NOUN
ejde-636	210	25	0	0	NUM
ejde-636	210	26	for	for	ADP
ejde-636	210	27	any	any	DET
ejde-636	210	28	ρ	ρ	NOUN
ejde-636	210	29	∈	∈	PROPN
ejde-636	210	30	[	[	X
ejde-636	210	31	αc	αc	NOUN
ejde-636	210	32	ρ0	ρ0	PROPN
ejde-636	210	33	,	,	PUNCT
ejde-636	210	34	ρ0	ρ0	PROPN
ejde-636	210	35	]	]	PUNCT
ejde-636	210	36	.	.	PUNCT
ejde-636	211	1	hence	hence	ADV
ejde-636	211	2	,	,	PUNCT
ejde-636	211	3	using	use	VERB
ejde-636	211	4	(	(	PUNCT
ejde-636	211	5	3.1	3.1	NUM
ejde-636	211	6	)	)	PUNCT
ejde-636	211	7	and	and	CCONJ
ejde-636	211	8	(	(	PUNCT
ejde-636	211	9	3.2	3.2	NUM
ejde-636	211	10	)	)	PUNCT
ejde-636	211	11	we	we	PRON
ejde-636	211	12	obtain	obtain	VERB
ejde-636	211	13	f(∥u∥22	f(∥u∥22	NUM
ejde-636	211	14	,	,	PUNCT
ejde-636	211	15	∥∆u∥22	∥∆u∥22	X
ejde-636	211	16	+	+	CCONJ
ejde-636	211	17	∥∇u∥22	∥∇u∥22	NUM
ejde-636	211	18	)	)	PUNCT
ejde-636	211	19	<	<	X
ejde-636	211	20	0	0	X
ejde-636	211	21	.	.	PUNCT
ejde-636	212	1	that	that	PRON
ejde-636	212	2	is	is	ADV
ejde-636	212	3	,	,	PUNCT
ejde-636	212	4	∥∆u∥22	∥∆u∥22	X
ejde-636	213	1	+	+	CCONJ
ejde-636	213	2	∥∇u∥22	∥∇u∥22	X
ejde-636	213	3	<	<	X
ejde-636	213	4	α	α	X
ejde-636	213	5	c	c	PROPN
ejde-636	213	6	ρ0	ρ0	PROPN
ejde-636	213	7	.	.	PUNCT
ejde-636	214	1	set	set	VERB
ejde-636	214	2	v	v	NOUN
ejde-636	214	3	=	=	SYM
ejde-636	214	4	√	√	NUM
ejde-636	214	5	θu	θu	NOUN
ejde-636	214	6	.	.	PUNCT
ejde-636	215	1	then	then	ADV
ejde-636	215	2	∥v∥22	∥v∥22	PROPN
ejde-636	215	3	=	=	SYM
ejde-636	215	4	θα	θα	NOUN
ejde-636	215	5	and	and	CCONJ
ejde-636	215	6	∥∆v∥22	∥∆v∥22	PUNCT
ejde-636	215	7	+	+	CCONJ
ejde-636	215	8	∥∇v∥22	∥∇v∥22	PROPN
ejde-636	215	9	<	<	X
ejde-636	215	10	ρ0	ρ0	PROPN
ejde-636	215	11	.	.	PUNCT
ejde-636	216	1	thus	thus	ADV
ejde-636	216	2	v	v	ADP
ejde-636	216	3	∈	∈	PROPN
ejde-636	216	4	v	v	NOUN
ejde-636	216	5	(	(	PUNCT
ejde-636	216	6	θα	θα	NOUN
ejde-636	216	7	)	)	PUNCT
ejde-636	216	8	.	.	PUNCT
ejde-636	217	1	a	a	DET
ejde-636	217	2	direct	direct	ADJ
ejde-636	217	3	calculation	calculation	NOUN
ejde-636	217	4	yields	yield	NOUN
ejde-636	217	5	mp	mp	PROPN
ejde-636	217	6	,	,	PUNCT
ejde-636	217	7	q(θα	q(θα	NOUN
ejde-636	217	8	)	)	PUNCT
ejde-636	217	9	≤	≤	NOUN
ejde-636	217	10	ep	ep	PROPN
ejde-636	217	11	,	,	PUNCT
ejde-636	217	12	q(v	q(v	PROPN
ejde-636	217	13	)	)	PUNCT
ejde-636	217	14	<	<	X
ejde-636	217	15	1	1	NUM
ejde-636	217	16	2	2	NUM
ejde-636	217	17	θ∥∆u∥22	θ∥∆u∥22	PROPN
ejde-636	217	18	+	+	CCONJ
ejde-636	217	19	1	1	NUM
ejde-636	217	20	2	2	NUM
ejde-636	217	21	θ∥∇u∥22	θ∥∇u∥22	PROPN
ejde-636	217	22	−	−	PROPN
ejde-636	217	23	µ	µ	X
ejde-636	217	24	q	q	NOUN
ejde-636	217	25	θ∥v∥qq	θ∥v∥qq	NOUN
ejde-636	217	26	−	−	PROPN
ejde-636	217	27	1	1	NUM
ejde-636	217	28	p	p	NOUN
ejde-636	217	29	θ∥v∥pp	θ∥v∥pp	PROPN
ejde-636	217	30	=	=	SYM
ejde-636	217	31	θep	θep	PROPN
ejde-636	217	32	,	,	PUNCT
ejde-636	217	33	q(u	q(u	ADJ
ejde-636	217	34	)	)	PUNCT
ejde-636	217	35	≤	≤	NOUN
ejde-636	217	36	θ(mp	θ(mp	NOUN
ejde-636	217	37	,	,	PUNCT
ejde-636	217	38	q(α	q(α	PROPN
ejde-636	217	39	)	)	PUNCT
ejde-636	217	40	+	+	NUM
ejde-636	217	41	ϵ	ϵ	X
ejde-636	217	42	)	)	PUNCT
ejde-636	217	43	.	.	PUNCT
ejde-636	218	1	because	because	SCONJ
ejde-636	218	2	of	of	ADP
ejde-636	218	3	the	the	DET
ejde-636	218	4	arbitrariness	arbitrariness	NOUN
ejde-636	218	5	of	of	ADP
ejde-636	218	6	ϵ	ϵ	NOUN
ejde-636	218	7	,	,	PUNCT
ejde-636	218	8	we	we	PRON
ejde-636	218	9	obtain	obtain	VERB
ejde-636	218	10	mp	mp	NOUN
ejde-636	218	11	,	,	PUNCT
ejde-636	218	12	q(θα	q(θα	NOUN
ejde-636	218	13	)	)	PUNCT
ejde-636	218	14	≤	≤	NOUN
ejde-636	218	15	θmp	θmp	PROPN
ejde-636	218	16	,	,	PUNCT
ejde-636	218	17	q(α	q(α	PROPN
ejde-636	218	18	)	)	PUNCT
ejde-636	218	19	.	.	PUNCT
ejde-636	219	1	if	if	SCONJ
ejde-636	219	2	mp	mp	PROPN
ejde-636	219	3	,	,	PUNCT
ejde-636	219	4	q(α	q(α	PROPN
ejde-636	219	5	)	)	PUNCT
ejde-636	219	6	is	be	AUX
ejde-636	219	7	attained	attain	VERB
ejde-636	219	8	,	,	PUNCT
ejde-636	219	9	we	we	PRON
ejde-636	219	10	can	can	AUX
ejde-636	219	11	choose	choose	VERB
ejde-636	219	12	ϵ	ϵ	X
ejde-636	219	13	=	=	SYM
ejde-636	219	14	0	0	NUM
ejde-636	219	15	.	.	PUNCT
ejde-636	220	1	□	□	PUNCT
ejde-636	220	2	3.2	3.2	NUM
ejde-636	220	3	.	.	PUNCT
ejde-636	221	1	proof	proof	NOUN
ejde-636	221	2	of	of	ADP
ejde-636	221	3	theorem	theorem	ADJ
ejde-636	221	4	1.2	1.2	NUM
ejde-636	221	5	.	.	PUNCT
ejde-636	222	1	we	we	PRON
ejde-636	222	2	define	define	VERB
ejde-636	222	3	mc	mc	PROPN
ejde-636	222	4	=	=	PUNCT
ejde-636	222	5	{	{	PUNCT
ejde-636	222	6	u	u	NOUN
ejde-636	222	7	∈	∈	PROPN
ejde-636	222	8	v	v	NOUN
ejde-636	222	9	(	(	PUNCT
ejde-636	222	10	c	c	NOUN
ejde-636	222	11	)	)	PUNCT
ejde-636	222	12	:	:	PUNCT
ejde-636	222	13	ep	ep	PROPN
ejde-636	222	14	,	,	PUNCT
ejde-636	222	15	q(u	q(u	ADJ
ejde-636	222	16	)	)	PUNCT
ejde-636	222	17	=	=	SYM
ejde-636	222	18	mp	mp	PROPN
ejde-636	222	19	,	,	PUNCT
ejde-636	222	20	q(c	q(c	PROPN
ejde-636	222	21	)	)	PUNCT
ejde-636	222	22	}	}	PUNCT
ejde-636	222	23	.	.	PUNCT
ejde-636	223	1	lemma	lemma	PROPN
ejde-636	223	2	3.2	3.2	NUM
ejde-636	223	3	.	.	PUNCT
ejde-636	224	1	let	let	VERB
ejde-636	224	2	2	2	NUM
ejde-636	224	3	<	<	X
ejde-636	224	4	q	q	X
ejde-636	224	5	<	<	X
ejde-636	224	6	2	2	NUM
ejde-636	224	7	+	+	SYM
ejde-636	224	8	4	4	NUM
ejde-636	224	9	n	n	NOUN
ejde-636	224	10	<	<	X
ejde-636	224	11	p	p	X
ejde-636	224	12	<	<	X
ejde-636	224	13	p	p	X
ejde-636	224	14	≤	≤	NUM
ejde-636	224	15	4∗.	4∗.	NUM
ejde-636	224	16	for	for	ADP
ejde-636	224	17	any	any	DET
ejde-636	224	18	c	c	PROPN
ejde-636	224	19	∈	∈	PROPN
ejde-636	224	20	(	(	PUNCT
ejde-636	224	21	0	0	NUM
ejde-636	224	22	,	,	PUNCT
ejde-636	224	23	c0	c0	NOUN
ejde-636	224	24	)	)	PUNCT
ejde-636	224	25	and	and	CCONJ
ejde-636	224	26	the	the	DET
ejde-636	224	27	sequence	sequence	NOUN
ejde-636	224	28	{	{	PUNCT
ejde-636	224	29	un	un	PROPN
ejde-636	224	30	}	}	PUNCT
ejde-636	224	31	⊂	⊂	PRON
ejde-636	224	32	bρ0	bρ0	VERB
ejde-636	224	33	such	such	DET
ejde-636	224	34	that	that	PRON
ejde-636	224	35	∥un∥2	∥un∥2	NUM
ejde-636	224	36	→	→	SYM
ejde-636	224	37	c	c	PROPN
ejde-636	224	38	and	and	CCONJ
ejde-636	224	39	ep	ep	PROPN
ejde-636	224	40	,	,	PUNCT
ejde-636	224	41	q(un	q(un	PROPN
ejde-636	224	42	)	)	PUNCT
ejde-636	224	43	→	→	SYM
ejde-636	224	44	mp	mp	PROPN
ejde-636	224	45	,	,	PUNCT
ejde-636	224	46	q(c	q(c	PROPN
ejde-636	224	47	)	)	PUNCT
ejde-636	224	48	,	,	PUNCT
ejde-636	224	49	there	there	PRON
ejde-636	224	50	exists	exist	VERB
ejde-636	224	51	a	a	DET
ejde-636	224	52	sequence	sequence	NOUN
ejde-636	224	53	{	{	PUNCT
ejde-636	224	54	yn	yn	PROPN
ejde-636	224	55	}	}	PUNCT
ejde-636	224	56	⊂	⊂	PROPN
ejde-636	224	57	rn	rn	PROPN
ejde-636	224	58	such	such	ADJ
ejde-636	224	59	that	that	PRON
ejde-636	224	60	for	for	ADP
ejde-636	224	61	some	some	DET
ejde-636	224	62	r	r	NOUN
ejde-636	224	63	>	>	X
ejde-636	224	64	0	0	NUM
ejde-636	224	65	it	it	PRON
ejde-636	224	66	holds∫	holds∫	VERB
ejde-636	224	67	br(yn	br(yn	PROPN
ejde-636	224	68	)	)	PUNCT
ejde-636	224	69	|un|2dx	|un|2dx	PRON
ejde-636	224	70	≥	≥	NUM
ejde-636	224	71	β	β	X
ejde-636	224	72	>	>	X
ejde-636	224	73	0	0	NUM
ejde-636	224	74	.	.	PUNCT
ejde-636	225	1	(	(	PUNCT
ejde-636	225	2	3.4	3.4	NUM
ejde-636	225	3	)	)	PUNCT
ejde-636	225	4	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	225	5	dispersion	dispersion	NOUN
ejde-636	225	6	nonlinear	nonlinear	NOUN
ejde-636	225	7	schrödinger	schrödinger	NOUN
ejde-636	225	8	equation	equation	NOUN
ejde-636	225	9	9	9	NUM
ejde-636	225	10	proof	proof	NOUN
ejde-636	225	11	.	.	PUNCT
ejde-636	226	1	by	by	ADP
ejde-636	226	2	way	way	NOUN
ejde-636	226	3	of	of	ADP
ejde-636	226	4	contradiction	contradiction	NOUN
ejde-636	226	5	,	,	PUNCT
ejde-636	226	6	we	we	PRON
ejde-636	226	7	assume	assume	VERB
ejde-636	226	8	that	that	SCONJ
ejde-636	226	9	(	(	PUNCT
ejde-636	226	10	3.4	3.4	NUM
ejde-636	226	11	)	)	PUNCT
ejde-636	226	12	does	do	AUX
ejde-636	226	13	not	not	PART
ejde-636	226	14	hold	hold	VERB
ejde-636	226	15	.	.	PUNCT
ejde-636	227	1	from	from	ADP
ejde-636	227	2	{	{	PUNCT
ejde-636	227	3	un	un	PROPN
ejde-636	227	4	}	}	PUNCT
ejde-636	227	5	⊂	⊂	NOUN
ejde-636	227	6	bρ0	bρ0	NOUN
ejde-636	227	7	and	and	CCONJ
ejde-636	227	8	∥un∥2	∥un∥2	PUNCT
ejde-636	227	9	→	→	SYM
ejde-636	227	10	c	c	NOUN
ejde-636	227	11	it	it	PRON
ejde-636	227	12	follows	follow	VERB
ejde-636	227	13	that	that	SCONJ
ejde-636	227	14	{	{	PUNCT
ejde-636	227	15	un	un	PROPN
ejde-636	227	16	}	}	PUNCT
ejde-636	227	17	is	be	AUX
ejde-636	227	18	bounded	bound	VERB
ejde-636	227	19	in	in	ADP
ejde-636	227	20	h2(rn	h2(rn	PROPN
ejde-636	227	21	)	)	PUNCT
ejde-636	227	22	.	.	PUNCT
ejde-636	228	1	for	for	ADP
ejde-636	228	2	2	2	NUM
ejde-636	228	3	<	<	X
ejde-636	228	4	q	q	X
ejde-636	228	5	<	<	X
ejde-636	228	6	2	2	NUM
ejde-636	228	7	+	+	SYM
ejde-636	228	8	4	4	NUM
ejde-636	228	9	n	n	NOUN
ejde-636	228	10	<	<	X
ejde-636	228	11	p	p	X
ejde-636	228	12	<	<	X
ejde-636	228	13	p	p	X
ejde-636	228	14	<	<	X
ejde-636	228	15	4∗	4∗	NOUN
ejde-636	228	16	,	,	PUNCT
ejde-636	228	17	by	by	ADP
ejde-636	228	18	lemma	lemma	PROPN
ejde-636	228	19	2.3	2.3	NUM
ejde-636	228	20	,	,	PUNCT
ejde-636	228	21	we	we	PRON
ejde-636	228	22	deduce	deduce	VERB
ejde-636	228	23	that	that	PRON
ejde-636	228	24	∥un∥qq	∥un∥qq	PROPN
ejde-636	228	25	→	→	SYM
ejde-636	228	26	0	0	NUM
ejde-636	228	27	and	and	CCONJ
ejde-636	228	28	∥un∥pp	∥un∥pp	PROPN
ejde-636	228	29	→	→	SYM
ejde-636	228	30	0	0	NUM
ejde-636	228	31	,	,	PUNCT
ejde-636	228	32	as	as	ADP
ejde-636	228	33	n	n	PROPN
ejde-636	228	34	→	→	SYM
ejde-636	228	35	∞.	∞.	PROPN
ejde-636	228	36	at	at	ADP
ejde-636	228	37	this	this	DET
ejde-636	228	38	point	point	NOUN
ejde-636	228	39	,	,	PUNCT
ejde-636	228	40	it	it	PRON
ejde-636	228	41	follows	follow	VERB
ejde-636	228	42	that	that	SCONJ
ejde-636	228	43	ep	ep	PROPN
ejde-636	228	44	,	,	PUNCT
ejde-636	228	45	q(un	q(un	PROPN
ejde-636	228	46	)	)	PUNCT
ejde-636	228	47	≥	≥	NOUN
ejde-636	228	48	on(1	on(1	NOUN
ejde-636	228	49	)	)	PUNCT
ejde-636	228	50	.	.	PUNCT
ejde-636	229	1	if	if	SCONJ
ejde-636	229	2	p	p	NOUN
ejde-636	229	3	=	=	SYM
ejde-636	229	4	4∗	4∗	NOUN
ejde-636	229	5	,	,	PUNCT
ejde-636	229	6	in	in	ADP
ejde-636	229	7	view	view	NOUN
ejde-636	229	8	of	of	ADP
ejde-636	229	9	f(c0	f(c0	NOUN
ejde-636	229	10	,	,	PUNCT
ejde-636	229	11	ρ0	ρ0	PROPN
ejde-636	229	12	)	)	PUNCT
ejde-636	229	13	=	=	SYM
ejde-636	229	14	0	0	NUM
ejde-636	229	15	,	,	PUNCT
ejde-636	229	16	a	a	DET
ejde-636	229	17	straightforward	straightforward	ADJ
ejde-636	229	18	computation	computation	NOUN
ejde-636	229	19	yields	yield	VERB
ejde-636	229	20	ep	ep	PROPN
ejde-636	229	21	,	,	PUNCT
ejde-636	229	22	q(un	q(un	PROPN
ejde-636	229	23	)	)	PUNCT
ejde-636	229	24	=	=	SYM
ejde-636	230	1	1	1	NUM
ejde-636	230	2	2	2	NUM
ejde-636	230	3	∥∆un∥22	∥∆un∥22	PUNCT
ejde-636	230	4	+	+	CCONJ
ejde-636	230	5	1	1	NUM
ejde-636	230	6	2	2	NUM
ejde-636	230	7	∥∇un∥22	∥∇un∥22	NOUN
ejde-636	230	8	−	−	PROPN
ejde-636	230	9	1	1	NUM
ejde-636	230	10	4∗	4∗	NOUN
ejde-636	230	11	∥un∥4	∥un∥4	ADP
ejde-636	230	12	∗	∗	NOUN
ejde-636	230	13	4∗	4∗	NOUN
ejde-636	230	14	+	+	CCONJ
ejde-636	230	15	on(1	on(1	NOUN
ejde-636	230	16	)	)	PUNCT
ejde-636	230	17	≥	≥	NOUN
ejde-636	230	18	1	1	NUM
ejde-636	230	19	2	2	NUM
ejde-636	230	20	∥∆un∥22	∥∆un∥22	PUNCT
ejde-636	230	21	+	+	CCONJ
ejde-636	230	22	1	1	NUM
ejde-636	230	23	2	2	NUM
ejde-636	230	24	∥∇un∥22	∥∇un∥22	NOUN
ejde-636	230	25	−	−	NUM
ejde-636	230	26	1	1	NUM
ejde-636	230	27	4∗	4∗	NUM
ejde-636	230	28	1	1	NUM
ejde-636	230	29	s4∗/2	s4∗/2	X
ejde-636	230	30	(	(	PUNCT
ejde-636	230	31	∥∆un∥22	∥∆un∥22	PROPN
ejde-636	230	32	+	+	NUM
ejde-636	230	33	∥∇un∥22	∥∇un∥22	X
ejde-636	230	34	)	)	PUNCT
ejde-636	230	35	4∗	4∗	NOUN
ejde-636	230	36	2	2	NUM
ejde-636	230	37	+	+	SYM
ejde-636	230	38	on(1	on(1	NOUN
ejde-636	230	39	)	)	PUNCT
ejde-636	230	40	≥	≥	NOUN
ejde-636	230	41	(	(	PUNCT
ejde-636	230	42	∥∆n∥22	∥∆n∥22	X
ejde-636	230	43	+	+	CCONJ
ejde-636	230	44	∥∇un∥22	∥∇un∥22	NUM
ejde-636	230	45	)	)	PUNCT
ejde-636	230	46	(	(	PUNCT
ejde-636	230	47	1	1	NUM
ejde-636	230	48	2	2	NUM
ejde-636	230	49	−	−	NUM
ejde-636	230	50	1	1	NUM
ejde-636	230	51	4∗	4∗	NUM
ejde-636	230	52	1	1	NUM
ejde-636	230	53	s4∗/2	s4∗/2	X
ejde-636	230	54	ρα2	ρα2	NOUN
ejde-636	230	55	0	0	NUM
ejde-636	230	56	)	)	PUNCT
ejde-636	231	1	+	+	CCONJ
ejde-636	231	2	on(1	on(1	NOUN
ejde-636	231	3	)	)	PUNCT
ejde-636	231	4	=	=	SYM
ejde-636	231	5	(	(	PUNCT
ejde-636	231	6	∥∆n∥22	∥∆n∥22	X
ejde-636	231	7	+	+	CCONJ
ejde-636	231	8	∥∇un∥22	∥∇un∥22	X
ejde-636	231	9	)	)	PUNCT
ejde-636	231	10	µ	µ	X
ejde-636	231	11	q	q	NOUN
ejde-636	231	12	cq	cq	PROPN
ejde-636	231	13	n	n	CCONJ
ejde-636	231	14	,	,	PUNCT
ejde-636	231	15	qρ	qρ	VERB
ejde-636	231	16	α0	α0	ADJ
ejde-636	231	17	0	0	NUM
ejde-636	232	1	cα1	cα1	NOUN
ejde-636	232	2	0	0	NUM
ejde-636	233	1	+	+	NUM
ejde-636	233	2	on(1	on(1	NOUN
ejde-636	233	3	)	)	PUNCT
ejde-636	233	4	>	>	X
ejde-636	233	5	0	0	X
ejde-636	233	6	.	.	PUNCT
ejde-636	234	1	both	both	DET
ejde-636	234	2	cases	case	NOUN
ejde-636	234	3	contradict	contradict	VERB
ejde-636	234	4	the	the	DET
ejde-636	234	5	factmp	factmp	NOUN
ejde-636	234	6	,	,	PUNCT
ejde-636	234	7	q(c	q(c	PROPN
ejde-636	234	8	)	)	PUNCT
ejde-636	234	9	<	<	X
ejde-636	234	10	0	0	X
ejde-636	234	11	.	.	PUNCT
ejde-636	235	1	thus	thus	ADV
ejde-636	235	2	,	,	PUNCT
ejde-636	235	3	we	we	PRON
ejde-636	235	4	arrive	arrive	VERB
ejde-636	235	5	at	at	ADP
ejde-636	235	6	the	the	DET
ejde-636	235	7	desired	desire	VERB
ejde-636	235	8	result	result	NOUN
ejde-636	235	9	.	.	PUNCT
ejde-636	236	1	□	□	PUNCT
ejde-636	236	2	proposition	proposition	NOUN
ejde-636	236	3	3.3	3.3	NUM
ejde-636	236	4	.	.	PUNCT
ejde-636	237	1	for	for	ADP
ejde-636	237	2	any	any	DET
ejde-636	237	3	c	c	PROPN
ejde-636	237	4	∈	∈	PROPN
ejde-636	237	5	(	(	PUNCT
ejde-636	237	6	0	0	NUM
ejde-636	237	7	,	,	PUNCT
ejde-636	237	8	c0	c0	NOUN
ejde-636	237	9	)	)	PUNCT
ejde-636	237	10	,	,	PUNCT
ejde-636	237	11	if	if	SCONJ
ejde-636	237	12	{	{	PUNCT
ejde-636	237	13	un	un	ADJ
ejde-636	237	14	}	}	PUNCT
ejde-636	237	15	⊂	⊂	PRON
ejde-636	237	16	bρ0	bρ0	PROPN
ejde-636	237	17	is	be	AUX
ejde-636	237	18	such	such	ADJ
ejde-636	237	19	that	that	PRON
ejde-636	237	20	∥un∥22	∥un∥22	PROPN
ejde-636	237	21	→	→	SYM
ejde-636	237	22	c	c	PROPN
ejde-636	237	23	and	and	CCONJ
ejde-636	237	24	ep	ep	PROPN
ejde-636	237	25	,	,	PUNCT
ejde-636	237	26	q(un	q(un	PROPN
ejde-636	237	27	)	)	PUNCT
ejde-636	237	28	→	→	SYM
ejde-636	237	29	mp	mp	PROPN
ejde-636	237	30	,	,	PUNCT
ejde-636	237	31	q(c	q(c	PROPN
ejde-636	237	32	)	)	PUNCT
ejde-636	237	33	,	,	PUNCT
ejde-636	237	34	then	then	ADV
ejde-636	237	35	,	,	PUNCT
ejde-636	237	36	up	up	ADP
ejde-636	237	37	to	to	ADP
ejde-636	237	38	translation	translation	NOUN
ejde-636	237	39	,	,	PUNCT
ejde-636	237	40	un	un	PROPN
ejde-636	237	41	−→	−→	PROPN
ejde-636	237	42	uc	uc	PROPN
ejde-636	237	43	∈	∈	PROPN
ejde-636	237	44	mc	mc	PROPN
ejde-636	237	45	in	in	ADP
ejde-636	237	46	h2(rn	h2(rn	PROPN
ejde-636	237	47	)	)	PUNCT
ejde-636	237	48	.	.	PUNCT
ejde-636	238	1	in	in	ADP
ejde-636	238	2	particular	particular	ADJ
ejde-636	238	3	,	,	PUNCT
ejde-636	238	4	the	the	DET
ejde-636	238	5	set	set	NOUN
ejde-636	238	6	mc	mc	PROPN
ejde-636	238	7	is	be	AUX
ejde-636	238	8	compact	compact	ADJ
ejde-636	238	9	in	in	ADP
ejde-636	238	10	h2(rn	h2(rn	PROPN
ejde-636	238	11	)	)	PUNCT
ejde-636	238	12	,	,	PUNCT
ejde-636	238	13	up	up	ADP
ejde-636	238	14	to	to	ADP
ejde-636	238	15	translation	translation	NOUN
ejde-636	238	16	.	.	PUNCT
ejde-636	239	1	the	the	DET
ejde-636	239	2	proof	proof	NOUN
ejde-636	239	3	of	of	ADP
ejde-636	239	4	the	the	DET
ejde-636	239	5	above	above	ADJ
ejde-636	239	6	proposition	proposition	NOUN
ejde-636	239	7	can	can	AUX
ejde-636	239	8	be	be	AUX
ejde-636	239	9	obtained	obtain	VERB
ejde-636	239	10	by	by	ADP
ejde-636	239	11	similar	similar	ADJ
ejde-636	239	12	arguments	argument	NOUN
ejde-636	239	13	as	as	ADP
ejde-636	239	14	in	in	ADP
ejde-636	239	15	[	[	X
ejde-636	239	16	27	27	NUM
ejde-636	239	17	]	]	PUNCT
ejde-636	239	18	(	(	PUNCT
ejde-636	239	19	see	see	VERB
ejde-636	239	20	also	also	ADV
ejde-636	239	21	[	[	X
ejde-636	239	22	18	18	NUM
ejde-636	239	23	]	]	NUM
ejde-636	239	24	)	)	PUNCT
ejde-636	239	25	.	.	PUNCT
ejde-636	240	1	proposition	proposition	NOUN
ejde-636	240	2	3.4	3.4	NUM
ejde-636	240	3	.	.	PUNCT
ejde-636	241	1	for	for	ADP
ejde-636	241	2	any	any	DET
ejde-636	241	3	c	c	PROPN
ejde-636	241	4	∈	∈	PROPN
ejde-636	241	5	(	(	PUNCT
ejde-636	241	6	0	0	NUM
ejde-636	241	7	,	,	PUNCT
ejde-636	241	8	c0	c0	NOUN
ejde-636	241	9	)	)	PUNCT
ejde-636	241	10	,	,	PUNCT
ejde-636	241	11	if	if	SCONJ
ejde-636	241	12	mp	mp	PROPN
ejde-636	241	13	,	,	PUNCT
ejde-636	241	14	q(c	q(c	PROPN
ejde-636	241	15	)	)	PUNCT
ejde-636	241	16	is	be	AUX
ejde-636	241	17	reached	reach	VERB
ejde-636	241	18	,	,	PUNCT
ejde-636	241	19	then	then	ADV
ejde-636	241	20	any	any	DET
ejde-636	241	21	ground	ground	NOUN
ejde-636	241	22	state	state	NOUN
ejde-636	241	23	is	be	AUX
ejde-636	241	24	contained	contain	VERB
ejde-636	241	25	in	in	ADP
ejde-636	241	26	v	v	NUM
ejde-636	241	27	(	(	PUNCT
ejde-636	241	28	c	c	NOUN
ejde-636	241	29	)	)	PUNCT
ejde-636	241	30	.	.	PUNCT
ejde-636	242	1	proof	proof	NOUN
ejde-636	242	2	.	.	PUNCT
ejde-636	243	1	for	for	ADP
ejde-636	243	2	any	any	DET
ejde-636	243	3	v	v	PROPN
ejde-636	243	4	∈	∈	PROPN
ejde-636	243	5	s(c	s(c	NOUN
ejde-636	243	6	)	)	PUNCT
ejde-636	243	7	and	and	CCONJ
ejde-636	243	8	s	s	PROPN
ejde-636	243	9	∈	∈	PROPN
ejde-636	243	10	(	(	PUNCT
ejde-636	243	11	0,∞	0,∞	NOUN
ejde-636	243	12	)	)	PUNCT
ejde-636	243	13	,	,	PUNCT
ejde-636	243	14	we	we	PRON
ejde-636	243	15	obtain	obtain	VERB
ejde-636	243	16	ψ′	ψ′	NUM
ejde-636	243	17	v(s	v(s	NOUN
ejde-636	243	18	)	)	PUNCT
ejde-636	243	19	=	=	PUNCT
ejde-636	244	1	2	2	NUM
ejde-636	244	2	s	s	NOUN
ejde-636	244	3	q(vs	q(vs	NUM
ejde-636	244	4	)	)	PUNCT
ejde-636	244	5	,	,	PUNCT
ejde-636	244	6	which	which	PRON
ejde-636	244	7	implies	imply	VERB
ejde-636	244	8	that	that	SCONJ
ejde-636	244	9	if	if	SCONJ
ejde-636	244	10	w	w	PROPN
ejde-636	244	11	∈	∈	PROPN
ejde-636	244	12	s(c	s(c	PROPN
ejde-636	244	13	)	)	PUNCT
ejde-636	244	14	is	be	AUX
ejde-636	244	15	a	a	DET
ejde-636	244	16	ground	ground	NOUN
ejde-636	244	17	state	state	NOUN
ejde-636	244	18	solution	solution	NOUN
ejde-636	244	19	,	,	PUNCT
ejde-636	244	20	then	then	ADV
ejde-636	244	21	there	there	PRON
ejde-636	244	22	exist	exist	VERB
ejde-636	244	23	v	v	ADP
ejde-636	244	24	∈	∈	PROPN
ejde-636	244	25	s(c	s(c	PROPN
ejde-636	244	26	)	)	PUNCT
ejde-636	244	27	and	and	CCONJ
ejde-636	244	28	s0	s0	PROPN
ejde-636	244	29	>	>	X
ejde-636	244	30	0	0	NUM
ejde-636	244	31	such	such	ADJ
ejde-636	244	32	that	that	DET
ejde-636	244	33	w	w	NOUN
ejde-636	244	34	=	=	PUNCT
ejde-636	244	35	vs0	vs0	NOUN
ejde-636	244	36	,	,	PUNCT
ejde-636	244	37	ep	ep	NOUN
ejde-636	244	38	,	,	PUNCT
ejde-636	244	39	q(w	q(w	NOUN
ejde-636	244	40	)	)	PUNCT
ejde-636	244	41	=	=	SYM
ejde-636	244	42	ψv(s0	ψv(s0	PROPN
ejde-636	244	43	)	)	PUNCT
ejde-636	244	44	and	and	CCONJ
ejde-636	244	45	ψ	ψ	X
ejde-636	244	46	′	′	NUM
ejde-636	244	47	v(s0	v(s0	NOUN
ejde-636	244	48	)	)	PUNCT
ejde-636	244	49	=	=	SYM
ejde-636	245	1	0	0	X
ejde-636	245	2	.	.	PUNCT
ejde-636	245	3	to	to	PART
ejde-636	245	4	conclude	conclude	VERB
ejde-636	245	5	the	the	DET
ejde-636	245	6	proof	proof	NOUN
ejde-636	245	7	,	,	PUNCT
ejde-636	245	8	it	it	PRON
ejde-636	245	9	suffices	suffice	VERB
ejde-636	245	10	to	to	PART
ejde-636	245	11	show	show	VERB
ejde-636	245	12	that	that	PRON
ejde-636	245	13	ψ′	ψ′	PROPN
ejde-636	245	14	v(s	v(s	NOUN
ejde-636	245	15	)	)	PUNCT
ejde-636	245	16	has	have	VERB
ejde-636	245	17	at	at	ADP
ejde-636	245	18	most	most	ADV
ejde-636	245	19	two	two	NUM
ejde-636	245	20	zeros	zero	NOUN
ejde-636	245	21	.	.	PUNCT
ejde-636	246	1	this	this	PRON
ejde-636	246	2	is	be	AUX
ejde-636	246	3	equivalent	equivalent	ADJ
ejde-636	246	4	to	to	ADP
ejde-636	246	5	showing	show	VERB
ejde-636	246	6	that	that	SCONJ
ejde-636	246	7	the	the	DET
ejde-636	246	8	function	function	NOUN
ejde-636	246	9	s	s	VERB
ejde-636	246	10	7→	7→	NUM
ejde-636	246	11	ψ′	ψ′	NUM
ejde-636	246	12	v(s	v(s	NOUN
ejde-636	246	13	)	)	PUNCT
ejde-636	247	1	s	s	AUX
ejde-636	247	2	has	have	VERB
ejde-636	247	3	at	at	ADP
ejde-636	247	4	most	most	ADJ
ejde-636	247	5	two	two	NUM
ejde-636	247	6	zeros	zero	NOUN
ejde-636	247	7	.	.	PUNCT
ejde-636	248	1	note	note	VERB
ejde-636	248	2	that	that	SCONJ
ejde-636	248	3	ξ(s	ξ(s	PROPN
ejde-636	248	4	)	)	PUNCT
ejde-636	248	5	=	=	PRON
ejde-636	248	6	ψ′	ψ′	NUM
ejde-636	248	7	v(s	v(s	NOUN
ejde-636	248	8	)	)	PUNCT
ejde-636	248	9	s	s	PART
ejde-636	249	1	=	=	PUNCT
ejde-636	249	2	2s2∥∆u∥22	2s2∥∆u∥22	PROPN
ejde-636	249	3	+	+	CCONJ
ejde-636	249	4	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	250	1	−	−	PROPN
ejde-636	250	2	s	s	NOUN
ejde-636	250	3	n(q−2	n(q−2	X
ejde-636	250	4	)	)	PUNCT
ejde-636	250	5	2	2	NUM
ejde-636	250	6	−2µn(q	−2µn(q	NOUN
ejde-636	250	7	−	−	NOUN
ejde-636	250	8	2	2	NUM
ejde-636	250	9	)	)	PUNCT
ejde-636	250	10	2q	2q	NOUN
ejde-636	250	11	∥u∥qq	∥u∥qq	ADP
ejde-636	250	12	−	−	PROPN
ejde-636	250	13	s	s	PART
ejde-636	250	14	n(p−2	n(p−2	PROPN
ejde-636	250	15	)	)	PUNCT
ejde-636	250	16	2	2	NUM
ejde-636	250	17	−2n(p−	−2n(p−	VERB
ejde-636	250	18	2	2	NUM
ejde-636	250	19	)	)	PUNCT
ejde-636	250	20	2p	2p	NUM
ejde-636	250	21	∥u∥pp	∥u∥pp	PROPN
ejde-636	250	22	and	and	CCONJ
ejde-636	250	23	ξ′(s	ξ′(s	NUM
ejde-636	250	24	)	)	PUNCT
ejde-636	250	25	=	=	SYM
ejde-636	250	26	s[4∥∆u∥22	s[4∥∆u∥22	PROPN
ejde-636	250	27	−	−	PROPN
ejde-636	250	28	s	s	NOUN
ejde-636	250	29	n(q−2	n(q−2	X
ejde-636	250	30	)	)	PUNCT
ejde-636	250	31	2	2	NUM
ejde-636	250	32	−4	−4	NOUN
ejde-636	250	33	·	·	PUNCT
ejde-636	250	34	µn(q	µn(q	PUNCT
ejde-636	250	35	−	−	PROPN
ejde-636	250	36	2	2	NUM
ejde-636	250	37	)	)	PUNCT
ejde-636	250	38	2q	2q	NOUN
ejde-636	250	39	(	(	PUNCT
ejde-636	250	40	n(q	n(q	PROPN
ejde-636	250	41	−	−	PROPN
ejde-636	250	42	2	2	NUM
ejde-636	250	43	)	)	SYM
ejde-636	250	44	2	2	NUM
ejde-636	250	45	−	−	PROPN
ejde-636	250	46	2)∥u∥qq	2)∥u∥qq	NOUN
ejde-636	250	47	−	−	NOUN
ejde-636	250	48	s	s	PART
ejde-636	250	49	n(p−2	n(p−2	PROPN
ejde-636	250	50	)	)	PUNCT
ejde-636	250	51	2	2	NUM
ejde-636	250	52	−4	−4	NOUN
ejde-636	250	53	·	·	PUNCT
ejde-636	250	54	n(p−	n(p−	X
ejde-636	250	55	2	2	NUM
ejde-636	250	56	)	)	PUNCT
ejde-636	250	57	2p	2p	NOUN
ejde-636	250	58	(	(	PUNCT
ejde-636	250	59	n(p−	n(p−	X
ejde-636	250	60	2	2	NUM
ejde-636	250	61	)	)	PUNCT
ejde-636	250	62	2	2	NUM
ejde-636	250	63	−	−	NOUN
ejde-636	251	1	2)∥u∥pp	2)∥u∥pp	NOUN
ejde-636	251	2	]	]	X
ejde-636	252	1	=	=	X
ejde-636	252	2	:	:	PUNCT
ejde-636	252	3	s[4∥∆u∥22	s[4∥∆u∥22	PROPN
ejde-636	252	4	−	−	PROPN
ejde-636	252	5	f(s	f(	NOUN
ejde-636	252	6	)	)	PUNCT
ejde-636	252	7	]	]	PUNCT
ejde-636	252	8	.	.	PUNCT
ejde-636	253	1	so	so	ADV
ejde-636	253	2	we	we	PRON
ejde-636	253	3	need	need	VERB
ejde-636	253	4	to	to	PART
ejde-636	253	5	show	show	VERB
ejde-636	253	6	that	that	SCONJ
ejde-636	253	7	ξ′(s	ξ′(s	PROPN
ejde-636	253	8	)	)	PUNCT
ejde-636	253	9	is	be	AUX
ejde-636	253	10	the	the	DET
ejde-636	253	11	unique	unique	ADJ
ejde-636	253	12	solution	solution	NOUN
ejde-636	253	13	.	.	PUNCT
ejde-636	254	1	since	since	SCONJ
ejde-636	254	2	2	2	NUM
ejde-636	254	3	<	<	X
ejde-636	254	4	q	q	X
ejde-636	254	5	<	<	X
ejde-636	254	6	2	2	NUM
ejde-636	254	7	+	+	SYM
ejde-636	254	8	4	4	NUM
ejde-636	254	9	n	n	NOUN
ejde-636	254	10	<	<	X
ejde-636	254	11	p	p	X
ejde-636	254	12	<	<	X
ejde-636	254	13	p	p	X
ejde-636	254	14	≤	≤	NUM
ejde-636	254	15	4∗	4∗	NOUN
ejde-636	254	16	,	,	PUNCT
ejde-636	254	17	n	n	X
ejde-636	254	18	≥	≥	NOUN
ejde-636	254	19	5	5	NUM
ejde-636	254	20	and	and	CCONJ
ejde-636	254	21	s	s	X
ejde-636	254	22	>	>	X
ejde-636	254	23	0	0	NUM
ejde-636	254	24	,	,	PUNCT
ejde-636	254	25	it	it	PRON
ejde-636	254	26	is	be	AUX
ejde-636	254	27	easy	easy	ADJ
ejde-636	254	28	to	to	PART
ejde-636	254	29	see	see	VERB
ejde-636	254	30	that	that	PRON
ejde-636	254	31	s	s	NOUN
ejde-636	254	32	→	→	SYM
ejde-636	254	33	f(s	f(s	X
ejde-636	254	34	)	)	PUNCT
ejde-636	254	35	is	be	AUX
ejde-636	254	36	a	a	DET
ejde-636	254	37	non	non	ADJ
ejde-636	254	38	-	-	ADJ
ejde-636	254	39	increasing	increasing	ADJ
ejde-636	254	40	function	function	NOUN
ejde-636	254	41	.	.	PUNCT
ejde-636	255	1	hence	hence	ADV
ejde-636	255	2	,	,	PUNCT
ejde-636	255	3	ξ′(s	ξ′(s	PROPN
ejde-636	255	4	)	)	PUNCT
ejde-636	255	5	has	have	VERB
ejde-636	255	6	a	a	DET
ejde-636	255	7	unique	unique	ADJ
ejde-636	255	8	solution	solution	NOUN
ejde-636	255	9	and	and	CCONJ
ejde-636	255	10	ξ(s	ξ(	NOUN
ejde-636	255	11	)	)	PUNCT
ejde-636	255	12	has	have	VERB
ejde-636	255	13	at	at	ADP
ejde-636	255	14	most	most	ADV
ejde-636	255	15	two	two	NUM
ejde-636	255	16	zeros	zero	NOUN
ejde-636	255	17	.	.	PUNCT
ejde-636	256	1	10	10	NUM
ejde-636	256	2	z.	z.	PROPN
ejde-636	256	3	ma	ma	PROPN
ejde-636	256	4	,	,	PUNCT
ejde-636	256	5	x.	x.	PROPN
ejde-636	256	6	chang	chang	PROPN
ejde-636	256	7	,	,	PUNCT
ejde-636	256	8	z.	z.	PROPN
ejde-636	256	9	feng	feng	PROPN
ejde-636	256	10	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	256	11	now	now	ADV
ejde-636	256	12	,	,	PUNCT
ejde-636	256	13	since	since	SCONJ
ejde-636	256	14	ψv(s	ψv(s	NUM
ejde-636	256	15	)	)	PUNCT
ejde-636	256	16	→	→	SYM
ejde-636	256	17	0−	0−	NUM
ejde-636	256	18	,	,	PUNCT
ejde-636	256	19	∥∆vs∥22	∥∆vs∥22	PROPN
ejde-636	256	20	+	+	CCONJ
ejde-636	256	21	∥∇vs∥22	∥∇vs∥22	PROPN
ejde-636	256	22	→	→	SYM
ejde-636	256	23	0	0	NUM
ejde-636	256	24	as	as	ADP
ejde-636	256	25	s→	s→	X
ejde-636	256	26	0	0	NUM
ejde-636	256	27	and	and	CCONJ
ejde-636	256	28	ψv(s	ψv(s	PUNCT
ejde-636	256	29	)	)	PUNCT
ejde-636	257	1	=	=	SYM
ejde-636	257	2	ep	ep	PROPN
ejde-636	257	3	,	,	PUNCT
ejde-636	257	4	q(vs	q(vs	PROPN
ejde-636	257	5	)	)	PUNCT
ejde-636	257	6	>	>	X
ejde-636	257	7	0	0	NUM
ejde-636	257	8	,	,	PUNCT
ejde-636	257	9	when	when	SCONJ
ejde-636	257	10	vs	vs	ADP
ejde-636	257	11	∈	∈	PROPN
ejde-636	257	12	∂v	∂v	PROPN
ejde-636	257	13	(	(	PUNCT
ejde-636	257	14	c	c	NOUN
ejde-636	257	15	)	)	PUNCT
ejde-636	257	16	,	,	PUNCT
ejde-636	257	17	ψ′	ψ′	PUNCT
ejde-636	257	18	v	v	X
ejde-636	257	19	has	have	VERB
ejde-636	257	20	a	a	DET
ejde-636	257	21	first	first	ADJ
ejde-636	257	22	zero	zero	NUM
ejde-636	257	23	s1	s1	NOUN
ejde-636	257	24	>	>	X
ejde-636	257	25	0	0	NUM
ejde-636	258	1	corresponding	correspond	VERB
ejde-636	258	2	to	to	ADP
ejde-636	258	3	a	a	DET
ejde-636	258	4	local	local	ADJ
ejde-636	258	5	minima	minima	NOUN
ejde-636	258	6	.	.	PUNCT
ejde-636	259	1	also	also	ADV
ejde-636	259	2	,	,	PUNCT
ejde-636	259	3	from	from	ADP
ejde-636	259	4	ψv(s1	ψv(s1	NOUN
ejde-636	259	5	)	)	PUNCT
ejde-636	259	6	<	<	X
ejde-636	259	7	0	0	NUM
ejde-636	259	8	,	,	PUNCT
ejde-636	259	9	ψv(s	ψv(s	PUNCT
ejde-636	259	10	)	)	PUNCT
ejde-636	259	11	>	>	X
ejde-636	259	12	0	0	PUNCT
ejde-636	260	1	when	when	SCONJ
ejde-636	260	2	vs	vs	ADP
ejde-636	260	3	∈	∈	PROPN
ejde-636	260	4	∂v	∂v	PROPN
ejde-636	260	5	(	(	PUNCT
ejde-636	260	6	c	c	NOUN
ejde-636	260	7	)	)	PUNCT
ejde-636	260	8	and	and	CCONJ
ejde-636	260	9	ψv(s	ψv(s	PUNCT
ejde-636	260	10	)	)	PUNCT
ejde-636	260	11	→	→	PUNCT
ejde-636	260	12	−∞	−∞	PUNCT
ejde-636	260	13	as	as	ADP
ejde-636	260	14	s→	s→	PROPN
ejde-636	260	15	∞	∞	PROPN
ejde-636	260	16	,	,	PUNCT
ejde-636	260	17	ψv	ψv	PROPN
ejde-636	260	18	has	have	AUX
ejde-636	260	19	a	a	DET
ejde-636	260	20	second	second	ADJ
ejde-636	260	21	zero	zero	NUM
ejde-636	260	22	s2	s2	NOUN
ejde-636	260	23	>	>	X
ejde-636	260	24	s1	s1	PROPN
ejde-636	260	25	corresponding	correspond	VERB
ejde-636	260	26	to	to	ADP
ejde-636	260	27	a	a	DET
ejde-636	260	28	local	local	ADJ
ejde-636	260	29	maxima	maxima	NOUN
ejde-636	260	30	.	.	PUNCT
ejde-636	261	1	in	in	ADP
ejde-636	261	2	particular	particular	ADJ
ejde-636	261	3	,	,	PUNCT
ejde-636	261	4	vs1	vs1	PROPN
ejde-636	261	5	∈	∈	PROPN
ejde-636	261	6	v	v	NOUN
ejde-636	261	7	(	(	PUNCT
ejde-636	261	8	c	c	NOUN
ejde-636	261	9	)	)	PUNCT
ejde-636	261	10	and	and	CCONJ
ejde-636	261	11	ep	ep	PROPN
ejde-636	261	12	,	,	PUNCT
ejde-636	261	13	q(vs1	q(vs1	PROPN
ejde-636	261	14	)	)	PUNCT
ejde-636	261	15	=	=	SYM
ejde-636	261	16	ψv(s1	ψv(s1	NOUN
ejde-636	261	17	)	)	PUNCT
ejde-636	261	18	<	<	X
ejde-636	261	19	0	0	X
ejde-636	261	20	.	.	PUNCT
ejde-636	262	1	thus	thus	ADV
ejde-636	262	2	,	,	PUNCT
ejde-636	262	3	if	if	SCONJ
ejde-636	262	4	mp	mp	PROPN
ejde-636	262	5	,	,	PUNCT
ejde-636	262	6	q(c	q(c	PROPN
ejde-636	262	7	)	)	PUNCT
ejde-636	262	8	is	be	AUX
ejde-636	262	9	achieved	achieve	VERB
ejde-636	262	10	,	,	PUNCT
ejde-636	262	11	it	it	PRON
ejde-636	262	12	is	be	AUX
ejde-636	262	13	a	a	DET
ejde-636	262	14	ground	ground	NOUN
ejde-636	262	15	state	state	NOUN
ejde-636	262	16	level	level	NOUN
ejde-636	262	17	.	.	PUNCT
ejde-636	263	1	□	□	PUNCT
ejde-636	263	2	proof	proof	NOUN
ejde-636	263	3	of	of	ADP
ejde-636	263	4	theorem	theorem	ADJ
ejde-636	263	5	1.2	1.2	NUM
ejde-636	263	6	.	.	PUNCT
ejde-636	264	1	the	the	DET
ejde-636	264	2	existence	existence	NOUN
ejde-636	264	3	of	of	ADP
ejde-636	264	4	a	a	DET
ejde-636	264	5	minimizer	minimizer	NOUN
ejde-636	264	6	for	for	ADP
ejde-636	264	7	ep	ep	PROPN
ejde-636	264	8	,	,	PUNCT
ejde-636	264	9	q	q	NOUN
ejde-636	264	10	on	on	ADP
ejde-636	264	11	v	v	NUM
ejde-636	264	12	(	(	PUNCT
ejde-636	264	13	c	c	NOUN
ejde-636	264	14	)	)	PUNCT
ejde-636	264	15	follows	follow	VERB
ejde-636	264	16	from	from	ADP
ejde-636	264	17	proposition	proposition	NOUN
ejde-636	264	18	3.3	3.3	NUM
ejde-636	264	19	.	.	PUNCT
ejde-636	265	1	by	by	ADP
ejde-636	265	2	proposition	proposition	NOUN
ejde-636	265	3	3.4	3.4	NUM
ejde-636	265	4	,	,	PUNCT
ejde-636	265	5	this	this	DET
ejde-636	265	6	local	local	ADJ
ejde-636	265	7	minimizer	minimizer	NOUN
ejde-636	265	8	is	be	AUX
ejde-636	265	9	a	a	DET
ejde-636	265	10	ground	ground	NOUN
ejde-636	265	11	state	state	NOUN
ejde-636	265	12	.	.	PUNCT
ejde-636	266	1	□	□	PUNCT
ejde-636	266	2	4	4	X
ejde-636	266	3	.	.	PUNCT
ejde-636	266	4	case	case	NOUN
ejde-636	266	5	p	p	NOUN
ejde-636	266	6	≤	≤	X
ejde-636	267	1	q	q	NOUN
ejde-636	267	2	<	<	X
ejde-636	267	3	p	p	X
ejde-636	267	4	<	<	X
ejde-636	267	5	4∗	4∗	NOUN
ejde-636	267	6	in	in	ADP
ejde-636	267	7	this	this	DET
ejde-636	267	8	section	section	NOUN
ejde-636	268	1	,	,	PUNCT
ejde-636	268	2	we	we	PRON
ejde-636	268	3	present	present	VERB
ejde-636	268	4	the	the	DET
ejde-636	268	5	proof	proof	NOUN
ejde-636	268	6	of	of	ADP
ejde-636	268	7	theorem	theorem	ADJ
ejde-636	268	8	1.3	1.3	NUM
ejde-636	268	9	.	.	PUNCT
ejde-636	268	10	4.1	4.1	NUM
ejde-636	268	11	.	.	PUNCT
ejde-636	269	1	monotonicity	monotonicity	NOUN
ejde-636	269	2	of	of	ADP
ejde-636	269	3	ground	ground	NOUN
ejde-636	269	4	state	state	NOUN
ejde-636	269	5	energy	energy	PROPN
ejde-636	269	6	mp	mp	PROPN
ejde-636	269	7	,	,	PUNCT
ejde-636	269	8	q(c	q(c	PROPN
ejde-636	269	9	)	)	PUNCT
ejde-636	269	10	.	.	PUNCT
ejde-636	270	1	we	we	PRON
ejde-636	270	2	start	start	VERB
ejde-636	270	3	by	by	ADP
ejde-636	270	4	showing	show	VERB
ejde-636	270	5	some	some	DET
ejde-636	270	6	properties	property	NOUN
ejde-636	270	7	of	of	ADP
ejde-636	270	8	qp	qp	NOUN
ejde-636	270	9	,	,	PUNCT
ejde-636	270	10	q(c	q(c	PROPN
ejde-636	270	11	)	)	PUNCT
ejde-636	270	12	and	and	CCONJ
ejde-636	270	13	the	the	DET
ejde-636	270	14	energy	energy	NOUN
ejde-636	270	15	functional	functional	ADJ
ejde-636	270	16	ep	ep	PROPN
ejde-636	270	17	,	,	PUNCT
ejde-636	270	18	q	q	PUNCT
ejde-636	270	19	restricted	restrict	VERB
ejde-636	270	20	on	on	ADP
ejde-636	270	21	it	it	PRON
ejde-636	270	22	.	.	PUNCT
ejde-636	271	1	for	for	ADP
ejde-636	271	2	any	any	DET
ejde-636	271	3	u	u	PROPN
ejde-636	271	4	∈	∈	PROPN
ejde-636	271	5	s(c	s(c	PROPN
ejde-636	271	6	)	)	PUNCT
ejde-636	271	7	and	and	CCONJ
ejde-636	271	8	s	s	PROPN
ejde-636	271	9	∈	∈	PROPN
ejde-636	271	10	(	(	PUNCT
ejde-636	271	11	0,+∞	0,+∞	NUM
ejde-636	271	12	)	)	PUNCT
ejde-636	271	13	,	,	PUNCT
ejde-636	271	14	we	we	PRON
ejde-636	271	15	define	define	VERB
ejde-636	271	16	us(x	us(x	NOUN
ejde-636	271	17	)	)	PUNCT
ejde-636	272	1	=	=	SYM
ejde-636	272	2	sn/4u	sn/4u	NOUN
ejde-636	272	3	(	(	PUNCT
ejde-636	272	4	√	√	NUM
ejde-636	272	5	sx	sx	PROPN
ejde-636	272	6	)	)	PUNCT
ejde-636	272	7	,	,	PUNCT
ejde-636	272	8	for	for	ADP
ejde-636	272	9	a.e	a.e	PROPN
ejde-636	272	10	.	.	PUNCT
ejde-636	272	11	x	x	PROPN
ejde-636	273	1	∈	∈	PROPN
ejde-636	273	2	rn	rn	PROPN
ejde-636	273	3	.	.	PUNCT
ejde-636	274	1	clearly	clearly	ADV
ejde-636	274	2	,	,	PUNCT
ejde-636	274	3	us	us	PROPN
ejde-636	274	4	∈	∈	PROPN
ejde-636	274	5	s(c	s(c	PROPN
ejde-636	274	6	)	)	PUNCT
ejde-636	274	7	for	for	ADP
ejde-636	274	8	any	any	DET
ejde-636	274	9	s	s	X
ejde-636	274	10	>	>	X
ejde-636	274	11	0	0	NUM
ejde-636	274	12	.	.	PUNCT
ejde-636	275	1	it	it	PRON
ejde-636	275	2	follows	follow	VERB
ejde-636	275	3	that	that	SCONJ
ejde-636	275	4	ep	ep	PROPN
ejde-636	275	5	,	,	PUNCT
ejde-636	275	6	q(us	q(us	PROPN
ejde-636	275	7	)	)	PUNCT
ejde-636	275	8	=	=	SYM
ejde-636	275	9	s2	s2	NOUN
ejde-636	275	10	2	2	NUM
ejde-636	275	11	∥∆u∥22	∥∆u∥22	PUNCT
ejde-636	275	12	+	+	SYM
ejde-636	275	13	s	s	X
ejde-636	275	14	2	2	NUM
ejde-636	275	15	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	275	16	−	−	PROPN
ejde-636	275	17	µ	µ	PRON
ejde-636	275	18	q	q	X
ejde-636	275	19	s	s	PROPN
ejde-636	275	20	n(q−2	n(q−2	X
ejde-636	275	21	)	)	PUNCT
ejde-636	275	22	4	4	NUM
ejde-636	275	23	∥u∥qq	∥u∥qq	ADP
ejde-636	275	24	−	−	PROPN
ejde-636	275	25	1	1	NUM
ejde-636	275	26	p	p	NOUN
ejde-636	275	27	s	s	X
ejde-636	275	28	n(p−2	n(p−2	PROPN
ejde-636	275	29	)	)	PUNCT
ejde-636	275	30	4	4	NUM
ejde-636	275	31	∥u∥pp	∥u∥pp	NUM
ejde-636	275	32	and	and	CCONJ
ejde-636	275	33	qp	qp	PROPN
ejde-636	275	34	,	,	PUNCT
ejde-636	275	35	q(us	q(us	NOUN
ejde-636	275	36	)	)	PUNCT
ejde-636	275	37	=	=	VERB
ejde-636	276	1	s2∥∆u∥22	s2∥∆u∥22	PROPN
ejde-636	276	2	+	+	CCONJ
ejde-636	276	3	s	s	NOUN
ejde-636	276	4	2	2	NUM
ejde-636	276	5	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	276	6	−	−	PROPN
ejde-636	276	7	µγqs	µγq	NOUN
ejde-636	276	8	n(q−2	n(q−2	X
ejde-636	276	9	)	)	PUNCT
ejde-636	276	10	4	4	NUM
ejde-636	276	11	∥u∥qq	∥u∥qq	ADP
ejde-636	276	12	−	−	PROPN
ejde-636	276	13	γps	γps	ADJ
ejde-636	276	14	n(p−2	n(p−2	PROPN
ejde-636	276	15	)	)	PUNCT
ejde-636	276	16	4	4	NUM
ejde-636	276	17	∥u∥pp	∥u∥pp	NOUN
ejde-636	276	18	.	.	PUNCT
ejde-636	277	1	then	then	ADV
ejde-636	277	2	,	,	PUNCT
ejde-636	277	3	we	we	PRON
ejde-636	277	4	have	have	VERB
ejde-636	277	5	the	the	DET
ejde-636	277	6	following	follow	VERB
ejde-636	277	7	properties	property	NOUN
ejde-636	277	8	for	for	ADP
ejde-636	277	9	ep	ep	PROPN
ejde-636	277	10	,	,	PUNCT
ejde-636	277	11	q(us	q(us	PROPN
ejde-636	277	12	)	)	PUNCT
ejde-636	277	13	and	and	CCONJ
ejde-636	277	14	qp	qp	PROPN
ejde-636	277	15	,	,	PUNCT
ejde-636	277	16	q(us	q(us	PROPN
ejde-636	277	17	)	)	PUNCT
ejde-636	277	18	.	.	PUNCT
ejde-636	278	1	lemma	lemma	PROPN
ejde-636	278	2	4.1	4.1	NUM
ejde-636	278	3	.	.	PUNCT
ejde-636	278	4	.	.	PUNCT
ejde-636	279	1	let	let	VERB
ejde-636	279	2	n	n	PRON
ejde-636	279	3	≥	≥	NOUN
ejde-636	279	4	5	5	NUM
ejde-636	279	5	,	,	PUNCT
ejde-636	279	6	c	c	NOUN
ejde-636	279	7	>	>	X
ejde-636	279	8	0	0	NUM
ejde-636	279	9	,	,	PUNCT
ejde-636	279	10	µ	µ	X
ejde-636	279	11	>	>	X
ejde-636	279	12	0	0	NUM
ejde-636	280	1	and	and	CCONJ
ejde-636	280	2	p	p	NOUN
ejde-636	280	3	≤	≤	NOUN
ejde-636	280	4	q	q	NOUN
ejde-636	281	1	<	<	X
ejde-636	281	2	p	p	X
ejde-636	281	3	<	<	X
ejde-636	281	4	4∗.	4∗.	NUM
ejde-636	281	5	when	when	SCONJ
ejde-636	281	6	q	q	NOUN
ejde-636	281	7	=	=	SYM
ejde-636	281	8	p	p	X
ejde-636	281	9	,	,	PUNCT
ejde-636	281	10	we	we	PRON
ejde-636	281	11	assume	assume	VERB
ejde-636	281	12	that	that	SCONJ
ejde-636	281	13	µc4	µc4	PROPN
ejde-636	281	14	/	/	SYM
ejde-636	281	15	n	n	NOUN
ejde-636	281	16	<	<	X
ejde-636	281	17	n+4	n+4	NUM
ejde-636	281	18	ncq	ncq	NOUN
ejde-636	281	19	n	n	CCONJ
ejde-636	281	20	,	,	PUNCT
ejde-636	281	21	q	q	PROPN
ejde-636	281	22	.	.	PUNCT
ejde-636	282	1	then	then	ADV
ejde-636	282	2	for	for	ADP
ejde-636	282	3	any	any	DET
ejde-636	282	4	u	u	PROPN
ejde-636	282	5	∈	∈	PROPN
ejde-636	282	6	s(c	s(c	PROPN
ejde-636	282	7	)	)	PUNCT
ejde-636	282	8	,	,	PUNCT
ejde-636	282	9	there	there	PRON
ejde-636	282	10	exists	exist	VERB
ejde-636	282	11	a	a	DET
ejde-636	282	12	unique	unique	ADJ
ejde-636	282	13	su	su	NOUN
ejde-636	282	14	∈	∈	PROPN
ejde-636	282	15	(	(	PUNCT
ejde-636	282	16	0,+∞	0,+∞	NUM
ejde-636	282	17	)	)	PUNCT
ejde-636	282	18	such	such	ADJ
ejde-636	282	19	that	that	SCONJ
ejde-636	282	20	usu	usu	PROPN
ejde-636	282	21	∈	∈	PROPN
ejde-636	282	22	qp	qp	PROPN
ejde-636	282	23	,	,	PUNCT
ejde-636	282	24	q(c	q(c	PROPN
ejde-636	282	25	)	)	PUNCT
ejde-636	282	26	and	and	CCONJ
ejde-636	282	27	su	su	PROPN
ejde-636	282	28	is	be	AUX
ejde-636	282	29	the	the	DET
ejde-636	282	30	unique	unique	ADJ
ejde-636	282	31	critical	critical	ADJ
ejde-636	282	32	point	point	NOUN
ejde-636	282	33	of	of	ADP
ejde-636	282	34	ep	ep	PROPN
ejde-636	282	35	,	,	PUNCT
ejde-636	282	36	q(us	q(us	PROPN
ejde-636	282	37	)	)	PUNCT
ejde-636	282	38	such	such	ADJ
ejde-636	282	39	that	that	SCONJ
ejde-636	282	40	ep	ep	PROPN
ejde-636	282	41	,	,	PUNCT
ejde-636	282	42	q(usu	q(usu	PROPN
ejde-636	282	43	)	)	PUNCT
ejde-636	282	44	=	=	SYM
ejde-636	282	45	maxs∈(0,+∞)ep	maxs∈(0,+∞)ep	VERB
ejde-636	282	46	,	,	PUNCT
ejde-636	282	47	q(us	q(us	PROPN
ejde-636	282	48	)	)	PUNCT
ejde-636	282	49	.	.	PUNCT
ejde-636	283	1	the	the	DET
ejde-636	283	2	function	function	NOUN
ejde-636	283	3	u	u	PROPN
ejde-636	283	4	7→	7→	PROPN
ejde-636	283	5	ep	ep	PROPN
ejde-636	283	6	,	,	PUNCT
ejde-636	283	7	q(usu	q(usu	PROPN
ejde-636	283	8	)	)	PUNCT
ejde-636	283	9	is	be	AUX
ejde-636	283	10	concave	concave	VERB
ejde-636	283	11	on	on	ADP
ejde-636	283	12	[	[	X
ejde-636	283	13	su,+∞	su,+∞	X
ejde-636	283	14	)	)	PUNCT
ejde-636	283	15	.	.	PUNCT
ejde-636	284	1	in	in	ADP
ejde-636	284	2	particular	particular	ADJ
ejde-636	284	3	,	,	PUNCT
ejde-636	284	4	if	if	SCONJ
ejde-636	284	5	qp	qp	NOUN
ejde-636	284	6	,	,	PUNCT
ejde-636	284	7	q(u	q(u	ADJ
ejde-636	284	8	)	)	PUNCT
ejde-636	284	9	≤	≤	NUM
ejde-636	284	10	0	0	NUM
ejde-636	284	11	,	,	PUNCT
ejde-636	284	12	then	then	ADV
ejde-636	284	13	su	su	PROPN
ejde-636	284	14	∈	∈	PROPN
ejde-636	284	15	(	(	PUNCT
ejde-636	284	16	0	0	NUM
ejde-636	284	17	,	,	PUNCT
ejde-636	284	18	1	1	NUM
ejde-636	284	19	]	]	PUNCT
ejde-636	284	20	.	.	PUNCT
ejde-636	285	1	moreover	moreover	ADV
ejde-636	285	2	,	,	PUNCT
ejde-636	285	3	the	the	DET
ejde-636	285	4	map	map	NOUN
ejde-636	285	5	u	u	PROPN
ejde-636	285	6	7→	7→	NUM
ejde-636	285	7	su	su	NOUN
ejde-636	285	8	is	be	AUX
ejde-636	285	9	of	of	ADP
ejde-636	285	10	class	class	NOUN
ejde-636	285	11	c1	c1	PROPN
ejde-636	285	12	.	.	PUNCT
ejde-636	286	1	since	since	SCONJ
ejde-636	286	2	the	the	DET
ejde-636	286	3	proof	proof	NOUN
ejde-636	286	4	is	be	AUX
ejde-636	286	5	similar	similar	ADJ
ejde-636	286	6	to	to	ADP
ejde-636	286	7	the	the	DET
ejde-636	286	8	one	one	NUM
ejde-636	286	9	of	of	ADP
ejde-636	286	10	[	[	X
ejde-636	286	11	28	28	NUM
ejde-636	286	12	,	,	PUNCT
ejde-636	286	13	lemma	lemma	PROPN
ejde-636	286	14	3.4	3.4	NUM
ejde-636	286	15	]	]	PUNCT
ejde-636	286	16	,	,	PUNCT
ejde-636	286	17	we	we	PRON
ejde-636	286	18	omit	omit	VERB
ejde-636	286	19	it	it	PRON
ejde-636	286	20	here	here	ADV
ejde-636	286	21	.	.	PUNCT
ejde-636	287	1	under	under	ADP
ejde-636	287	2	the	the	DET
ejde-636	287	3	same	same	ADJ
ejde-636	287	4	assumptions	assumption	NOUN
ejde-636	287	5	described	describe	VERB
ejde-636	287	6	in	in	ADP
ejde-636	287	7	lemma	lemma	PROPN
ejde-636	287	8	4.1	4.1	NUM
ejde-636	287	9	,	,	PUNCT
ejde-636	287	10	we	we	PRON
ejde-636	287	11	can	can	AUX
ejde-636	287	12	obtain	obtain	VERB
ejde-636	287	13	the	the	DET
ejde-636	287	14	following	follow	VERB
ejde-636	287	15	results	result	NOUN
ejde-636	287	16	concerning	concern	VERB
ejde-636	287	17	the	the	DET
ejde-636	287	18	nehari	nehari	NOUN
ejde-636	287	19	-	-	PUNCT
ejde-636	287	20	pohozaev	pohozaev	NOUN
ejde-636	287	21	’s	’s	PART
ejde-636	287	22	type	type	NOUN
ejde-636	287	23	set	set	VERB
ejde-636	287	24	qp	qp	NOUN
ejde-636	287	25	,	,	PUNCT
ejde-636	287	26	q(c	q(c	PROPN
ejde-636	287	27	)	)	PUNCT
ejde-636	287	28	and	and	CCONJ
ejde-636	287	29	the	the	DET
ejde-636	287	30	constrained	constrained	ADJ
ejde-636	287	31	functional	functional	ADJ
ejde-636	287	32	ep	ep	PROPN
ejde-636	287	33	,	,	PUNCT
ejde-636	287	34	q.	q.	PROPN
ejde-636	287	35	lemma	lemma	PROPN
ejde-636	287	36	4.2	4.2	NUM
ejde-636	287	37	.	.	PUNCT
ejde-636	288	1	let	let	VERB
ejde-636	288	2	n	n	PRON
ejde-636	288	3	≥	≥	NOUN
ejde-636	288	4	5	5	NUM
ejde-636	288	5	,	,	PUNCT
ejde-636	288	6	c	c	NOUN
ejde-636	288	7	>	>	X
ejde-636	288	8	0	0	NUM
ejde-636	288	9	,	,	PUNCT
ejde-636	288	10	µ	µ	X
ejde-636	288	11	>	>	X
ejde-636	288	12	0	0	NUM
ejde-636	289	1	and	and	CCONJ
ejde-636	289	2	p	p	NOUN
ejde-636	289	3	≤	≤	NOUN
ejde-636	289	4	q	q	NOUN
ejde-636	290	1	<	<	X
ejde-636	290	2	p	p	X
ejde-636	290	3	<	<	X
ejde-636	290	4	4∗.	4∗.	NUM
ejde-636	290	5	when	when	SCONJ
ejde-636	290	6	q	q	NOUN
ejde-636	290	7	=	=	SYM
ejde-636	290	8	p	p	X
ejde-636	290	9	,	,	PUNCT
ejde-636	290	10	we	we	PRON
ejde-636	290	11	assume	assume	VERB
ejde-636	290	12	that	that	SCONJ
ejde-636	290	13	µc4	µc4	PROPN
ejde-636	290	14	/	/	SYM
ejde-636	290	15	n	n	NOUN
ejde-636	290	16	<	<	X
ejde-636	290	17	n+4	n+4	NUM
ejde-636	290	18	ncq	ncq	NOUN
ejde-636	290	19	n	n	CCONJ
ejde-636	290	20	,	,	PUNCT
ejde-636	290	21	q	q	PROPN
ejde-636	290	22	.	.	PUNCT
ejde-636	291	1	then	then	ADV
ejde-636	291	2	we	we	PRON
ejde-636	291	3	have	have	VERB
ejde-636	291	4	(	(	PUNCT
ejde-636	291	5	1	1	X
ejde-636	291	6	)	)	PUNCT
ejde-636	291	7	qp	qp	NOUN
ejde-636	291	8	,	,	PUNCT
ejde-636	291	9	q(c	q(c	PROPN
ejde-636	291	10	)	)	PUNCT
ejde-636	291	11	̸=	̸=	NOUN
ejde-636	291	12	∅	∅	NOUN
ejde-636	291	13	;	;	PUNCT
ejde-636	291	14	(	(	PUNCT
ejde-636	291	15	2	2	X
ejde-636	291	16	)	)	PUNCT
ejde-636	291	17	infu∈qp	infu∈qp	NOUN
ejde-636	291	18	,	,	PUNCT
ejde-636	291	19	q(c	q(c	PROPN
ejde-636	291	20	)	)	PUNCT
ejde-636	291	21	∥∆u∥22	∥∆u∥22	PUNCT
ejde-636	292	1	+	+	CCONJ
ejde-636	292	2	1	1	NUM
ejde-636	292	3	2∥∇u∥	2∥∇u∥	NUM
ejde-636	292	4	2	2	NUM
ejde-636	292	5	2	2	NUM
ejde-636	292	6	>	>	SYM
ejde-636	292	7	0	0	NUM
ejde-636	292	8	and	and	CCONJ
ejde-636	292	9	infu∈qp	infu∈qp	NOUN
ejde-636	292	10	,	,	PUNCT
ejde-636	292	11	q(c	q(c	PROPN
ejde-636	292	12	)	)	PUNCT
ejde-636	292	13	∥∆u∥22	∥∆u∥22	X
ejde-636	292	14	>	>	X
ejde-636	292	15	0	0	NUM
ejde-636	292	16	;	;	PUNCT
ejde-636	292	17	(	(	PUNCT
ejde-636	292	18	3	3	X
ejde-636	292	19	)	)	PUNCT
ejde-636	292	20	infu∈qp	infu∈qp	NOUN
ejde-636	292	21	,	,	PUNCT
ejde-636	292	22	q(c)ep	q(c)ep	PRON
ejde-636	292	23	,	,	PUNCT
ejde-636	292	24	q(u	q(u	ADJ
ejde-636	292	25	)	)	PUNCT
ejde-636	292	26	>	>	X
ejde-636	292	27	0	0	NUM
ejde-636	292	28	;	;	PUNCT
ejde-636	292	29	(	(	PUNCT
ejde-636	292	30	4	4	X
ejde-636	292	31	)	)	PUNCT
ejde-636	292	32	ep	ep	NOUN
ejde-636	292	33	,	,	PUNCT
ejde-636	292	34	q	q	X
ejde-636	292	35	is	be	AUX
ejde-636	292	36	coercive	coercive	ADJ
ejde-636	292	37	on	on	ADP
ejde-636	292	38	qp	qp	NOUN
ejde-636	292	39	,	,	PUNCT
ejde-636	292	40	q(c	q(c	PROPN
ejde-636	292	41	)	)	PUNCT
ejde-636	292	42	.	.	PUNCT
ejde-636	293	1	proof	proof	NOUN
ejde-636	293	2	.	.	PUNCT
ejde-636	294	1	(	(	PUNCT
ejde-636	294	2	1	1	X
ejde-636	294	3	)	)	PUNCT
ejde-636	294	4	by	by	ADP
ejde-636	294	5	lemma	lemma	PROPN
ejde-636	294	6	4.1	4.1	NUM
ejde-636	294	7	,	,	PUNCT
ejde-636	294	8	for	for	ADP
ejde-636	294	9	any	any	DET
ejde-636	294	10	u	u	PROPN
ejde-636	294	11	∈	∈	PROPN
ejde-636	294	12	s(c	s(c	PROPN
ejde-636	294	13	)	)	PUNCT
ejde-636	294	14	,	,	PUNCT
ejde-636	294	15	there	there	PRON
ejde-636	294	16	always	always	ADV
ejde-636	294	17	exists	exist	VERB
ejde-636	294	18	su	su	PROPN
ejde-636	294	19	>	>	X
ejde-636	294	20	0	0	NUM
ejde-636	294	21	such	such	ADJ
ejde-636	294	22	that	that	SCONJ
ejde-636	294	23	usu	usu	PROPN
ejde-636	294	24	∈	∈	PROPN
ejde-636	294	25	qp	qp	PROPN
ejde-636	294	26	,	,	PUNCT
ejde-636	294	27	q(c	q(c	PROPN
ejde-636	294	28	)	)	PUNCT
ejde-636	294	29	,	,	PUNCT
ejde-636	294	30	it	it	PRON
ejde-636	294	31	follows	follow	VERB
ejde-636	294	32	that	that	SCONJ
ejde-636	294	33	qp	qp	NOUN
ejde-636	294	34	,	,	PUNCT
ejde-636	294	35	q(c	q(c	X
ejde-636	294	36	)	)	PUNCT
ejde-636	294	37	̸=	̸=	PROPN
ejde-636	294	38	∅.	∅.	ADV
ejde-636	294	39	(	(	PUNCT
ejde-636	294	40	2	2	NUM
ejde-636	294	41	)	)	PUNCT
ejde-636	294	42	for	for	ADP
ejde-636	294	43	any	any	DET
ejde-636	294	44	u	u	PROPN
ejde-636	294	45	∈	∈	PROPN
ejde-636	294	46	qp	qp	NOUN
ejde-636	294	47	,	,	PUNCT
ejde-636	294	48	q(c	q(c	PROPN
ejde-636	294	49	)	)	PUNCT
ejde-636	294	50	,	,	PUNCT
ejde-636	294	51	using	use	VERB
ejde-636	294	52	the	the	DET
ejde-636	294	53	gagliardo	gagliardo	NOUN
ejde-636	294	54	-	-	PUNCT
ejde-636	294	55	nirenberg	nirenberg	PROPN
ejde-636	294	56	inequality	inequality	NOUN
ejde-636	294	57	yields	yield	NOUN
ejde-636	294	58	∥∆u∥22	∥∆u∥22	VERB
ejde-636	295	1	+	+	CCONJ
ejde-636	295	2	1	1	NUM
ejde-636	295	3	2	2	NUM
ejde-636	295	4	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	295	5	=	=	PUNCT
ejde-636	295	6	µγq∥u∥qq	µγq∥u∥qq	X
ejde-636	295	7	+	+	CCONJ
ejde-636	295	8	γp∥u∥pp	γp∥u∥pp	PROPN
ejde-636	295	9	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	295	10	dispersion	dispersion	NOUN
ejde-636	295	11	nonlinear	nonlinear	NOUN
ejde-636	295	12	schrödinger	schrödinger	NOUN
ejde-636	295	13	equation	equation	NOUN
ejde-636	295	14	11	11	NUM
ejde-636	295	15	≤	≤	NUM
ejde-636	295	16	µγqc	µγqc	PROPN
ejde-636	295	17	q	q	PROPN
ejde-636	295	18	n	n	CCONJ
ejde-636	295	19	,	,	PUNCT
ejde-636	295	20	q	q	NOUN
ejde-636	295	21	(	(	PUNCT
ejde-636	295	22	√	√	INTJ
ejde-636	295	23	c)q(1−γq)(∥∆u∥2	c)q(1−γq)(∥∆u∥2	NOUN
ejde-636	295	24	+	+	CCONJ
ejde-636	295	25	1	1	NUM
ejde-636	295	26	2	2	NUM
ejde-636	295	27	∥∇u∥22	∥∇u∥22	NUM
ejde-636	295	28	)	)	PUNCT
ejde-636	295	29	qγq	qγq	NOUN
ejde-636	295	30	2	2	NUM
ejde-636	295	31	+	+	CCONJ
ejde-636	295	32	γpc	γpc	X
ejde-636	295	33	p	p	NOUN
ejde-636	295	34	n	n	NOUN
ejde-636	295	35	,	,	PUNCT
ejde-636	295	36	p	p	X
ejde-636	295	37	(	(	PUNCT
ejde-636	295	38	√	√	PROPN
ejde-636	295	39	c)p(1−γp)(∥∆u∥2	c)p(1−γp)(∥∆u∥2	NOUN
ejde-636	295	40	+	+	CCONJ
ejde-636	295	41	1	1	NUM
ejde-636	295	42	2	2	NUM
ejde-636	295	43	∥∇u∥22	∥∇u∥22	NOUN
ejde-636	295	44	)	)	PUNCT
ejde-636	296	1	pγp	pγp	NOUN
ejde-636	296	2	2	2	NUM
ejde-636	296	3	.	.	PUNCT
ejde-636	297	1	if	if	SCONJ
ejde-636	297	2	p	p	X
ejde-636	297	3	<	<	X
ejde-636	297	4	q	q	X
ejde-636	297	5	<	<	X
ejde-636	297	6	p	p	X
ejde-636	297	7	,	,	PUNCT
ejde-636	297	8	then	then	ADV
ejde-636	297	9	pγp	pγp	ADV
ejde-636	297	10	>	>	X
ejde-636	297	11	qγq	qγq	X
ejde-636	297	12	>	>	X
ejde-636	297	13	2	2	NUM
ejde-636	297	14	.	.	PUNCT
ejde-636	298	1	if	if	SCONJ
ejde-636	298	2	p	p	NOUN
ejde-636	298	3	=	=	X
ejde-636	298	4	q	q	X
ejde-636	298	5	<	<	X
ejde-636	298	6	p	p	NOUN
ejde-636	298	7	and	and	CCONJ
ejde-636	298	8	µc4	µc4	PROPN
ejde-636	298	9	/	/	SYM
ejde-636	298	10	n	n	NOUN
ejde-636	298	11	<	<	X
ejde-636	298	12	n+4	n+4	NUM
ejde-636	298	13	ncq	ncq	NOUN
ejde-636	298	14	n	n	CCONJ
ejde-636	298	15	,	,	PUNCT
ejde-636	298	16	q	q	X
ejde-636	298	17	,	,	PUNCT
ejde-636	298	18	then	then	ADV
ejde-636	298	19	pγp	pγp	VERB
ejde-636	298	20	>	>	X
ejde-636	298	21	qγq	qγq	X
ejde-636	298	22	=	=	SYM
ejde-636	298	23	2	2	NUM
ejde-636	298	24	and	and	CCONJ
ejde-636	298	25	µncq	µncq	NOUN
ejde-636	298	26	n	n	CCONJ
ejde-636	298	27	,	,	PUNCT
ejde-636	298	28	q	q	PROPN
ejde-636	298	29	n+4	n+4	NUM
ejde-636	298	30	c4	c4	NOUN
ejde-636	298	31	/	/	SYM
ejde-636	298	32	n	n	CCONJ
ejde-636	298	33	<	<	X
ejde-636	298	34	1	1	NUM
ejde-636	298	35	.	.	PUNCT
ejde-636	298	36	in	in	ADP
ejde-636	298	37	either	either	DET
ejde-636	298	38	case	case	NOUN
ejde-636	298	39	,	,	PUNCT
ejde-636	298	40	there	there	PRON
ejde-636	298	41	exists	exist	VERB
ejde-636	298	42	a	a	DET
ejde-636	298	43	constant	constant	ADJ
ejde-636	298	44	c	c	NOUN
ejde-636	298	45	>	>	X
ejde-636	298	46	0	0	NUM
ejde-636	298	47	such	such	ADJ
ejde-636	298	48	that	that	SCONJ
ejde-636	298	49	∥∆u∥22	∥∆u∥22	VERB
ejde-636	298	50	+	+	NUM
ejde-636	298	51	1	1	NUM
ejde-636	298	52	2∥∇u∥	2∥∇u∥	NUM
ejde-636	298	53	2	2	NUM
ejde-636	298	54	2	2	NUM
ejde-636	298	55	≥	≥	NOUN
ejde-636	298	56	c	c	NOUN
ejde-636	298	57	,	,	PUNCT
ejde-636	298	58	which	which	PRON
ejde-636	298	59	implies	imply	VERB
ejde-636	298	60	infu∈qp	infu∈qp	NOUN
ejde-636	298	61	,	,	PUNCT
ejde-636	298	62	q(c	q(c	X
ejde-636	298	63	)	)	PUNCT
ejde-636	298	64	∥∆u∥22	∥∆u∥22	PUNCT
ejde-636	299	1	+	+	CCONJ
ejde-636	299	2	1	1	NUM
ejde-636	299	3	2∥∇u∥	2∥∇u∥	NUM
ejde-636	299	4	2	2	NUM
ejde-636	299	5	2	2	NUM
ejde-636	299	6	>	>	X
ejde-636	299	7	0	0	NUM
ejde-636	299	8	.	.	PUNCT
ejde-636	300	1	by	by	ADP
ejde-636	300	2	a	a	DET
ejde-636	300	3	similar	similar	ADJ
ejde-636	300	4	argument	argument	NOUN
ejde-636	300	5	,	,	PUNCT
ejde-636	300	6	we	we	PRON
ejde-636	300	7	can	can	AUX
ejde-636	300	8	deduce	deduce	VERB
ejde-636	300	9	that	that	DET
ejde-636	300	10	infu∈qp	infu∈qp	NOUN
ejde-636	300	11	,	,	PUNCT
ejde-636	300	12	q(c	q(c	PROPN
ejde-636	300	13	)	)	PUNCT
ejde-636	300	14	∥∆u∥22	∥∆u∥22	X
ejde-636	300	15	>	>	X
ejde-636	300	16	0	0	X
ejde-636	300	17	.	.	PUNCT
ejde-636	301	1	(	(	PUNCT
ejde-636	301	2	3	3	X
ejde-636	301	3	)	)	PUNCT
ejde-636	301	4	for	for	ADP
ejde-636	301	5	eachy	eachy	NOUN
ejde-636	301	6	u	u	PROPN
ejde-636	301	7	∈	∈	PROPN
ejde-636	301	8	qp	qp	NOUN
ejde-636	301	9	,	,	PUNCT
ejde-636	301	10	q(c	q(c	PROPN
ejde-636	301	11	)	)	PUNCT
ejde-636	301	12	,	,	PUNCT
ejde-636	301	13	we	we	PRON
ejde-636	301	14	have	have	VERB
ejde-636	301	15	ep	ep	NOUN
ejde-636	301	16	,	,	PUNCT
ejde-636	301	17	q(u	q(u	ADJ
ejde-636	301	18	)	)	PUNCT
ejde-636	301	19	=	=	SYM
ejde-636	301	20	qγq	qγq	NOUN
ejde-636	301	21	−	−	NUM
ejde-636	301	22	2	2	NUM
ejde-636	301	23	2qγq	2qγq	NUM
ejde-636	301	24	∥∆u∥22	∥∆u∥22	VERB
ejde-636	301	25	+	+	X
ejde-636	301	26	qγq	qγq	NOUN
ejde-636	301	27	−	−	NOUN
ejde-636	301	28	1	1	NUM
ejde-636	301	29	2qγq	2qγq	PROPN
ejde-636	301	30	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	302	1	+	+	CCONJ
ejde-636	302	2	pγp	pγp	ADJ
ejde-636	302	3	−	−	NOUN
ejde-636	302	4	qγq	qγq	NOUN
ejde-636	302	5	pqγq	pqγq	NOUN
ejde-636	302	6	∥u∥pp	∥u∥pp	PROPN
ejde-636	302	7	.	.	PUNCT
ejde-636	303	1	(	(	PUNCT
ejde-636	303	2	4.1	4.1	NUM
ejde-636	303	3	)	)	PUNCT
ejde-636	303	4	from	from	ADP
ejde-636	303	5	(	(	PUNCT
ejde-636	303	6	2	2	X
ejde-636	303	7	)	)	PUNCT
ejde-636	303	8	it	it	PRON
ejde-636	303	9	follows	follow	VERB
ejde-636	303	10	that	that	SCONJ
ejde-636	303	11	infu∈qp	infu∈qp	NOUN
ejde-636	303	12	,	,	PUNCT
ejde-636	303	13	q(c)ep	q(c)ep	PRON
ejde-636	303	14	,	,	PUNCT
ejde-636	303	15	q(u	q(u	ADJ
ejde-636	303	16	)	)	PUNCT
ejde-636	303	17	>	>	X
ejde-636	303	18	0	0	X
ejde-636	303	19	.	.	PUNCT
ejde-636	304	1	(	(	PUNCT
ejde-636	304	2	4	4	NUM
ejde-636	304	3	)	)	PUNCT
ejde-636	304	4	by	by	ADP
ejde-636	304	5	(	(	PUNCT
ejde-636	304	6	4.1	4.1	NUM
ejde-636	304	7	)	)	PUNCT
ejde-636	304	8	,	,	PUNCT
ejde-636	304	9	it	it	PRON
ejde-636	304	10	is	be	AUX
ejde-636	304	11	easily	easily	ADV
ejde-636	304	12	seen	see	VERB
ejde-636	304	13	that	that	SCONJ
ejde-636	304	14	(	(	PUNCT
ejde-636	304	15	4	4	X
ejde-636	304	16	)	)	PUNCT
ejde-636	304	17	holds	hold	VERB
ejde-636	304	18	.	.	PUNCT
ejde-636	305	1	□	□	PUNCT
ejde-636	305	2	for	for	ADP
ejde-636	305	3	any	any	DET
ejde-636	305	4	fixed	fix	VERB
ejde-636	305	5	c	c	NOUN
ejde-636	305	6	>	>	X
ejde-636	305	7	0	0	PROPN
ejde-636	305	8	,	,	PUNCT
ejde-636	305	9	lemma	lemma	PROPN
ejde-636	305	10	4.2	4.2	NUM
ejde-636	305	11	indicates	indicate	VERB
ejde-636	305	12	that	that	SCONJ
ejde-636	305	13	mp	mp	PROPN
ejde-636	305	14	,	,	PUNCT
ejde-636	305	15	q(c	q(c	PROPN
ejde-636	305	16	)	)	PUNCT
ejde-636	305	17	=	=	SYM
ejde-636	305	18	inf	inf	NOUN
ejde-636	305	19	u∈qp	u∈qp	NOUN
ejde-636	305	20	,	,	PUNCT
ejde-636	305	21	q(c	q(c	PROPN
ejde-636	305	22	)	)	PUNCT
ejde-636	305	23	ep	ep	PROPN
ejde-636	305	24	,	,	PUNCT
ejde-636	305	25	q(u	q(u	NOUN
ejde-636	305	26	)	)	PUNCT
ejde-636	305	27	is	be	AUX
ejde-636	305	28	well	well	ADV
ejde-636	305	29	-	-	PUNCT
ejde-636	305	30	defined	define	VERB
ejde-636	305	31	and	and	CCONJ
ejde-636	305	32	strictly	strictly	ADV
ejde-636	305	33	positive	positive	ADJ
ejde-636	305	34	.	.	PUNCT
ejde-636	306	1	we	we	PRON
ejde-636	306	2	now	now	ADV
ejde-636	306	3	analyze	analyze	VERB
ejde-636	306	4	the	the	DET
ejde-636	306	5	behaviors	behavior	NOUN
ejde-636	306	6	of	of	ADP
ejde-636	306	7	mp	mp	PROPN
ejde-636	306	8	,	,	PUNCT
ejde-636	306	9	q(c	q(c	PROPN
ejde-636	306	10	)	)	PUNCT
ejde-636	306	11	when	when	SCONJ
ejde-636	306	12	c	c	X
ejde-636	306	13	>	>	X
ejde-636	306	14	0	0	NUM
ejde-636	306	15	varies	varie	NOUN
ejde-636	306	16	.	.	PUNCT
ejde-636	307	1	lemma	lemma	PROPN
ejde-636	307	2	4.3	4.3	NUM
ejde-636	307	3	.	.	PUNCT
ejde-636	308	1	let	let	VERB
ejde-636	308	2	p	p	NOUN
ejde-636	308	3	≤	≤	NOUN
ejde-636	309	1	p	p	NOUN
ejde-636	309	2	<	<	X
ejde-636	309	3	q	q	X
ejde-636	309	4	<	<	X
ejde-636	309	5	4∗.	4∗.	NUM
ejde-636	309	6	when	when	SCONJ
ejde-636	309	7	q	q	NOUN
ejde-636	309	8	=	=	SYM
ejde-636	309	9	p	p	X
ejde-636	309	10	,	,	PUNCT
ejde-636	309	11	we	we	PRON
ejde-636	309	12	assume	assume	VERB
ejde-636	309	13	that	that	SCONJ
ejde-636	309	14	µc4	µc4	PROPN
ejde-636	309	15	/	/	SYM
ejde-636	309	16	n	n	NOUN
ejde-636	309	17	<	<	X
ejde-636	309	18	n+4	n+4	NUM
ejde-636	309	19	ncq	ncq	NOUN
ejde-636	309	20	n	n	CCONJ
ejde-636	309	21	,	,	PUNCT
ejde-636	309	22	q	q	PROPN
ejde-636	309	23	.	.	PUNCT
ejde-636	310	1	then	then	ADV
ejde-636	310	2	the	the	DET
ejde-636	310	3	function	function	NOUN
ejde-636	310	4	c	c	PROPN
ejde-636	310	5	7→	7→	PROPN
ejde-636	310	6	mp	mp	PROPN
ejde-636	310	7	,	,	PUNCT
ejde-636	310	8	q(c	q(c	PROPN
ejde-636	310	9	)	)	PUNCT
ejde-636	310	10	is	be	AUX
ejde-636	310	11	continuous	continuous	ADJ
ejde-636	310	12	for	for	ADP
ejde-636	310	13	c	c	PROPN
ejde-636	310	14	∈	∈	PROPN
ejde-636	310	15	(	(	PUNCT
ejde-636	310	16	0,+∞	0,+∞	NUM
ejde-636	310	17	)	)	PUNCT
ejde-636	310	18	.	.	PUNCT
ejde-636	311	1	proof	proof	NOUN
ejde-636	311	2	.	.	PUNCT
ejde-636	312	1	we	we	PRON
ejde-636	312	2	define	define	VERB
ejde-636	312	3	γ(c	γ(c	PROPN
ejde-636	312	4	)	)	PUNCT
ejde-636	312	5	=	=	SYM
ejde-636	312	6	inf	inf	PROPN
ejde-636	312	7	u∈s(c	u∈s(c	NOUN
ejde-636	312	8	)	)	PUNCT
ejde-636	312	9	max	max	PROPN
ejde-636	312	10	s>0	s>0	PROPN
ejde-636	312	11	ep	ep	PROPN
ejde-636	312	12	,	,	PUNCT
ejde-636	312	13	q(us	q(us	PROPN
ejde-636	312	14	)	)	PUNCT
ejde-636	312	15	.	.	PUNCT
ejde-636	313	1	(	(	PUNCT
ejde-636	313	2	4.2	4.2	NUM
ejde-636	313	3	)	)	PUNCT
ejde-636	313	4	to	to	PART
ejde-636	313	5	prove	prove	VERB
ejde-636	313	6	γ(c	γ(c	PROPN
ejde-636	313	7	)	)	PUNCT
ejde-636	313	8	=	=	SYM
ejde-636	313	9	mp	mp	PROPN
ejde-636	313	10	,	,	PUNCT
ejde-636	313	11	q(c	q(c	PROPN
ejde-636	313	12	)	)	PUNCT
ejde-636	313	13	,	,	PUNCT
ejde-636	313	14	for	for	ADP
ejde-636	313	15	any	any	DET
ejde-636	313	16	u	u	PROPN
ejde-636	313	17	∈	∈	PROPN
ejde-636	313	18	qp	qp	NOUN
ejde-636	313	19	,	,	PUNCT
ejde-636	313	20	q(c	q(c	PROPN
ejde-636	313	21	)	)	PUNCT
ejde-636	313	22	we	we	PRON
ejde-636	313	23	have	have	VERB
ejde-636	313	24	ep	ep	NOUN
ejde-636	313	25	,	,	PUNCT
ejde-636	313	26	q(u	q(u	ADJ
ejde-636	313	27	)	)	PUNCT
ejde-636	313	28	=	=	SYM
ejde-636	313	29	maxs>0ep	maxs>0ep	NOUN
ejde-636	313	30	,	,	PUNCT
ejde-636	313	31	q(us	q(us	PROPN
ejde-636	313	32	)	)	PUNCT
ejde-636	313	33	,	,	PUNCT
ejde-636	313	34	which	which	PRON
ejde-636	313	35	implies	imply	VERB
ejde-636	313	36	that	that	PRON
ejde-636	313	37	γ(c	γ(c	PROPN
ejde-636	313	38	)	)	PUNCT
ejde-636	313	39	≤	≤	PROPN
ejde-636	313	40	mp	mp	PROPN
ejde-636	313	41	,	,	PUNCT
ejde-636	313	42	q(c	q(c	PROPN
ejde-636	313	43	)	)	PUNCT
ejde-636	313	44	.	.	PUNCT
ejde-636	314	1	on	on	ADP
ejde-636	314	2	the	the	DET
ejde-636	314	3	other	other	ADJ
ejde-636	314	4	hand	hand	NOUN
ejde-636	314	5	,	,	PUNCT
ejde-636	314	6	for	for	ADP
ejde-636	314	7	any	any	DET
ejde-636	314	8	u	u	PROPN
ejde-636	314	9	∈	∈	PROPN
ejde-636	314	10	s(c	s(c	PROPN
ejde-636	314	11	)	)	PUNCT
ejde-636	314	12	,	,	PUNCT
ejde-636	314	13	by	by	ADP
ejde-636	314	14	lemma	lemma	PROPN
ejde-636	314	15	4.1	4.1	NUM
ejde-636	314	16	there	there	ADV
ejde-636	314	17	exists	exist	VERB
ejde-636	314	18	su	su	PROPN
ejde-636	314	19	>	>	X
ejde-636	314	20	0	0	NUM
ejde-636	315	1	such	such	ADJ
ejde-636	315	2	that	that	SCONJ
ejde-636	315	3	usu	usu	PROPN
ejde-636	315	4	∈	∈	PROPN
ejde-636	315	5	qp	qp	PROPN
ejde-636	315	6	,	,	PUNCT
ejde-636	315	7	q(c	q(c	PROPN
ejde-636	315	8	)	)	PUNCT
ejde-636	315	9	and	and	CCONJ
ejde-636	315	10	maxs>0ep	maxs>0ep	NOUN
ejde-636	315	11	,	,	PUNCT
ejde-636	315	12	q(us	q(us	NOUN
ejde-636	315	13	)	)	PUNCT
ejde-636	316	1	=	=	SYM
ejde-636	316	2	ep	ep	PROPN
ejde-636	316	3	,	,	PUNCT
ejde-636	316	4	q(usu	q(usu	PROPN
ejde-636	316	5	)	)	PUNCT
ejde-636	316	6	≥	≥	PROPN
ejde-636	316	7	mp	mp	PROPN
ejde-636	316	8	,	,	PUNCT
ejde-636	316	9	q(c	q(c	PROPN
ejde-636	316	10	)	)	PUNCT
ejde-636	316	11	.	.	PUNCT
ejde-636	317	1	thus	thus	ADV
ejde-636	317	2	,	,	PUNCT
ejde-636	317	3	we	we	PRON
ejde-636	317	4	have	have	VERB
ejde-636	317	5	γ(c	γ(c	PROPN
ejde-636	317	6	)	)	PUNCT
ejde-636	318	1	=	=	SYM
ejde-636	318	2	mp	mp	PROPN
ejde-636	318	3	,	,	PUNCT
ejde-636	318	4	q(c	q(c	PROPN
ejde-636	318	5	)	)	PUNCT
ejde-636	318	6	.	.	PUNCT
ejde-636	319	1	for	for	ADP
ejde-636	319	2	each	each	DET
ejde-636	319	3	fixed	fix	VERB
ejde-636	319	4	c	c	PROPN
ejde-636	319	5	>	>	X
ejde-636	319	6	0	0	NUM
ejde-636	319	7	,	,	PUNCT
ejde-636	319	8	taking	take	VERB
ejde-636	319	9	{	{	PUNCT
ejde-636	319	10	cn	cn	ADJ
ejde-636	319	11	}	}	PUNCT
ejde-636	319	12	⊂	⊂	PRON
ejde-636	319	13	r+	r+	VERB
ejde-636	319	14	such	such	ADJ
ejde-636	319	15	that	that	SCONJ
ejde-636	319	16	cn	cn	PROPN
ejde-636	319	17	→	→	SYM
ejde-636	319	18	c	c	X
ejde-636	319	19	,	,	PUNCT
ejde-636	319	20	we	we	PRON
ejde-636	319	21	shall	shall	AUX
ejde-636	319	22	prove	prove	VERB
ejde-636	319	23	limn→∞mp	limn→∞mp	PROPN
ejde-636	319	24	,	,	PUNCT
ejde-636	319	25	q(cn	q(cn	PROPN
ejde-636	319	26	)	)	PUNCT
ejde-636	319	27	=	=	SYM
ejde-636	319	28	mp	mp	PROPN
ejde-636	319	29	,	,	PUNCT
ejde-636	319	30	q(c	q(c	PROPN
ejde-636	319	31	)	)	PUNCT
ejde-636	319	32	.	.	PUNCT
ejde-636	320	1	for	for	ADP
ejde-636	320	2	any	any	DET
ejde-636	320	3	ϵ	ϵ	X
ejde-636	320	4	>	>	X
ejde-636	320	5	0	0	NUM
ejde-636	320	6	,	,	PUNCT
ejde-636	320	7	by	by	ADP
ejde-636	320	8	the	the	DET
ejde-636	320	9	definition	definition	NOUN
ejde-636	320	10	of	of	ADP
ejde-636	320	11	mp	mp	PROPN
ejde-636	320	12	,	,	PUNCT
ejde-636	320	13	q(c	q(c	PROPN
ejde-636	320	14	)	)	PUNCT
ejde-636	320	15	there	there	PRON
ejde-636	320	16	exists	exist	VERB
ejde-636	320	17	v	v	ADP
ejde-636	320	18	∈	∈	PROPN
ejde-636	320	19	qp	qp	NOUN
ejde-636	320	20	,	,	PUNCT
ejde-636	320	21	q(c	q(c	PROPN
ejde-636	320	22	)	)	PUNCT
ejde-636	320	23	such	such	ADJ
ejde-636	320	24	that	that	SCONJ
ejde-636	320	25	ep	ep	PROPN
ejde-636	320	26	,	,	PUNCT
ejde-636	320	27	q(v	q(v	PROPN
ejde-636	320	28	)	)	PUNCT
ejde-636	320	29	≤	≤	PROPN
ejde-636	320	30	mp	mp	PROPN
ejde-636	320	31	,	,	PUNCT
ejde-636	320	32	q(c	q(c	PROPN
ejde-636	320	33	)	)	PUNCT
ejde-636	320	34	+	+	CCONJ
ejde-636	320	35	ϵ	ϵ	SYM
ejde-636	320	36	2	2	NUM
ejde-636	320	37	.	.	PUNCT
ejde-636	321	1	set	set	VERB
ejde-636	321	2	vn	vn	INTJ
ejde-636	321	3	:	:	PUNCT
ejde-636	321	4	=	=	PUNCT
ejde-636	321	5	√	√	ADP
ejde-636	321	6	cn	cn	INTJ
ejde-636	321	7	c	c	NOUN
ejde-636	321	8	v	v	ADP
ejde-636	321	9	∈	∈	PROPN
ejde-636	321	10	s(cn	s(cn	NUM
ejde-636	321	11	)	)	PUNCT
ejde-636	321	12	.	.	PUNCT
ejde-636	322	1	from	from	ADP
ejde-636	322	2	the	the	DET
ejde-636	322	3	fact	fact	NOUN
ejde-636	322	4	µc4	µc4	PROPN
ejde-636	322	5	/	/	SYM
ejde-636	322	6	n	n	NOUN
ejde-636	322	7	<	<	X
ejde-636	322	8	n+4	n+4	NUM
ejde-636	322	9	ncp	ncp	PROPN
ejde-636	322	10	n	n	CCONJ
ejde-636	322	11	,	,	PUNCT
ejde-636	322	12	p	p	X
ejde-636	322	13	,	,	PUNCT
ejde-636	322	14	cn	cn	PROPN
ejde-636	322	15	→	→	SYM
ejde-636	322	16	c	c	PROPN
ejde-636	322	17	and	and	CCONJ
ejde-636	322	18	lemma	lemma	PROPN
ejde-636	322	19	2.7	2.7	NUM
ejde-636	322	20	,	,	PUNCT
ejde-636	322	21	it	it	PRON
ejde-636	322	22	follows	follow	VERB
ejde-636	322	23	that	that	SCONJ
ejde-636	322	24	mp	mp	PROPN
ejde-636	322	25	,	,	PUNCT
ejde-636	322	26	q(cn	q(cn	PROPN
ejde-636	322	27	)	)	PUNCT
ejde-636	322	28	≤	≤	NOUN
ejde-636	322	29	max	max	PROPN
ejde-636	322	30	s>0	s>0	PROPN
ejde-636	322	31	ep	ep	PROPN
ejde-636	322	32	,	,	PUNCT
ejde-636	322	33	q((vn)s	q((vn)s	NUM
ejde-636	322	34	)	)	PUNCT
ejde-636	322	35	=	=	SYM
ejde-636	323	1	max	max	PROPN
ejde-636	323	2	s>0	s>0	PROPN
ejde-636	323	3	(	(	PUNCT
ejde-636	323	4	s2	s2	NOUN
ejde-636	323	5	2	2	NUM
ejde-636	323	6	∥∆vn∥22	∥∆vn∥22	PROPN
ejde-636	323	7	+	+	SYM
ejde-636	323	8	s	s	NOUN
ejde-636	323	9	2	2	NUM
ejde-636	323	10	∥∇vn∥22	∥∇vn∥22	PROPN
ejde-636	323	11	−	−	PROPN
ejde-636	323	12	µ	µ	X
ejde-636	323	13	q	q	X
ejde-636	323	14	s	s	PROPN
ejde-636	323	15	n(q−2	n(q−2	X
ejde-636	323	16	)	)	PUNCT
ejde-636	323	17	4	4	NUM
ejde-636	323	18	∥vn∥qq	∥vn∥qq	NOUN
ejde-636	323	19	−	−	NOUN
ejde-636	323	20	1	1	NUM
ejde-636	323	21	p	p	NOUN
ejde-636	323	22	s	s	X
ejde-636	323	23	n(p−2	n(p−2	PROPN
ejde-636	323	24	)	)	PUNCT
ejde-636	323	25	4	4	NUM
ejde-636	323	26	∥vn∥pp	∥vn∥pp	NOUN
ejde-636	323	27	)	)	PUNCT
ejde-636	323	28	≤	≤	NUM
ejde-636	324	1	max	max	PROPN
ejde-636	324	2	s>0	s>0	PROPN
ejde-636	324	3	(	(	PUNCT
ejde-636	324	4	s2	s2	PROPN
ejde-636	324	5	2	2	NUM
ejde-636	324	6	∥∆v∥22	∥∆v∥22	PUNCT
ejde-636	324	7	+	+	SYM
ejde-636	324	8	s	s	NUM
ejde-636	324	9	2	2	NUM
ejde-636	324	10	∥∇v∥22	∥∇v∥22	PROPN
ejde-636	324	11	−	−	PROPN
ejde-636	324	12	µ	µ	PROPN
ejde-636	324	13	q	q	X
ejde-636	324	14	s	s	PROPN
ejde-636	324	15	n(q−2	n(q−2	X
ejde-636	324	16	)	)	PUNCT
ejde-636	324	17	4	4	NUM
ejde-636	324	18	∥v∥qq	∥v∥qq	X
ejde-636	324	19	−	−	NOUN
ejde-636	324	20	1	1	NUM
ejde-636	324	21	p	p	NOUN
ejde-636	324	22	s	s	X
ejde-636	324	23	n(p−2	n(p−2	PROPN
ejde-636	324	24	)	)	PUNCT
ejde-636	324	25	4	4	NUM
ejde-636	324	26	∥v∥pp	∥v∥pp	NOUN
ejde-636	324	27	)	)	PUNCT
ejde-636	325	1	+	+	CCONJ
ejde-636	325	2	ϵ	ϵ	SYM
ejde-636	325	3	2	2	NUM
ejde-636	325	4	=	=	SYM
ejde-636	325	5	max	max	PROPN
ejde-636	325	6	s>0	s>0	PROPN
ejde-636	325	7	ep	ep	PROPN
ejde-636	325	8	,	,	PUNCT
ejde-636	325	9	q((v)s	q((v)s	PROPN
ejde-636	325	10	)	)	PUNCT
ejde-636	326	1	+	+	CCONJ
ejde-636	326	2	ϵ	ϵ	SYM
ejde-636	326	3	2	2	X
ejde-636	326	4	=	=	SYM
ejde-636	326	5	ep	ep	PROPN
ejde-636	326	6	,	,	PUNCT
ejde-636	326	7	q(v	q(v	PROPN
ejde-636	326	8	)	)	PUNCT
ejde-636	327	1	+	+	CCONJ
ejde-636	327	2	ϵ	ϵ	SYM
ejde-636	327	3	2	2	NUM
ejde-636	327	4	≤	≤	NOUN
ejde-636	327	5	mp	mp	NOUN
ejde-636	327	6	,	,	PUNCT
ejde-636	327	7	q(c	q(c	PROPN
ejde-636	327	8	)	)	PUNCT
ejde-636	328	1	+	+	CCONJ
ejde-636	328	2	ϵ.	ϵ.	NOUN
ejde-636	328	3	that	that	ADV
ejde-636	328	4	is	be	AUX
ejde-636	328	5	,	,	PUNCT
ejde-636	328	6	lim	lim	PROPN
ejde-636	328	7	sup	sup	PROPN
ejde-636	328	8	n→∞	n→∞	NUM
ejde-636	328	9	mp	mp	PROPN
ejde-636	328	10	,	,	PUNCT
ejde-636	328	11	q(cn	q(cn	PROPN
ejde-636	328	12	)	)	PUNCT
ejde-636	328	13	≤	≤	PROPN
ejde-636	328	14	mp	mp	PROPN
ejde-636	328	15	,	,	PUNCT
ejde-636	328	16	q(c	q(c	PROPN
ejde-636	328	17	)	)	PUNCT
ejde-636	328	18	.	.	PUNCT
ejde-636	329	1	(	(	PUNCT
ejde-636	329	2	4.3	4.3	NUM
ejde-636	329	3	)	)	PUNCT
ejde-636	329	4	12	12	NUM
ejde-636	329	5	z.	z.	PROPN
ejde-636	329	6	ma	ma	PROPN
ejde-636	329	7	,	,	PUNCT
ejde-636	329	8	x.	x.	PROPN
ejde-636	329	9	chang	chang	PROPN
ejde-636	329	10	,	,	PUNCT
ejde-636	330	1	z.	z.	PROPN
ejde-636	330	2	feng	feng	PROPN
ejde-636	330	3	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	330	4	then	then	ADV
ejde-636	330	5	we	we	PRON
ejde-636	330	6	take	take	VERB
ejde-636	330	7	{	{	PUNCT
ejde-636	330	8	un	un	PROPN
ejde-636	330	9	}	}	PUNCT
ejde-636	330	10	⊂	⊂	PROPN
ejde-636	330	11	qp	qp	PROPN
ejde-636	330	12	,	,	PUNCT
ejde-636	330	13	q(cn	q(cn	PROPN
ejde-636	330	14	)	)	PUNCT
ejde-636	331	1	such	such	ADJ
ejde-636	331	2	that	that	SCONJ
ejde-636	331	3	ep	ep	PROPN
ejde-636	331	4	,	,	PUNCT
ejde-636	331	5	q(un	q(un	PROPN
ejde-636	331	6	)	)	PUNCT
ejde-636	331	7	≤	≤	PROPN
ejde-636	331	8	mp	mp	PROPN
ejde-636	331	9	,	,	PUNCT
ejde-636	331	10	q(cn	q(cn	PROPN
ejde-636	331	11	)	)	PUNCT
ejde-636	331	12	+	+	CCONJ
ejde-636	331	13	ϵ	ϵ	ADP
ejde-636	331	14	2	2	NUM
ejde-636	331	15	.	.	PUNCT
ejde-636	332	1	(	(	PUNCT
ejde-636	332	2	4.4	4.4	NUM
ejde-636	332	3	)	)	PUNCT
ejde-636	332	4	in	in	ADP
ejde-636	332	5	view	view	NOUN
ejde-636	332	6	of	of	ADP
ejde-636	332	7	qp	qp	NOUN
ejde-636	332	8	,	,	PUNCT
ejde-636	332	9	q(un	q(un	PROPN
ejde-636	332	10	)	)	PUNCT
ejde-636	332	11	=	=	SYM
ejde-636	332	12	0	0	NUM
ejde-636	332	13	,	,	PUNCT
ejde-636	332	14	for	for	ADP
ejde-636	332	15	n	n	CCONJ
ejde-636	332	16	large	large	ADJ
ejde-636	332	17	enough	enough	ADV
ejde-636	332	18	,	,	PUNCT
ejde-636	332	19	from	from	ADP
ejde-636	332	20	(	(	PUNCT
ejde-636	332	21	4.3	4.3	NUM
ejde-636	332	22	)	)	PUNCT
ejde-636	332	23	and	and	CCONJ
ejde-636	332	24	(	(	PUNCT
ejde-636	332	25	4.4	4.4	NUM
ejde-636	332	26	)	)	PUNCT
ejde-636	332	27	it	it	PRON
ejde-636	332	28	follows	follow	VERB
ejde-636	332	29	that(1	that(1	PROPN
ejde-636	332	30	2	2	NUM
ejde-636	332	31	−	−	NUM
ejde-636	332	32	1	1	NUM
ejde-636	332	33	qγq	qγq	NOUN
ejde-636	332	34	)	)	PUNCT
ejde-636	332	35	∥∆un∥22	∥∆un∥22	PUNCT
ejde-636	333	1	+	+	CCONJ
ejde-636	333	2	1	1	NUM
ejde-636	333	3	2	2	NUM
ejde-636	333	4	(	(	PUNCT
ejde-636	333	5	1−	1−	NUM
ejde-636	333	6	1	1	NUM
ejde-636	333	7	qγq	qγq	NOUN
ejde-636	333	8	)	)	PUNCT
ejde-636	334	1	∥∇un∥22	∥∇un∥22	PROPN
ejde-636	335	1	+	+	CCONJ
ejde-636	335	2	(	(	PUNCT
ejde-636	335	3	γp	γp	PROPN
ejde-636	335	4	qγq	qγq	NOUN
ejde-636	335	5	−	−	NUM
ejde-636	335	6	1	1	NUM
ejde-636	335	7	p	p	NOUN
ejde-636	335	8	)	)	PUNCT
ejde-636	335	9	∥un∥pp	∥un∥pp	NOUN
ejde-636	335	10	≤	≤	NOUN
ejde-636	335	11	ep	ep	PROPN
ejde-636	335	12	,	,	PUNCT
ejde-636	335	13	q(un	q(un	PROPN
ejde-636	335	14	)	)	PUNCT
ejde-636	335	15	≤	≤	PROPN
ejde-636	335	16	mp	mp	PROPN
ejde-636	335	17	,	,	PUNCT
ejde-636	335	18	q(cn	q(cn	PROPN
ejde-636	335	19	)	)	PUNCT
ejde-636	336	1	+	+	CCONJ
ejde-636	336	2	ϵ	ϵ	SYM
ejde-636	336	3	2	2	NUM
ejde-636	336	4	≤	≤	NOUN
ejde-636	336	5	mp	mp	NOUN
ejde-636	336	6	,	,	PUNCT
ejde-636	336	7	q(c	q(c	PROPN
ejde-636	336	8	)	)	PUNCT
ejde-636	337	1	+	+	CCONJ
ejde-636	337	2	3ϵ	3ϵ	NOUN
ejde-636	337	3	4	4	NUM
ejde-636	337	4	.	.	PUNCT
ejde-636	338	1	if	if	SCONJ
ejde-636	338	2	pγp	pγp	ADV
ejde-636	338	3	>	>	X
ejde-636	338	4	qγq	qγq	X
ejde-636	338	5	>	>	X
ejde-636	339	1	2	2	NUM
ejde-636	339	2	,	,	PUNCT
ejde-636	339	3	we	we	PRON
ejde-636	339	4	can	can	AUX
ejde-636	339	5	derive	derive	VERB
ejde-636	339	6	that	that	SCONJ
ejde-636	339	7	{	{	PUNCT
ejde-636	339	8	un	un	PROPN
ejde-636	339	9	}	}	PUNCT
ejde-636	339	10	is	be	AUX
ejde-636	339	11	bounded	bound	VERB
ejde-636	339	12	in	in	ADP
ejde-636	339	13	h2(rn	h2(rn	PROPN
ejde-636	339	14	)	)	PUNCT
ejde-636	339	15	.	.	PUNCT
ejde-636	340	1	if	if	SCONJ
ejde-636	340	2	pγp	pγp	ADV
ejde-636	340	3	>	>	X
ejde-636	340	4	qγq	qγq	X
ejde-636	340	5	=	=	SYM
ejde-636	340	6	2	2	NUM
ejde-636	340	7	,	,	PUNCT
ejde-636	340	8	recalling	recall	VERB
ejde-636	340	9	lemma	lemma	PROPN
ejde-636	340	10	4.2	4.2	NUM
ejde-636	340	11	(	(	PUNCT
ejde-636	340	12	2	2	NUM
ejde-636	340	13	)	)	PUNCT
ejde-636	340	14	,	,	PUNCT
ejde-636	340	15	we	we	PRON
ejde-636	340	16	can	can	AUX
ejde-636	340	17	see	see	VERB
ejde-636	340	18	the	the	DET
ejde-636	340	19	same	same	ADJ
ejde-636	340	20	result	result	NOUN
ejde-636	340	21	.	.	PUNCT
ejde-636	341	1	without	without	ADP
ejde-636	341	2	loss	loss	NOUN
ejde-636	341	3	of	of	ADP
ejde-636	341	4	generality	generality	NOUN
ejde-636	341	5	,	,	PUNCT
ejde-636	341	6	as	as	ADP
ejde-636	341	7	n→	n→	ADV
ejde-636	341	8	∞	∞	NUM
ejde-636	341	9	we	we	PRON
ejde-636	341	10	assume	assume	VERB
ejde-636	341	11	that	that	SCONJ
ejde-636	341	12	∥∆un∥22	∥∆un∥22	PROPN
ejde-636	341	13	→	→	SYM
ejde-636	341	14	c1	c1	PROPN
ejde-636	341	15	,	,	PUNCT
ejde-636	341	16	∥∇un∥22	∥∇un∥22	PROPN
ejde-636	341	17	→	→	SYM
ejde-636	341	18	c2	c2	PROPN
ejde-636	341	19	,	,	PUNCT
ejde-636	341	20	∥un∥qq	∥un∥qq	PROPN
ejde-636	341	21	→	→	SYM
ejde-636	341	22	c3	c3	PROPN
ejde-636	341	23	,	,	PUNCT
ejde-636	341	24	∥un∥pp	∥un∥pp	PROPN
ejde-636	341	25	→	→	SYM
ejde-636	341	26	c4	c4	NOUN
ejde-636	341	27	.	.	PUNCT
ejde-636	342	1	if	if	SCONJ
ejde-636	342	2	follows	follow	VERB
ejde-636	342	3	from	from	ADP
ejde-636	342	4	lemma	lemma	PROPN
ejde-636	342	5	4.2	4.2	NUM
ejde-636	342	6	(	(	PUNCT
ejde-636	342	7	2	2	NUM
ejde-636	342	8	)	)	PUNCT
ejde-636	342	9	that	that	PRON
ejde-636	342	10	c1	c1	PROPN
ejde-636	342	11	>	>	X
ejde-636	342	12	0	0	PROPN
ejde-636	342	13	,	,	PUNCT
ejde-636	342	14	c2	c2	PROPN
ejde-636	342	15	≥	≥	NUM
ejde-636	342	16	0	0	NUM
ejde-636	342	17	,	,	PUNCT
ejde-636	342	18	and	and	CCONJ
ejde-636	342	19	c3	c3	PROPN
ejde-636	342	20	≥	≥	PROPN
ejde-636	342	21	0	0	NUM
ejde-636	342	22	,	,	PUNCT
ejde-636	342	23	c4	c4	NOUN
ejde-636	342	24	≥	≥	NOUN
ejde-636	342	25	0	0	NUM
ejde-636	342	26	with	with	ADP
ejde-636	342	27	c3	c3	PROPN
ejde-636	342	28	+	+	CCONJ
ejde-636	342	29	c4	c4	PROPN
ejde-636	342	30	>	>	X
ejde-636	342	31	0	0	X
ejde-636	342	32	.	.	PUNCT
ejde-636	343	1	let	let	VERB
ejde-636	343	2	ũn	ũn	PRON
ejde-636	343	3	:	:	PUNCT
ejde-636	343	4	=	=	SYM
ejde-636	343	5	√	√	PROPN
ejde-636	343	6	c	c	PROPN
ejde-636	343	7	cn	cn	PROPN
ejde-636	343	8	un	un	PROPN
ejde-636	343	9	.	.	PROPN
ejde-636	343	10	clearly	clearly	ADV
ejde-636	343	11	,	,	PUNCT
ejde-636	343	12	ũn	ũn	NOUN
ejde-636	343	13	∈	∈	PROPN
ejde-636	343	14	s(c	s(c	NOUN
ejde-636	343	15	)	)	PUNCT
ejde-636	343	16	.	.	PUNCT
ejde-636	344	1	from	from	ADP
ejde-636	344	2	lemma	lemma	PROPN
ejde-636	344	3	2.7	2.7	NUM
ejde-636	344	4	it	it	PRON
ejde-636	344	5	follows	follow	VERB
ejde-636	344	6	that	that	SCONJ
ejde-636	344	7	mp	mp	NOUN
ejde-636	344	8	,	,	PUNCT
ejde-636	344	9	q(c	q(c	PROPN
ejde-636	344	10	)	)	PUNCT
ejde-636	344	11	≤	≤	PUNCT
ejde-636	344	12	max	max	PROPN
ejde-636	344	13	s>0	s>0	PROPN
ejde-636	344	14	ep	ep	PROPN
ejde-636	344	15	,	,	PUNCT
ejde-636	344	16	q((ũn)s	q((ũn)s	NOUN
ejde-636	344	17	)	)	PUNCT
ejde-636	345	1	=	=	SYM
ejde-636	345	2	max	max	PROPN
ejde-636	345	3	s>0	s>0	PROPN
ejde-636	346	1	[	[	X
ejde-636	346	2	s2	s2	NOUN
ejde-636	346	3	2	2	NUM
ejde-636	346	4	(	(	PUNCT
ejde-636	346	5	c	c	PROPN
ejde-636	346	6	cn	cn	PROPN
ejde-636	346	7	)	)	PUNCT
ejde-636	346	8	∥∆un∥22	∥∆un∥22	PROPN
ejde-636	347	1	+	+	SYM
ejde-636	347	2	s	s	X
ejde-636	347	3	2	2	NUM
ejde-636	347	4	(	(	PUNCT
ejde-636	347	5	c	c	PROPN
ejde-636	347	6	cn	cn	PROPN
ejde-636	347	7	)	)	PUNCT
ejde-636	347	8	∥∇un∥22	∥∇un∥22	PROPN
ejde-636	347	9	−	−	PROPN
ejde-636	347	10	µ	µ	PRON
ejde-636	347	11	q	q	X
ejde-636	347	12	s	s	PROPN
ejde-636	347	13	n(q−2	n(q−2	PRON
ejde-636	347	14	)	)	PUNCT
ejde-636	347	15	4	4	NUM
ejde-636	347	16	(	(	PUNCT
ejde-636	347	17	c	c	NOUN
ejde-636	347	18	cn	cn	PROPN
ejde-636	347	19	)	)	PUNCT
ejde-636	347	20	q	q	PROPN
ejde-636	347	21	2	2	NUM
ejde-636	347	22	∥un∥qq	∥un∥qq	NOUN
ejde-636	347	23	−	−	NOUN
ejde-636	347	24	1	1	NUM
ejde-636	347	25	p	p	NOUN
ejde-636	347	26	s	s	X
ejde-636	347	27	n(p−2	n(p−2	PROPN
ejde-636	347	28	)	)	PUNCT
ejde-636	347	29	4	4	NUM
ejde-636	347	30	(	(	PUNCT
ejde-636	347	31	c	c	NOUN
ejde-636	347	32	cn	cn	PROPN
ejde-636	347	33	)	)	PUNCT
ejde-636	347	34	p/2∥un∥pp	p/2∥un∥pp	PROPN
ejde-636	347	35	]	]	PUNCT
ejde-636	347	36	≤	≤	NUM
ejde-636	347	37	max	max	PROPN
ejde-636	347	38	s>0	s>0	PROPN
ejde-636	347	39	(	(	PUNCT
ejde-636	347	40	s2	s2	PROPN
ejde-636	347	41	2	2	NUM
ejde-636	347	42	∥∆un∥22	∥∆un∥22	PUNCT
ejde-636	347	43	+	+	SYM
ejde-636	347	44	s	s	X
ejde-636	347	45	2	2	NUM
ejde-636	347	46	∥∇un∥22	∥∇un∥22	PROPN
ejde-636	347	47	−	−	PROPN
ejde-636	347	48	µ	µ	PROPN
ejde-636	347	49	q	q	X
ejde-636	347	50	s	s	PROPN
ejde-636	347	51	n(q−2	n(q−2	X
ejde-636	347	52	)	)	PUNCT
ejde-636	347	53	4	4	NUM
ejde-636	347	54	∥un∥qq	∥un∥qq	NOUN
ejde-636	347	55	−	−	NOUN
ejde-636	347	56	1	1	NUM
ejde-636	347	57	p	p	NOUN
ejde-636	347	58	s	s	X
ejde-636	347	59	n(p−2	n(p−2	PROPN
ejde-636	347	60	)	)	PUNCT
ejde-636	347	61	4	4	NUM
ejde-636	347	62	∥un∥pp	∥un∥pp	NOUN
ejde-636	347	63	)	)	PUNCT
ejde-636	348	1	+	+	CCONJ
ejde-636	348	2	3ϵ	3ϵ	NOUN
ejde-636	348	3	4	4	NUM
ejde-636	348	4	=	=	SYM
ejde-636	348	5	max	max	PROPN
ejde-636	348	6	s>0	s>0	PROPN
ejde-636	348	7	ep	ep	PROPN
ejde-636	348	8	,	,	PUNCT
ejde-636	348	9	q((un)s	q((un)s	PROPN
ejde-636	348	10	)	)	PUNCT
ejde-636	349	1	+	+	NUM
ejde-636	349	2	3ϵ	3ϵ	NOUN
ejde-636	349	3	4	4	NUM
ejde-636	349	4	=	=	SYM
ejde-636	349	5	ep	ep	PROPN
ejde-636	349	6	,	,	PUNCT
ejde-636	349	7	q(un	q(un	PROPN
ejde-636	349	8	)	)	PUNCT
ejde-636	350	1	+	+	NUM
ejde-636	350	2	3ϵ	3ϵ	NOUN
ejde-636	350	3	4	4	NUM
ejde-636	350	4	≤	≤	NOUN
ejde-636	350	5	mp	mp	PROPN
ejde-636	350	6	,	,	PUNCT
ejde-636	350	7	q(cn	q(cn	PROPN
ejde-636	350	8	)	)	PUNCT
ejde-636	351	1	+	+	CCONJ
ejde-636	351	2	ϵ.	ϵ.	NOUN
ejde-636	351	3	that	that	ADV
ejde-636	351	4	is	be	AUX
ejde-636	351	5	,	,	PUNCT
ejde-636	351	6	mp	mp	PROPN
ejde-636	351	7	,	,	PUNCT
ejde-636	351	8	q(c	q(c	PROPN
ejde-636	351	9	)	)	PUNCT
ejde-636	351	10	≤	≤	PROPN
ejde-636	351	11	lim	lim	PROPN
ejde-636	351	12	inf	inf	PROPN
ejde-636	351	13	n→∞	n→∞	NUM
ejde-636	351	14	mp	mp	PROPN
ejde-636	351	15	,	,	PUNCT
ejde-636	351	16	q(cn	q(cn	PROPN
ejde-636	351	17	)	)	PUNCT
ejde-636	351	18	.	.	PUNCT
ejde-636	352	1	hence	hence	ADV
ejde-636	352	2	,	,	PUNCT
ejde-636	352	3	we	we	PRON
ejde-636	352	4	arrive	arrive	VERB
ejde-636	352	5	at	at	ADP
ejde-636	352	6	the	the	DET
ejde-636	352	7	desired	desire	VERB
ejde-636	352	8	result	result	NOUN
ejde-636	352	9	.	.	PUNCT
ejde-636	353	1	□	□	PUNCT
ejde-636	353	2	lemma	lemma	PROPN
ejde-636	353	3	4.4	4.4	NUM
ejde-636	353	4	.	.	PUNCT
ejde-636	354	1	let	let	VERB
ejde-636	354	2	p	p	NOUN
ejde-636	354	3	≤	≤	NOUN
ejde-636	355	1	p	p	NOUN
ejde-636	355	2	<	<	X
ejde-636	355	3	q	q	X
ejde-636	355	4	<	<	X
ejde-636	355	5	4∗.	4∗.	NUM
ejde-636	355	6	when	when	SCONJ
ejde-636	355	7	q	q	NOUN
ejde-636	355	8	=	=	SYM
ejde-636	355	9	p	p	X
ejde-636	355	10	,	,	PUNCT
ejde-636	355	11	we	we	PRON
ejde-636	355	12	assume	assume	VERB
ejde-636	355	13	that	that	SCONJ
ejde-636	355	14	µc4	µc4	PROPN
ejde-636	355	15	/	/	SYM
ejde-636	355	16	n	n	NOUN
ejde-636	355	17	<	<	X
ejde-636	355	18	n+4	n+4	NUM
ejde-636	355	19	ncq	ncq	NOUN
ejde-636	355	20	n	n	CCONJ
ejde-636	355	21	,	,	PUNCT
ejde-636	355	22	q	q	PROPN
ejde-636	355	23	.	.	PUNCT
ejde-636	356	1	then	then	ADV
ejde-636	356	2	the	the	DET
ejde-636	356	3	function	function	NOUN
ejde-636	356	4	c	c	PROPN
ejde-636	356	5	7→	7→	PROPN
ejde-636	356	6	mp	mp	PROPN
ejde-636	356	7	,	,	PUNCT
ejde-636	356	8	q(c	q(c	PROPN
ejde-636	356	9	)	)	PUNCT
ejde-636	356	10	is	be	AUX
ejde-636	356	11	non	non	ADJ
ejde-636	356	12	-	-	ADJ
ejde-636	356	13	increasing	increase	VERB
ejde-636	356	14	for	for	ADP
ejde-636	356	15	c	c	PROPN
ejde-636	356	16	∈	∈	PROPN
ejde-636	356	17	(	(	PUNCT
ejde-636	356	18	0,+∞	0,+∞	NUM
ejde-636	356	19	)	)	PUNCT
ejde-636	356	20	.	.	PUNCT
ejde-636	357	1	proof	proof	NOUN
ejde-636	357	2	.	.	PUNCT
ejde-636	358	1	for	for	ADP
ejde-636	358	2	0	0	NUM
ejde-636	358	3	<	<	X
ejde-636	358	4	c1	c1	PROPN
ejde-636	358	5	<	<	X
ejde-636	358	6	c2	c2	PROPN
ejde-636	358	7	<	<	X
ejde-636	358	8	+	+	PROPN
ejde-636	358	9	∞	∞	PROPN
ejde-636	358	10	,	,	PUNCT
ejde-636	358	11	we	we	PRON
ejde-636	358	12	shall	shall	AUX
ejde-636	358	13	prove	prove	VERB
ejde-636	358	14	that	that	SCONJ
ejde-636	358	15	mp	mp	NOUN
ejde-636	358	16	,	,	PUNCT
ejde-636	358	17	q(c2	q(c2	ADJ
ejde-636	358	18	)	)	PUNCT
ejde-636	358	19	≤	≤	PROPN
ejde-636	358	20	mp	mp	PROPN
ejde-636	358	21	,	,	PUNCT
ejde-636	358	22	q(c1	q(c1	PROPN
ejde-636	358	23	)	)	PUNCT
ejde-636	358	24	.	.	PUNCT
ejde-636	359	1	according	accord	VERB
ejde-636	359	2	to	to	ADP
ejde-636	359	3	the	the	DET
ejde-636	359	4	definition	definition	NOUN
ejde-636	359	5	of	of	ADP
ejde-636	359	6	γ(c	γ(c	PROPN
ejde-636	359	7	)	)	PUNCT
ejde-636	359	8	in	in	ADP
ejde-636	359	9	4.2	4.2	NUM
ejde-636	359	10	,	,	PUNCT
ejde-636	359	11	for	for	ADP
ejde-636	359	12	any	any	DET
ejde-636	359	13	ϵ	ϵ	X
ejde-636	359	14	>	>	X
ejde-636	359	15	0	0	PUNCT
ejde-636	359	16	there	there	PRON
ejde-636	359	17	exists	exist	VERB
ejde-636	359	18	u1	u1	PROPN
ejde-636	359	19	∈	∈	PROPN
ejde-636	359	20	qp	qp	PROPN
ejde-636	359	21	,	,	PUNCT
ejde-636	359	22	q(c1	q(c1	PROPN
ejde-636	359	23	)	)	PUNCT
ejde-636	359	24	such	such	ADJ
ejde-636	359	25	that	that	SCONJ
ejde-636	359	26	ep	ep	PROPN
ejde-636	359	27	,	,	PUNCT
ejde-636	359	28	q(u1	q(u1	NOUN
ejde-636	359	29	)	)	PUNCT
ejde-636	359	30	≤	≤	NOUN
ejde-636	359	31	mp	mp	PROPN
ejde-636	359	32	,	,	PUNCT
ejde-636	359	33	q(c1	q(c1	PROPN
ejde-636	359	34	)	)	PUNCT
ejde-636	360	1	+	+	CCONJ
ejde-636	360	2	ϵ	ϵ	SYM
ejde-636	360	3	2	2	NUM
ejde-636	360	4	and	and	CCONJ
ejde-636	360	5	max	max	PROPN
ejde-636	360	6	λ>0	λ>0	PROPN
ejde-636	360	7	ep	ep	PROPN
ejde-636	360	8	,	,	PUNCT
ejde-636	360	9	q((u1)λ	q((u1)λ	NOUN
ejde-636	360	10	)	)	PUNCT
ejde-636	360	11	=	=	SYM
ejde-636	360	12	e(u1	e(u1	NOUN
ejde-636	360	13	)	)	PUNCT
ejde-636	360	14	.	.	PUNCT
ejde-636	361	1	for	for	ADP
ejde-636	361	2	κ	κ	PROPN
ejde-636	361	3	>	>	X
ejde-636	361	4	0	0	PUNCT
ejde-636	362	1	and	and	CCONJ
ejde-636	362	2	λ	λ	PROPN
ejde-636	362	3	∈	∈	PROPN
ejde-636	362	4	(	(	PUNCT
ejde-636	362	5	0	0	NUM
ejde-636	362	6	,	,	PUNCT
ejde-636	362	7	1	1	NUM
ejde-636	362	8	)	)	PUNCT
ejde-636	362	9	,	,	PUNCT
ejde-636	362	10	we	we	PRON
ejde-636	362	11	define	define	VERB
ejde-636	362	12	wκ	wκ	PRON
ejde-636	363	1	λ	λ	X
ejde-636	364	1	:	:	PUNCT
ejde-636	365	1	=	=	SYM
ejde-636	365	2	uκ1	uκ1	PROPN
ejde-636	365	3	+	+	CCONJ
ejde-636	365	4	(	(	PUNCT
ejde-636	365	5	vκ0	vκ0	NOUN
ejde-636	365	6	)	)	PUNCT
ejde-636	365	7	λ	λ	NOUN
ejde-636	365	8	.	.	PUNCT
ejde-636	366	1	we	we	PRON
ejde-636	366	2	choose	choose	VERB
ejde-636	366	3	uκ1	uκ1	PROPN
ejde-636	366	4	∈	∈	PROPN
ejde-636	366	5	h2(rn	h2(rn	PROPN
ejde-636	366	6	)	)	PUNCT
ejde-636	366	7	such	such	ADJ
ejde-636	366	8	that	that	DET
ejde-636	366	9	suppuκ1	suppuκ1	NOUN
ejde-636	367	1	⊂	⊂	X
ejde-636	367	2	b	b	PROPN
ejde-636	367	3	1	1	NUM
ejde-636	367	4	κ	κ	NOUN
ejde-636	367	5	(	(	PUNCT
ejde-636	367	6	0	0	NUM
ejde-636	367	7	)	)	PUNCT
ejde-636	367	8	and	and	CCONJ
ejde-636	367	9	∥uκ1	∥uκ1	VERB
ejde-636	367	10	−	−	PROPN
ejde-636	367	11	u1∥	u1∥	NOUN
ejde-636	367	12	=	=	SYM
ejde-636	367	13	o(κ	o(κ	PROPN
ejde-636	367	14	)	)	PUNCT
ejde-636	367	15	,	,	PUNCT
ejde-636	367	16	while	while	SCONJ
ejde-636	367	17	vδ0	vδ0	NOUN
ejde-636	367	18	:	:	PUNCT
ejde-636	367	19	=	=	SYM
ejde-636	367	20	(	(	PUNCT
ejde-636	367	21	c2	c2	PROPN
ejde-636	367	22	−	−	PROPN
ejde-636	367	23	∥uκ1∥22)1/2	∥uκ1∥22)1/2	NUM
ejde-636	367	24	vκ	vκ	PROPN
ejde-636	367	25	∥vκ∥2	∥vκ∥2	VERB
ejde-636	367	26	,	,	PUNCT
ejde-636	367	27	where	where	SCONJ
ejde-636	367	28	vκ	vκ	ADP
ejde-636	367	29	∈	∈	PROPN
ejde-636	367	30	c∞	c∞	PROPN
ejde-636	367	31	0	0	NUM
ejde-636	367	32	(	(	PUNCT
ejde-636	367	33	rn	rn	PROPN
ejde-636	367	34	)	)	PUNCT
ejde-636	367	35	such	such	ADJ
ejde-636	367	36	that	that	SCONJ
ejde-636	367	37	supp	supp	PROPN
ejde-636	367	38	vκ	vκ	PROPN
ejde-636	367	39	⊂	⊂	PROPN
ejde-636	367	40	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	367	41	dispersion	dispersion	NOUN
ejde-636	367	42	nonlinear	nonlinear	NOUN
ejde-636	367	43	schrödinger	schrödinger	NOUN
ejde-636	367	44	equation	equation	NOUN
ejde-636	367	45	13	13	NUM
ejde-636	367	46	b	b	SYM
ejde-636	367	47	2	2	NUM
ejde-636	367	48	κ+1(0)\b	κ+1(0)\b	PROPN
ejde-636	367	49	2	2	NUM
ejde-636	367	50	κ	κ	NOUN
ejde-636	367	51	(	(	PUNCT
ejde-636	367	52	0	0	NUM
ejde-636	367	53	)	)	PUNCT
ejde-636	367	54	.	.	PUNCT
ejde-636	368	1	it	it	PRON
ejde-636	368	2	is	be	AUX
ejde-636	368	3	obvious	obvious	ADJ
ejde-636	368	4	that	that	SCONJ
ejde-636	368	5	dist(supp(vκ0	dist(supp(vκ0	NOUN
ejde-636	368	6	)	)	PUNCT
ejde-636	368	7	λ	λ	PROPN
ejde-636	368	8	,	,	PUNCT
ejde-636	368	9	suppu	suppu	NOUN
ejde-636	368	10	κ	κ	NOUN
ejde-636	368	11	1	1	NUM
ejde-636	368	12	)	)	PUNCT
ejde-636	368	13	≥	≥	NOUN
ejde-636	368	14	1	1	NUM
ejde-636	368	15	κ	κ	NOUN
ejde-636	368	16	(	(	PUNCT
ejde-636	368	17	2√	2√	PROPN
ejde-636	368	18	λ	λ	NOUN
ejde-636	368	19	−	−	NOUN
ejde-636	368	20	1	1	NUM
ejde-636	368	21	)	)	PUNCT
ejde-636	368	22	>	>	X
ejde-636	369	1	0	0	X
ejde-636	369	2	.	.	PUNCT
ejde-636	370	1	hence	hence	ADV
ejde-636	370	2	,	,	PUNCT
ejde-636	370	3	∥wκ	∥wκ	PROPN
ejde-636	370	4	λ∥22	λ∥22	PROPN
ejde-636	370	5	=	=	SYM
ejde-636	370	6	c2	c2	PROPN
ejde-636	370	7	.	.	PUNCT
ejde-636	371	1	by	by	ADP
ejde-636	371	2	a	a	DET
ejde-636	371	3	standard	standard	ADJ
ejde-636	371	4	argument	argument	NOUN
ejde-636	371	5	,	,	PUNCT
ejde-636	371	6	as	as	ADP
ejde-636	371	7	λ	λ	PROPN
ejde-636	371	8	,	,	PUNCT
ejde-636	371	9	κ→	κ→	PROPN
ejde-636	371	10	0	0	NUM
ejde-636	371	11	,	,	PUNCT
ejde-636	371	12	we	we	PRON
ejde-636	371	13	derive	derive	VERB
ejde-636	371	14	∥∆wκ	∥∆wκ	PROPN
ejde-636	371	15	λ∥22	λ∥22	PROPN
ejde-636	371	16	→	→	SYM
ejde-636	371	17	∥∆u1∥22	∥∆u1∥22	PROPN
ejde-636	371	18	,	,	PUNCT
ejde-636	371	19	∥∇wκ	∥∇wκ	NOUN
ejde-636	371	20	λ∥22	λ∥22	PROPN
ejde-636	371	21	→	→	SYM
ejde-636	371	22	∥∇u1∥22	∥∇u1∥22	PROPN
ejde-636	371	23	,	,	PUNCT
ejde-636	371	24	∥wκ	∥wκ	PROPN
ejde-636	371	25	λ∥qq	λ∥qq	PROPN
ejde-636	371	26	→	→	SYM
ejde-636	371	27	∥u1∥qq	∥u1∥qq	PROPN
ejde-636	371	28	,	,	PUNCT
ejde-636	371	29	∥wκ	∥wκ	PROPN
ejde-636	371	30	λ∥pp	λ∥pp	PROPN
ejde-636	371	31	→	→	SYM
ejde-636	371	32	∥u1||pp	∥u1||pp	NOUN
ejde-636	371	33	.	.	PUNCT
ejde-636	372	1	letting	let	VERB
ejde-636	372	2	(	(	PUNCT
ejde-636	372	3	wκ	wκ	INTJ
ejde-636	372	4	λ)t	λ)t	NOUN
ejde-636	372	5	=	=	PUNCT
ejde-636	372	6	tn/4wκ	tn/4wκ	NUM
ejde-636	372	7	λ	λ	PROPN
ejde-636	372	8	(	(	PUNCT
ejde-636	372	9	√	√	NUM
ejde-636	372	10	tx	tx	PROPN
ejde-636	372	11	)	)	PUNCT
ejde-636	372	12	,	,	PUNCT
ejde-636	372	13	by	by	ADP
ejde-636	372	14	lemma	lemma	PROPN
ejde-636	372	15	2.7	2.7	NUM
ejde-636	372	16	again	again	ADV
ejde-636	372	17	,	,	PUNCT
ejde-636	372	18	we	we	PRON
ejde-636	372	19	can	can	AUX
ejde-636	372	20	deduce	deduce	VERB
ejde-636	372	21	that	that	PRON
ejde-636	372	22	for	for	ADP
ejde-636	372	23	λ	λ	PROPN
ejde-636	372	24	,	,	PUNCT
ejde-636	372	25	κ	κ	X
ejde-636	372	26	>	>	X
ejde-636	372	27	0	0	PUNCT
ejde-636	373	1	small	small	ADJ
ejde-636	373	2	enough	enough	ADV
ejde-636	373	3	,	,	PUNCT
ejde-636	373	4	it	it	PRON
ejde-636	373	5	holds	hold	VERB
ejde-636	373	6	mp	mp	NOUN
ejde-636	373	7	,	,	PUNCT
ejde-636	373	8	q(c2	q(c2	ADJ
ejde-636	373	9	)	)	PUNCT
ejde-636	373	10	≤	≤	NUM
ejde-636	373	11	max	max	PROPN
ejde-636	373	12	t>0	t>0	NOUN
ejde-636	373	13	ep	ep	PROPN
ejde-636	373	14	,	,	PUNCT
ejde-636	373	15	q((w	q((w	NOUN
ejde-636	373	16	κ	κ	NOUN
ejde-636	373	17	λ)t	λ)t	NOUN
ejde-636	373	18	)	)	PUNCT
ejde-636	373	19	≤	≤	NUM
ejde-636	373	20	max	max	PROPN
ejde-636	373	21	t>0	t>0	NOUN
ejde-636	373	22	ep	ep	PROPN
ejde-636	373	23	,	,	PUNCT
ejde-636	373	24	q((u1)t	q((u1)t	NUM
ejde-636	373	25	)	)	PUNCT
ejde-636	374	1	+	+	CCONJ
ejde-636	374	2	ϵ	ϵ	SYM
ejde-636	374	3	2	2	X
ejde-636	374	4	=	=	SYM
ejde-636	374	5	ep	ep	PROPN
ejde-636	374	6	,	,	PUNCT
ejde-636	374	7	q(u1	q(u1	PROPN
ejde-636	374	8	)	)	PUNCT
ejde-636	375	1	+	+	CCONJ
ejde-636	375	2	ϵ	ϵ	SYM
ejde-636	375	3	2	2	NUM
ejde-636	375	4	≤	≤	PROPN
ejde-636	375	5	mp	mp	PROPN
ejde-636	375	6	,	,	PUNCT
ejde-636	375	7	q(c1	q(c1	PROPN
ejde-636	375	8	)	)	PUNCT
ejde-636	376	1	+	+	CCONJ
ejde-636	376	2	ϵ.	ϵ.	NOUN
ejde-636	376	3	□	□	PUNCT
ejde-636	376	4	lemma	lemma	PROPN
ejde-636	376	5	4.5	4.5	NUM
ejde-636	376	6	.	.	PUNCT
ejde-636	377	1	let	let	VERB
ejde-636	377	2	p	p	PRON
ejde-636	377	3	≤	≤	ADJ
ejde-636	378	1	q	q	NOUN
ejde-636	378	2	<	<	X
ejde-636	378	3	p	p	X
ejde-636	378	4	<	<	X
ejde-636	378	5	4∗.	4∗.	NUM
ejde-636	378	6	assume	assume	VERB
ejde-636	378	7	that	that	SCONJ
ejde-636	378	8	uc	uc	PROPN
ejde-636	378	9	∈	∈	PROPN
ejde-636	378	10	s(c	s(c	PROPN
ejde-636	378	11	)	)	PUNCT
ejde-636	378	12	solves	solve	VERB
ejde-636	378	13	∆2u−∆u+	∆2u−∆u+	PROPN
ejde-636	378	14	ωcu	ωcu	PROPN
ejde-636	378	15	=	=	PUNCT
ejde-636	379	1	µ|u|q−2u+	µ|u|q−2u+	PROPN
ejde-636	379	2	|u|p−2u	|u|p−2u	PROPN
ejde-636	379	3	.	.	PUNCT
ejde-636	380	1	(	(	PUNCT
ejde-636	380	2	4.5	4.5	NUM
ejde-636	380	3	)	)	PUNCT
ejde-636	380	4	then	then	ADV
ejde-636	380	5	there	there	PRON
ejde-636	380	6	exists	exist	VERB
ejde-636	380	7	c∗	c∗	PROPN
ejde-636	380	8	>	>	X
ejde-636	380	9	0	0	NUM
ejde-636	381	1	such	such	ADJ
ejde-636	381	2	that	that	PRON
ejde-636	381	3	ωc	ωc	INTJ
ejde-636	381	4	>	>	X
ejde-636	381	5	0	0	NUM
ejde-636	381	6	for	for	ADP
ejde-636	381	7	any	any	DET
ejde-636	381	8	c	c	PROPN
ejde-636	381	9	∈	∈	PROPN
ejde-636	381	10	(	(	PUNCT
ejde-636	381	11	0	0	NUM
ejde-636	381	12	,	,	PUNCT
ejde-636	381	13	c∗	c∗	NOUN
ejde-636	381	14	)	)	PUNCT
ejde-636	381	15	.	.	PUNCT
ejde-636	382	1	proof	proof	NOUN
ejde-636	382	2	.	.	PUNCT
ejde-636	383	1	by	by	ADP
ejde-636	383	2	(	(	PUNCT
ejde-636	383	3	4.5	4.5	NUM
ejde-636	383	4	)	)	PUNCT
ejde-636	383	5	we	we	PRON
ejde-636	383	6	deduce	deduce	VERB
ejde-636	383	7	qp	qp	NOUN
ejde-636	383	8	,	,	PUNCT
ejde-636	383	9	q(u	q(u	ADJ
ejde-636	383	10	)	)	PUNCT
ejde-636	383	11	=	=	SYM
ejde-636	383	12	0	0	NUM
ejde-636	384	1	and	and	CCONJ
ejde-636	384	2	∥∆uc∥22	∥∆uc∥22	PROPN
ejde-636	385	1	+	+	CCONJ
ejde-636	385	2	∥∇uc∥22	∥∇uc∥22	PROPN
ejde-636	385	3	+	+	CCONJ
ejde-636	385	4	ωc∥uc∥22	ωc∥uc∥22	PROPN
ejde-636	385	5	−	−	PROPN
ejde-636	385	6	µ∥uc∥qq	µ∥uc∥qq	NOUN
ejde-636	385	7	−	−	PROPN
ejde-636	386	1	∥uc∥pp	∥uc∥pp	PROPN
ejde-636	387	1	=	=	NOUN
ejde-636	388	1	0	0	X
ejde-636	388	2	.	.	PUNCT
ejde-636	389	1	then	then	ADV
ejde-636	389	2	ωcγqc	ωcγqc	NOUN
ejde-636	389	3	=	=	PUNCT
ejde-636	389	4	(	(	PUNCT
ejde-636	389	5	1−	1−	NUM
ejde-636	389	6	γq)∥∆uc∥22	γq)∥∆uc∥22	PROPN
ejde-636	389	7	+	+	CCONJ
ejde-636	389	8	(	(	PUNCT
ejde-636	389	9	1	1	NUM
ejde-636	389	10	2	2	NUM
ejde-636	389	11	−	−	NOUN
ejde-636	389	12	γq)∥∇uc∥22	γq)∥∇uc∥22	PRON
ejde-636	389	13	−	−	PROPN
ejde-636	389	14	(	(	PUNCT
ejde-636	389	15	γp	γp	NOUN
ejde-636	389	16	−	−	PROPN
ejde-636	389	17	γq)∥uc∥pp	γq)∥uc∥pp	PROPN
ejde-636	389	18	.	.	PUNCT
ejde-636	390	1	(	(	PUNCT
ejde-636	390	2	4.6	4.6	NUM
ejde-636	390	3	)	)	PUNCT
ejde-636	390	4	for	for	ADP
ejde-636	390	5	small	small	ADJ
ejde-636	390	6	c	c	PROPN
ejde-636	390	7	>	>	X
ejde-636	390	8	0	0	NUM
ejde-636	390	9	,	,	PUNCT
ejde-636	390	10	using	use	VERB
ejde-636	390	11	the	the	DET
ejde-636	390	12	gagliardo	gagliardo	NOUN
ejde-636	390	13	-	-	PUNCT
ejde-636	390	14	nirenberg	nirenberg	PROPN
ejde-636	390	15	inequality	inequality	NOUN
ejde-636	390	16	leads	lead	VERB
ejde-636	390	17	to	to	ADP
ejde-636	390	18	∥∆uc∥22	∥∆uc∥22	PUNCT
ejde-636	390	19	=	=	PRON
ejde-636	390	20	γpc	γpc	X
ejde-636	390	21	q	q	NOUN
ejde-636	390	22	n	n	CCONJ
ejde-636	390	23	,	,	PUNCT
ejde-636	390	24	q∥∆uc∥	q∥∆uc∥	ADP
ejde-636	390	25	qγq	qγq	PROPN
ejde-636	390	26	2	2	NUM
ejde-636	390	27	(	(	PUNCT
ejde-636	390	28	√	√	NUM
ejde-636	390	29	c)q(1−γq	c)q(1−γq	NOUN
ejde-636	390	30	)	)	PUNCT
ejde-636	390	31	+	+	CCONJ
ejde-636	390	32	γpc	γpc	X
ejde-636	390	33	p	p	NOUN
ejde-636	390	34	n	n	CCONJ
ejde-636	390	35	,	,	PUNCT
ejde-636	390	36	p∥∆uc∥	p∥∆uc∥	VERB
ejde-636	390	37	pγp	pγp	ADJ
ejde-636	390	38	2	2	NUM
ejde-636	390	39	(	(	PUNCT
ejde-636	390	40	√	√	NUM
ejde-636	390	41	c)p(1−γp	c)p(1−γp	NUM
ejde-636	390	42	)	)	PUNCT
ejde-636	390	43	≤	≤	NUM
ejde-636	390	44	γp	γp	VERB
ejde-636	390	45	max{cq	max{cq	PROPN
ejde-636	390	46	n	n	CCONJ
ejde-636	390	47	,	,	PUNCT
ejde-636	390	48	q	q	NOUN
ejde-636	390	49	,	,	PUNCT
ejde-636	390	50	c	c	PROPN
ejde-636	390	51	p	p	NOUN
ejde-636	390	52	n	n	NOUN
ejde-636	390	53	,	,	PUNCT
ejde-636	390	54	p	p	NOUN
ejde-636	390	55	}	}	PUNCT
ejde-636	390	56	(	(	PUNCT
ejde-636	390	57	√	√	NUM
ejde-636	390	58	c)q(1−γq)(∥∆uc∥	c)q(1−γq)(∥∆uc∥	NOUN
ejde-636	390	59	qγq	qγq	NOUN
ejde-636	390	60	2	2	NUM
ejde-636	390	61	+	+	CCONJ
ejde-636	390	62	∥∆uc∥	∥∆uc∥	VERB
ejde-636	390	63	pγp	pγp	ADJ
ejde-636	390	64	2	2	NUM
ejde-636	390	65	)	)	PUNCT
ejde-636	390	66	.	.	PUNCT
ejde-636	391	1	then	then	ADV
ejde-636	391	2	,	,	PUNCT
ejde-636	391	3	for	for	ADP
ejde-636	391	4	p	p	PROPN
ejde-636	391	5	≤	≤	ADJ
ejde-636	391	6	q	q	NOUN
ejde-636	391	7	<	<	X
ejde-636	391	8	p	p	X
ejde-636	391	9	<	<	X
ejde-636	391	10	4∗	4∗	NOUN
ejde-636	391	11	,	,	PUNCT
ejde-636	391	12	as	as	ADP
ejde-636	391	13	c→	c→	X
ejde-636	391	14	0	0	NUM
ejde-636	391	15	we	we	PRON
ejde-636	391	16	obtain∫	obtain∫	VERB
ejde-636	391	17	rn	rn	PROPN
ejde-636	391	18	|∆uc|2dx→	|∆uc|2dx→	PROPN
ejde-636	391	19	∞.	∞.	PROPN
ejde-636	391	20	(	(	PUNCT
ejde-636	391	21	4.7	4.7	NUM
ejde-636	391	22	)	)	PUNCT
ejde-636	391	23	on	on	ADP
ejde-636	391	24	the	the	DET
ejde-636	391	25	other	other	ADJ
ejde-636	391	26	hand	hand	NOUN
ejde-636	391	27	,	,	PUNCT
ejde-636	391	28	we	we	PRON
ejde-636	391	29	from	from	ADP
ejde-636	391	30	(	(	PUNCT
ejde-636	391	31	2.1	2.1	NUM
ejde-636	391	32	)	)	PUNCT
ejde-636	391	33	and	and	CCONJ
ejde-636	391	34	(	(	PUNCT
ejde-636	391	35	4.6	4.6	NUM
ejde-636	391	36	)	)	PUNCT
ejde-636	391	37	derive	derive	ADJ
ejde-636	391	38	ωcγqc	ωcγqc	NOUN
ejde-636	391	39	=	=	SYM
ejde-636	391	40	(	(	PUNCT
ejde-636	391	41	1−	1−	NUM
ejde-636	391	42	γq)∥∆uc∥22	γq)∥∆uc∥22	PROPN
ejde-636	391	43	+	+	CCONJ
ejde-636	391	44	(	(	PUNCT
ejde-636	391	45	1	1	NUM
ejde-636	391	46	2	2	NUM
ejde-636	391	47	−	−	NOUN
ejde-636	391	48	γq)∥∇uc∥22	γq)∥∇uc∥22	PRON
ejde-636	391	49	−	−	PROPN
ejde-636	392	1	(	(	PUNCT
ejde-636	392	2	γp	γp	PROPN
ejde-636	392	3	−	−	PROPN
ejde-636	392	4	γq)∥uc∥pp	γq)∥uc∥pp	PROPN
ejde-636	392	5	>	>	X
ejde-636	392	6	(	(	PUNCT
ejde-636	392	7	1−	1−	NUM
ejde-636	392	8	γq)∥∆uc∥22	γq)∥∆uc∥22	PROPN
ejde-636	392	9	+	+	CCONJ
ejde-636	393	1	(	(	PUNCT
ejde-636	393	2	1	1	NUM
ejde-636	393	3	2	2	NUM
ejde-636	393	4	−	−	NOUN
ejde-636	393	5	γq	γq	NOUN
ejde-636	393	6	)	)	PUNCT
ejde-636	393	7	√	√	ADP
ejde-636	394	1	c∥∆uc∥2	c∥∆uc∥2	NOUN
ejde-636	394	2	.	.	PUNCT
ejde-636	395	1	from	from	ADP
ejde-636	395	2	(	(	PUNCT
ejde-636	395	3	4.7	4.7	NUM
ejde-636	395	4	)	)	PUNCT
ejde-636	395	5	,	,	PUNCT
ejde-636	395	6	it	it	PRON
ejde-636	395	7	follows	follow	VERB
ejde-636	395	8	that	that	SCONJ
ejde-636	395	9	ωc	ωc	PROPN
ejde-636	395	10	>	>	X
ejde-636	395	11	0	0	PUNCT
ejde-636	396	1	if	if	SCONJ
ejde-636	396	2	c	c	PROPN
ejde-636	396	3	>	>	X
ejde-636	396	4	0	0	NUM
ejde-636	396	5	is	be	AUX
ejde-636	396	6	small	small	ADJ
ejde-636	396	7	enough	enough	ADV
ejde-636	396	8	.	.	PUNCT
ejde-636	397	1	□	□	PUNCT
ejde-636	397	2	lemma	lemma	PROPN
ejde-636	397	3	4.6	4.6	NUM
ejde-636	397	4	.	.	PUNCT
ejde-636	398	1	let	let	VERB
ejde-636	398	2	p	p	PRON
ejde-636	398	3	≤	≤	ADJ
ejde-636	399	1	q	q	NOUN
ejde-636	399	2	<	<	X
ejde-636	399	3	p	p	X
ejde-636	399	4	<	<	X
ejde-636	399	5	4∗	4∗	NOUN
ejde-636	399	6	and	and	CCONJ
ejde-636	399	7	c	c	NOUN
ejde-636	399	8	∈	∈	PROPN
ejde-636	399	9	(	(	PUNCT
ejde-636	399	10	0	0	NUM
ejde-636	399	11	,	,	PUNCT
ejde-636	399	12	c∗	c∗	NOUN
ejde-636	399	13	)	)	PUNCT
ejde-636	399	14	.	.	PUNCT
ejde-636	400	1	when	when	SCONJ
ejde-636	400	2	q	q	NOUN
ejde-636	400	3	=	=	SYM
ejde-636	400	4	p	p	X
ejde-636	400	5	,	,	PUNCT
ejde-636	400	6	we	we	PRON
ejde-636	400	7	assume	assume	VERB
ejde-636	400	8	that	that	SCONJ
ejde-636	400	9	µc4	µc4	PROPN
ejde-636	400	10	/	/	SYM
ejde-636	400	11	n	n	NOUN
ejde-636	400	12	<	<	X
ejde-636	400	13	n+4	n+4	NUM
ejde-636	400	14	ncq	ncq	NOUN
ejde-636	400	15	n	n	CCONJ
ejde-636	400	16	,	,	PUNCT
ejde-636	400	17	q	q	PROPN
ejde-636	400	18	.	.	PUNCT
ejde-636	400	19	suppose	suppose	VERB
ejde-636	400	20	that	that	SCONJ
ejde-636	400	21	u	u	PROPN
ejde-636	400	22	∈	∈	PROPN
ejde-636	400	23	s(c	s(c	PROPN
ejde-636	400	24	)	)	PUNCT
ejde-636	400	25	such	such	ADJ
ejde-636	400	26	that	that	SCONJ
ejde-636	400	27	ep	ep	PROPN
ejde-636	400	28	,	,	PUNCT
ejde-636	400	29	q(u	q(u	ADJ
ejde-636	400	30	)	)	PUNCT
ejde-636	400	31	=	=	SYM
ejde-636	400	32	mp	mp	PROPN
ejde-636	400	33	,	,	PUNCT
ejde-636	400	34	q(c	q(c	PROPN
ejde-636	400	35	)	)	PUNCT
ejde-636	400	36	and	and	CCONJ
ejde-636	400	37	∆2u−∆u+	∆2u−∆u+	NOUN
ejde-636	400	38	ωu	ωu	PROPN
ejde-636	401	1	=	=	SYM
ejde-636	401	2	µ|u|q−2u+	µ|u|q−2u+	PROPN
ejde-636	401	3	|u|p−2u	|u|p−2u	PROPN
ejde-636	401	4	.	.	PUNCT
ejde-636	402	1	then	then	ADV
ejde-636	402	2	the	the	DET
ejde-636	402	3	function	function	NOUN
ejde-636	402	4	c	c	PROPN
ejde-636	402	5	7→	7→	PROPN
ejde-636	402	6	mp	mp	PROPN
ejde-636	402	7	,	,	PUNCT
ejde-636	402	8	q(c	q(c	PROPN
ejde-636	402	9	)	)	PUNCT
ejde-636	402	10	is	be	AUX
ejde-636	402	11	strictly	strictly	ADV
ejde-636	402	12	decreasing	decrease	VERB
ejde-636	402	13	in	in	ADP
ejde-636	402	14	a	a	DET
ejde-636	402	15	right	right	ADJ
ejde-636	402	16	neighborhood	neighborhood	NOUN
ejde-636	402	17	of	of	ADP
ejde-636	402	18	c.	c.	NOUN
ejde-636	402	19	proof	proof	NOUN
ejde-636	402	20	.	.	PUNCT
ejde-636	403	1	by	by	ADP
ejde-636	403	2	lemma	lemma	PROPN
ejde-636	403	3	4.5	4.5	NUM
ejde-636	403	4	,	,	PUNCT
ejde-636	403	5	we	we	PRON
ejde-636	403	6	know	know	VERB
ejde-636	403	7	that	that	SCONJ
ejde-636	403	8	ω	ω	PROPN
ejde-636	403	9	>	>	X
ejde-636	403	10	0	0	X
ejde-636	403	11	.	.	PUNCT
ejde-636	404	1	set	set	VERB
ejde-636	404	2	uλ	uλ	ADP
ejde-636	404	3	,	,	PUNCT
ejde-636	404	4	t(x	t(x	PROPN
ejde-636	404	5	)	)	PUNCT
ejde-636	405	1	=	=	SYM
ejde-636	405	2	tn/4	tn/4	NOUN
ejde-636	405	3	√	√	VERB
ejde-636	405	4	λu	λu	PROPN
ejde-636	405	5	(	(	PUNCT
ejde-636	405	6	√	√	PROPN
ejde-636	405	7	tx	tx	PROPN
ejde-636	405	8	)	)	PUNCT
ejde-636	405	9	for	for	ADP
ejde-636	405	10	λ	λ	PROPN
ejde-636	405	11	,	,	PUNCT
ejde-636	405	12	t	t	X
ejde-636	405	13	>	>	X
ejde-636	405	14	0	0	X
ejde-636	405	15	.	.	PUNCT
ejde-636	406	1	we	we	PRON
ejde-636	406	2	define	define	VERB
ejde-636	406	3	k(λ	k(λ	NOUN
ejde-636	406	4	,	,	PUNCT
ejde-636	406	5	t	t	PROPN
ejde-636	406	6	)	)	PUNCT
ejde-636	406	7	=	=	SYM
ejde-636	406	8	ep	ep	PROPN
ejde-636	406	9	,	,	PUNCT
ejde-636	406	10	q(uλ	q(uλ	PROPN
ejde-636	406	11	,	,	PUNCT
ejde-636	406	12	t	t	NOUN
ejde-636	406	13	)	)	PUNCT
ejde-636	406	14	=	=	SYM
ejde-636	406	15	t2	t2	PROPN
ejde-636	406	16	2	2	NUM
ejde-636	406	17	λ∥∆u∥22	λ∥∆u∥22	PROPN
ejde-636	406	18	+	+	PROPN
ejde-636	406	19	t	t	PROPN
ejde-636	406	20	2	2	NUM
ejde-636	406	21	λ∥∇u∥22−	λ∥∇u∥22−	NOUN
ejde-636	406	22	µ	µ	X
ejde-636	406	23	·	·	SYM
ejde-636	406	24	t	t	PROPN
ejde-636	406	25	n(q−2	n(q−2	PUNCT
ejde-636	406	26	)	)	PUNCT
ejde-636	406	27	4	4	NUM
ejde-636	406	28	q	q	NOUN
ejde-636	406	29	λ	λ	X
ejde-636	406	30	q	q	PROPN
ejde-636	406	31	2	2	NUM
ejde-636	406	32	∥u∥qq−	∥u∥qq−	X
ejde-636	406	33	t	t	NOUN
ejde-636	406	34	n(p−2	n(p−2	NUM
ejde-636	406	35	)	)	PUNCT
ejde-636	406	36	4	4	NUM
ejde-636	407	1	p	p	NOUN
ejde-636	407	2	λp/2∥u∥pp	λp/2∥u∥pp	PROPN
ejde-636	407	3	and	and	CCONJ
ejde-636	407	4	m(λ	m(λ	PROPN
ejde-636	407	5	,	,	PUNCT
ejde-636	407	6	t	t	PROPN
ejde-636	407	7	)	)	PUNCT
ejde-636	407	8	=	=	SYM
ejde-636	407	9	qp	qp	PROPN
ejde-636	407	10	,	,	PUNCT
ejde-636	407	11	q(uλ	q(uλ	PROPN
ejde-636	407	12	,	,	PUNCT
ejde-636	407	13	t	t	PROPN
ejde-636	407	14	)	)	PUNCT
ejde-636	407	15	=	=	VERB
ejde-636	407	16	t2λ∥∆u∥22	t2λ∥∆u∥22	NOUN
ejde-636	408	1	+	+	CCONJ
ejde-636	408	2	t	t	PROPN
ejde-636	408	3	2	2	NUM
ejde-636	408	4	λ∥∇u∥22	λ∥∇u∥22	PROPN
ejde-636	408	5	−	−	PROPN
ejde-636	408	6	µγqt	µγqt	ADP
ejde-636	408	7	n(q−2	n(q−2	PUNCT
ejde-636	408	8	)	)	PUNCT
ejde-636	408	9	4	4	NUM
ejde-636	408	10	λ	λ	NOUN
ejde-636	408	11	q	q	NOUN
ejde-636	408	12	2	2	NUM
ejde-636	408	13	∥u∥qq	∥u∥qq	ADP
ejde-636	408	14	−	−	PROPN
ejde-636	408	15	γpt	γpt	PROPN
ejde-636	408	16	n(p−2	n(p−2	PROPN
ejde-636	408	17	)	)	PUNCT
ejde-636	408	18	4	4	NUM
ejde-636	408	19	λp/2∥u∥pp	λp/2∥u∥pp	PROPN
ejde-636	408	20	.	.	PUNCT
ejde-636	409	1	14	14	NUM
ejde-636	409	2	z.	z.	PROPN
ejde-636	409	3	ma	ma	PROPN
ejde-636	409	4	,	,	PUNCT
ejde-636	409	5	x.	x.	PROPN
ejde-636	409	6	chang	chang	PROPN
ejde-636	409	7	,	,	PUNCT
ejde-636	409	8	z.	z.	PROPN
ejde-636	409	9	feng	feng	PROPN
ejde-636	409	10	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	409	11	by	by	ADP
ejde-636	409	12	a	a	DET
ejde-636	409	13	direct	direct	ADJ
ejde-636	409	14	calculation	calculation	NOUN
ejde-636	409	15	,	,	PUNCT
ejde-636	409	16	we	we	PRON
ejde-636	409	17	have	have	VERB
ejde-636	409	18	∂k	∂k	PROPN
ejde-636	409	19	∂λ	∂λ	PROPN
ejde-636	409	20	(	(	PUNCT
ejde-636	409	21	1	1	NUM
ejde-636	409	22	,	,	PUNCT
ejde-636	409	23	1	1	NUM
ejde-636	409	24	)	)	PUNCT
ejde-636	409	25	=	=	SYM
ejde-636	409	26	1	1	NUM
ejde-636	409	27	2	2	NUM
ejde-636	409	28	∥∆u∥22	∥∆u∥22	VERB
ejde-636	409	29	+	+	CCONJ
ejde-636	409	30	1	1	NUM
ejde-636	409	31	2	2	NUM
ejde-636	409	32	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	409	33	−	−	PROPN
ejde-636	409	34	µ	µ	SYM
ejde-636	409	35	2	2	NUM
ejde-636	409	36	∥u∥qq	∥u∥qq	ADP
ejde-636	409	37	−	−	PROPN
ejde-636	409	38	1	1	NUM
ejde-636	409	39	2	2	NUM
ejde-636	409	40	∥u∥pp	∥u∥pp	NOUN
ejde-636	409	41	=	=	SYM
ejde-636	409	42	−1	−1	NOUN
ejde-636	409	43	2	2	NUM
ejde-636	409	44	ωc	ωc	NOUN
ejde-636	409	45	,	,	PUNCT
ejde-636	409	46	∂k	∂k	PROPN
ejde-636	409	47	∂t	∂t	PROPN
ejde-636	409	48	(	(	PUNCT
ejde-636	409	49	1	1	NUM
ejde-636	409	50	,	,	PUNCT
ejde-636	409	51	1	1	NUM
ejde-636	409	52	)	)	PUNCT
ejde-636	409	53	=	=	PRON
ejde-636	410	1	∥∆u∥22	∥∆u∥22	X
ejde-636	410	2	+	+	CCONJ
ejde-636	410	3	1	1	NUM
ejde-636	410	4	2	2	NUM
ejde-636	410	5	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	410	6	−	−	ADP
ejde-636	410	7	µγq∥u∥qq	µγq∥u∥qq	ADP
ejde-636	410	8	−	−	NOUN
ejde-636	410	9	γp∥u∥pp	γp∥u∥pp	PROPN
ejde-636	410	10	=	=	SYM
ejde-636	410	11	0	0	NUM
ejde-636	410	12	,	,	PUNCT
ejde-636	410	13	∂2k	∂2k	NOUN
ejde-636	410	14	∂t2	∂t2	NOUN
ejde-636	410	15	(	(	PUNCT
ejde-636	410	16	1	1	NUM
ejde-636	410	17	,	,	PUNCT
ejde-636	410	18	1	1	NUM
ejde-636	410	19	)	)	PUNCT
ejde-636	410	20	=	=	PRON
ejde-636	410	21	∥∆u∥22	∥∆u∥22	VERB
ejde-636	410	22	−	−	PROPN
ejde-636	411	1	µγq	µγq	PROPN
ejde-636	412	1	(	(	PUNCT
ejde-636	412	2	n(q	n(q	PROPN
ejde-636	412	3	−	−	PROPN
ejde-636	412	4	2	2	NUM
ejde-636	412	5	)	)	PUNCT
ejde-636	412	6	4	4	NUM
ejde-636	412	7	−	−	PROPN
ejde-636	412	8	1)∥u∥qq	1)∥u∥qq	NOUN
ejde-636	412	9	−	−	NOUN
ejde-636	413	1	γp	γp	PROPN
ejde-636	413	2	(	(	PUNCT
ejde-636	413	3	n(p−	n(p−	X
ejde-636	413	4	2	2	NUM
ejde-636	413	5	)	)	PUNCT
ejde-636	413	6	4	4	NUM
ejde-636	413	7	−	−	NOUN
ejde-636	413	8	1)∥u∥pp	1)∥u∥pp	NUM
ejde-636	413	9	<	<	X
ejde-636	413	10	0	0	NUM
ejde-636	413	11	,	,	PUNCT
ejde-636	413	12	which	which	PRON
ejde-636	413	13	yields	yield	VERB
ejde-636	413	14	for	for	ADP
ejde-636	413	15	δt	δt	NOUN
ejde-636	413	16	small	small	ADJ
ejde-636	413	17	enough	enough	ADV
ejde-636	413	18	and	and	CCONJ
ejde-636	413	19	δλ	δλ	X
ejde-636	413	20	>	>	X
ejde-636	413	21	0	0	PROPN
ejde-636	413	22	,	,	PUNCT
ejde-636	413	23	k(1	k(1	PROPN
ejde-636	413	24	+	+	CCONJ
ejde-636	413	25	δλ	δλ	PROPN
ejde-636	413	26	,	,	PUNCT
ejde-636	413	27	1	1	NUM
ejde-636	413	28	+	+	NUM
ejde-636	413	29	δt	δt	NOUN
ejde-636	413	30	)	)	PUNCT
ejde-636	413	31	<	<	X
ejde-636	413	32	k(1	k(1	PROPN
ejde-636	413	33	,	,	PUNCT
ejde-636	413	34	1	1	NUM
ejde-636	413	35	)	)	PUNCT
ejde-636	413	36	for	for	ADP
ejde-636	413	37	ω	ω	PROPN
ejde-636	413	38	>	>	X
ejde-636	413	39	0	0	PROPN
ejde-636	413	40	.	.	PUNCT
ejde-636	413	41	(	(	PUNCT
ejde-636	413	42	4.8	4.8	NUM
ejde-636	413	43	)	)	PUNCT
ejde-636	413	44	in	in	ADP
ejde-636	413	45	addition	addition	NOUN
ejde-636	413	46	,	,	PUNCT
ejde-636	413	47	we	we	PRON
ejde-636	413	48	observe	observe	VERB
ejde-636	413	49	that	that	SCONJ
ejde-636	413	50	m(1	m(1	NOUN
ejde-636	413	51	,	,	PUNCT
ejde-636	413	52	1	1	NUM
ejde-636	413	53	)	)	PUNCT
ejde-636	413	54	=	=	SYM
ejde-636	413	55	qp	qp	NOUN
ejde-636	413	56	,	,	PUNCT
ejde-636	413	57	q(u	q(u	ADJ
ejde-636	413	58	)	)	PUNCT
ejde-636	413	59	=	=	PRON
ejde-636	413	60	∥∆u∥22	∥∆u∥22	X
ejde-636	414	1	+	+	CCONJ
ejde-636	414	2	1	1	NUM
ejde-636	414	3	2	2	NUM
ejde-636	414	4	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	414	5	−	−	ADP
ejde-636	414	6	µγq∥u∥qq	µγq∥u∥qq	ADP
ejde-636	414	7	−	−	NOUN
ejde-636	414	8	γp∥u∥pp	γp∥u∥pp	PROPN
ejde-636	414	9	=	=	SYM
ejde-636	415	1	0	0	X
ejde-636	415	2	.	.	PUNCT
ejde-636	416	1	we	we	PRON
ejde-636	416	2	now	now	ADV
ejde-636	416	3	claim	claim	VERB
ejde-636	416	4	that	that	SCONJ
ejde-636	416	5	∂m	∂m	PROPN
ejde-636	416	6	∂t	∂t	PROPN
ejde-636	416	7	(	(	PUNCT
ejde-636	416	8	1	1	NUM
ejde-636	416	9	,	,	PUNCT
ejde-636	416	10	1	1	NUM
ejde-636	416	11	)	)	PUNCT
ejde-636	416	12	=	=	SYM
ejde-636	417	1	2∥∆u∥22	2∥∆u∥22	NUM
ejde-636	418	1	+	+	CCONJ
ejde-636	418	2	1	1	NUM
ejde-636	418	3	2	2	NUM
ejde-636	418	4	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	418	5	−	−	PROPN
ejde-636	418	6	µγq	µγq	NOUN
ejde-636	418	7	n(q	n(q	PROPN
ejde-636	418	8	−	−	PROPN
ejde-636	418	9	2	2	NUM
ejde-636	418	10	)	)	PUNCT
ejde-636	418	11	4	4	NUM
ejde-636	418	12	∥u∥qq	∥u∥qq	ADP
ejde-636	418	13	−	−	PROPN
ejde-636	418	14	γp	γp	NOUN
ejde-636	418	15	n(p−	n(p−	X
ejde-636	418	16	2	2	NUM
ejde-636	418	17	)	)	PUNCT
ejde-636	418	18	4	4	NUM
ejde-636	418	19	∥u∥pp	∥u∥pp	NOUN
ejde-636	418	20	̸=	̸=	PROPN
ejde-636	418	21	0	0	NUM
ejde-636	418	22	.	.	PUNCT
ejde-636	419	1	otherwise	otherwise	ADV
ejde-636	419	2	,	,	PUNCT
ejde-636	419	3	we	we	PRON
ejde-636	419	4	assume	assume	VERB
ejde-636	419	5	that	that	SCONJ
ejde-636	419	6	∂m	∂m	PROPN
ejde-636	419	7	∂t	∂t	PROPN
ejde-636	419	8	(	(	PUNCT
ejde-636	419	9	1	1	NUM
ejde-636	419	10	,	,	PUNCT
ejde-636	419	11	1	1	NUM
ejde-636	419	12	)	)	PUNCT
ejde-636	419	13	=	=	PRON
ejde-636	420	1	∥∆u∥22	∥∆u∥22	X
ejde-636	420	2	+	+	NUM
ejde-636	420	3	1	1	NUM
ejde-636	420	4	4	4	NUM
ejde-636	420	5	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	420	6	−	−	PROPN
ejde-636	420	7	µγq	µγq	NOUN
ejde-636	420	8	n(q	n(q	PROPN
ejde-636	420	9	−	−	PROPN
ejde-636	420	10	2	2	NUM
ejde-636	420	11	)	)	PUNCT
ejde-636	420	12	8	8	NUM
ejde-636	420	13	∥u∥qq	∥u∥qq	ADP
ejde-636	420	14	−	−	PROPN
ejde-636	420	15	γp	γp	NOUN
ejde-636	420	16	n(p−	n(p−	X
ejde-636	420	17	2	2	NUM
ejde-636	420	18	)	)	PUNCT
ejde-636	420	19	8	8	NUM
ejde-636	420	20	∥u∥pp	∥u∥pp	NOUN
ejde-636	420	21	=	=	SYM
ejde-636	421	1	0	0	PROPN
ejde-636	421	2	.	.	PUNCT
ejde-636	422	1	then	then	ADV
ejde-636	422	2	for	for	ADP
ejde-636	422	3	any	any	DET
ejde-636	422	4	p	p	NOUN
ejde-636	422	5	≤	≤	NOUN
ejde-636	422	6	q	q	NOUN
ejde-636	422	7	<	<	X
ejde-636	422	8	p	p	X
ejde-636	422	9	<	<	X
ejde-636	422	10	4∗	4∗	NOUN
ejde-636	422	11	,	,	PUNCT
ejde-636	422	12	we	we	PRON
ejde-636	422	13	have	have	VERB
ejde-636	422	14	that	that	PRON
ejde-636	422	15	1	1	NUM
ejde-636	422	16	4	4	NUM
ejde-636	422	17	∥∇u∥22	∥∇u∥22	NOUN
ejde-636	422	18	=	=	PUNCT
ejde-636	422	19	µγq(1−	µγq(1−	PROPN
ejde-636	422	20	n(q	n(q	PROPN
ejde-636	422	21	−	−	PROPN
ejde-636	422	22	2	2	NUM
ejde-636	422	23	)	)	PUNCT
ejde-636	422	24	8	8	NUM
ejde-636	422	25	)	)	PUNCT
ejde-636	422	26	∥u∥qq	∥u∥qq	X
ejde-636	423	1	+	+	PUNCT
ejde-636	423	2	γp(1−	γp(1−	NOUN
ejde-636	423	3	n(p−	n(p−	X
ejde-636	423	4	2	2	NUM
ejde-636	423	5	)	)	PUNCT
ejde-636	423	6	8	8	NUM
ejde-636	423	7	)	)	PUNCT
ejde-636	423	8	∥u∥pp	∥u∥pp	PROPN
ejde-636	423	9	,	,	PUNCT
ejde-636	423	10	which	which	PRON
ejde-636	423	11	is	be	AUX
ejde-636	423	12	impossible	impossible	ADJ
ejde-636	423	13	.	.	PUNCT
ejde-636	424	1	according	accord	VERB
ejde-636	424	2	to	to	ADP
ejde-636	424	3	the	the	DET
ejde-636	424	4	implicit	implicit	ADJ
ejde-636	424	5	function	function	NOUN
ejde-636	424	6	theorem	theorem	VERB
ejde-636	424	7	,	,	PUNCT
ejde-636	424	8	we	we	PRON
ejde-636	424	9	deduce	deduce	VERB
ejde-636	424	10	that	that	SCONJ
ejde-636	424	11	there	there	PRON
ejde-636	424	12	exists	exist	VERB
ejde-636	424	13	ϵ	ϵ	X
ejde-636	424	14	>	>	X
ejde-636	424	15	0	0	PUNCT
ejde-636	424	16	and	and	CCONJ
ejde-636	424	17	a	a	DET
ejde-636	424	18	continuous	continuous	ADJ
ejde-636	424	19	function	function	NOUN
ejde-636	424	20	g	g	NOUN
ejde-636	424	21	:	:	PUNCT
ejde-636	425	1	[	[	X
ejde-636	425	2	1−ϵ	1−ϵ	NUM
ejde-636	425	3	,	,	PUNCT
ejde-636	425	4	1+ϵ	1+ϵ	NUM
ejde-636	425	5	]	]	X
ejde-636	425	6	7→	7→	NUM
ejde-636	425	7	r	r	NOUN
ejde-636	425	8	satisfying	satisfy	VERB
ejde-636	425	9	g(1	g(1	NOUN
ejde-636	425	10	)	)	PUNCT
ejde-636	426	1	=	=	PUNCT
ejde-636	426	2	1	1	NUM
ejde-636	426	3	such	such	ADJ
ejde-636	426	4	that	that	SCONJ
ejde-636	426	5	m(λ	m(λ	PROPN
ejde-636	426	6	,	,	PUNCT
ejde-636	426	7	g(λ	g(λ	PROPN
ejde-636	426	8	)	)	PUNCT
ejde-636	426	9	)	)	PUNCT
ejde-636	427	1	=	=	SYM
ejde-636	427	2	0	0	NUM
ejde-636	427	3	for	for	ADP
ejde-636	427	4	λ	λ	PROPN
ejde-636	427	5	∈	∈	PROPN
ejde-636	427	6	[	[	X
ejde-636	427	7	1−	1−	NUM
ejde-636	427	8	ϵ	ϵ	NUM
ejde-636	427	9	,	,	PUNCT
ejde-636	427	10	1	1	NUM
ejde-636	427	11	+	+	CCONJ
ejde-636	427	12	ϵ	ϵ	X
ejde-636	427	13	]	]	X
ejde-636	427	14	.	.	PUNCT
ejde-636	428	1	this	this	PRON
ejde-636	428	2	together	together	ADV
ejde-636	428	3	with	with	ADP
ejde-636	428	4	(	(	PUNCT
ejde-636	428	5	4.8	4.8	NUM
ejde-636	428	6	)	)	PUNCT
ejde-636	428	7	gives	give	VERB
ejde-636	428	8	mp	mp	PROPN
ejde-636	428	9	,	,	PUNCT
ejde-636	428	10	q((1	q((1	PROPN
ejde-636	428	11	+	+	X
ejde-636	428	12	ϵ)c	ϵ)c	X
ejde-636	428	13	)	)	PUNCT
ejde-636	428	14	≤	≤	NOUN
ejde-636	428	15	ep	ep	PROPN
ejde-636	428	16	,	,	PUNCT
ejde-636	428	17	q(u1+ϵ,g(1+ϵ	q(u1+ϵ,g(1+ϵ	NOUN
ejde-636	428	18	)	)	PUNCT
ejde-636	428	19	)	)	PUNCT
ejde-636	429	1	<	<	X
ejde-636	429	2	ep	ep	PROPN
ejde-636	429	3	,	,	PUNCT
ejde-636	429	4	q(u	q(u	ADJ
ejde-636	429	5	)	)	PUNCT
ejde-636	429	6	=	=	SYM
ejde-636	429	7	mp	mp	PROPN
ejde-636	429	8	,	,	PUNCT
ejde-636	429	9	q(c	q(c	PROPN
ejde-636	429	10	)	)	PUNCT
ejde-636	429	11	.	.	PUNCT
ejde-636	430	1	we	we	PRON
ejde-636	430	2	have	have	AUX
ejde-636	430	3	arrived	arrive	VERB
ejde-636	430	4	at	at	ADP
ejde-636	430	5	the	the	DET
ejde-636	430	6	desired	desire	VERB
ejde-636	430	7	result	result	NOUN
ejde-636	430	8	.	.	PUNCT
ejde-636	431	1	□	□	PUNCT
ejde-636	431	2	4.2	4.2	NUM
ejde-636	431	3	.	.	PUNCT
ejde-636	431	4	ground	ground	NOUN
ejde-636	431	5	states	state	NOUN
ejde-636	431	6	.	.	PUNCT
ejde-636	432	1	in	in	ADP
ejde-636	432	2	this	this	DET
ejde-636	432	3	subsection	subsection	NOUN
ejde-636	432	4	,	,	PUNCT
ejde-636	432	5	before	before	ADP
ejde-636	432	6	presenting	present	VERB
ejde-636	432	7	the	the	DET
ejde-636	432	8	proof	proof	NOUN
ejde-636	432	9	of	of	ADP
ejde-636	432	10	theorem	theorem	NOUN
ejde-636	432	11	1.3	1.3	NUM
ejde-636	432	12	,	,	PUNCT
ejde-636	432	13	we	we	PRON
ejde-636	432	14	show	show	VERB
ejde-636	432	15	the	the	DET
ejde-636	432	16	minimizer	minimizer	NOUN
ejde-636	432	17	of	of	ADP
ejde-636	432	18	ep	ep	PROPN
ejde-636	432	19	,	,	PUNCT
ejde-636	432	20	q(u	q(u	ADJ
ejde-636	432	21	)	)	PUNCT
ejde-636	432	22	constrained	constrain	VERB
ejde-636	432	23	on	on	ADP
ejde-636	432	24	qp	qp	NOUN
ejde-636	432	25	,	,	PUNCT
ejde-636	432	26	q(c	q(c	PROPN
ejde-636	432	27	)	)	PUNCT
ejde-636	432	28	.	.	PUNCT
ejde-636	433	1	for	for	ADP
ejde-636	433	2	convenience	convenience	NOUN
ejde-636	433	3	,	,	PUNCT
ejde-636	433	4	we	we	PRON
ejde-636	433	5	set	set	VERB
ejde-636	433	6	f(s	f(s	ADV
ejde-636	433	7	)	)	PUNCT
ejde-636	434	1	=	=	VERB
ejde-636	434	2	µ|s|q−2s+	µ|s|q−2s+	NOUN
ejde-636	434	3	|s|p−2s	|s|p−2s	PROPN
ejde-636	434	4	,	,	PUNCT
ejde-636	434	5	f	f	PROPN
ejde-636	434	6	(	(	PUNCT
ejde-636	434	7	s	s	X
ejde-636	434	8	)	)	PUNCT
ejde-636	434	9	=	=	SYM
ejde-636	434	10	µ	µ	DET
ejde-636	434	11	q	q	NOUN
ejde-636	434	12	|s|	|s|	PROPN
ejde-636	434	13	q	q	NOUN
ejde-636	434	14	+	+	NUM
ejde-636	434	15	1	1	NUM
ejde-636	434	16	p	p	NOUN
ejde-636	434	17	|s|	|s|	PROPN
ejde-636	434	18	p	p	NOUN
ejde-636	434	19	and	and	CCONJ
ejde-636	434	20	h(s	h(	NOUN
ejde-636	434	21	)	)	PUNCT
ejde-636	434	22	=	=	PRON
ejde-636	434	23	f(s)s−	f(s)s−	ADV
ejde-636	434	24	2f	2f	X
ejde-636	434	25	(	(	PUNCT
ejde-636	434	26	s	s	NOUN
ejde-636	434	27	)	)	PUNCT
ejde-636	434	28	.	.	PUNCT
ejde-636	435	1	lemma	lemma	PROPN
ejde-636	435	2	4.7	4.7	NUM
ejde-636	435	3	.	.	PUNCT
ejde-636	436	1	let	let	VERB
ejde-636	436	2	p	p	PRON
ejde-636	436	3	≤	≤	ADJ
ejde-636	437	1	q	q	NOUN
ejde-636	437	2	<	<	X
ejde-636	437	3	p	p	X
ejde-636	437	4	<	<	X
ejde-636	437	5	4∗	4∗	NOUN
ejde-636	437	6	and	and	CCONJ
ejde-636	437	7	c	c	NOUN
ejde-636	437	8	∈	∈	PROPN
ejde-636	437	9	(	(	PUNCT
ejde-636	437	10	0	0	NUM
ejde-636	437	11	,	,	PUNCT
ejde-636	437	12	c∗	c∗	NOUN
ejde-636	437	13	)	)	PUNCT
ejde-636	437	14	.	.	PUNCT
ejde-636	438	1	when	when	SCONJ
ejde-636	438	2	q	q	NOUN
ejde-636	438	3	=	=	SYM
ejde-636	438	4	p	p	X
ejde-636	438	5	,	,	PUNCT
ejde-636	438	6	we	we	PRON
ejde-636	438	7	assume	assume	VERB
ejde-636	438	8	that	that	SCONJ
ejde-636	438	9	µc4	µc4	PROPN
ejde-636	438	10	/	/	SYM
ejde-636	438	11	n	n	NOUN
ejde-636	438	12	<	<	X
ejde-636	438	13	n+4	n+4	NUM
ejde-636	438	14	ncq	ncq	NOUN
ejde-636	438	15	n	n	CCONJ
ejde-636	438	16	,	,	PUNCT
ejde-636	438	17	q	q	PROPN
ejde-636	438	18	.	.	PUNCT
ejde-636	439	1	then	then	ADV
ejde-636	439	2	there	there	PRON
ejde-636	439	3	exists	exist	VERB
ejde-636	439	4	u0	u0	PROPN
ejde-636	439	5	∈	∈	PROPN
ejde-636	439	6	qp	qp	NOUN
ejde-636	439	7	,	,	PUNCT
ejde-636	439	8	q(c	q(c	PROPN
ejde-636	439	9	)	)	PUNCT
ejde-636	439	10	such	such	ADJ
ejde-636	439	11	that	that	SCONJ
ejde-636	439	12	ep	ep	PROPN
ejde-636	439	13	,	,	PUNCT
ejde-636	439	14	q(u0	q(u0	PROPN
ejde-636	439	15	)	)	PUNCT
ejde-636	439	16	=	=	SYM
ejde-636	439	17	mp	mp	PROPN
ejde-636	439	18	,	,	PUNCT
ejde-636	439	19	q(c	q(c	PROPN
ejde-636	439	20	)	)	PUNCT
ejde-636	439	21	.	.	PUNCT
ejde-636	440	1	proof	proof	NOUN
ejde-636	440	2	.	.	PUNCT
ejde-636	441	1	using	use	VERB
ejde-636	441	2	the	the	DET
ejde-636	441	3	ekeland	ekeland	NOUN
ejde-636	441	4	variational	variational	ADJ
ejde-636	441	5	principle	principle	NOUN
ejde-636	441	6	,	,	PUNCT
ejde-636	441	7	there	there	PRON
ejde-636	441	8	exists	exist	VERB
ejde-636	441	9	a	a	DET
ejde-636	441	10	minimizing	minimize	VERB
ejde-636	441	11	sequence	sequence	NOUN
ejde-636	441	12	{	{	PUNCT
ejde-636	441	13	un	un	PROPN
ejde-636	441	14	}	}	PUNCT
ejde-636	441	15	⊂	⊂	PROPN
ejde-636	441	16	qp	qp	NOUN
ejde-636	441	17	,	,	PUNCT
ejde-636	441	18	q(c	q(c	PROPN
ejde-636	441	19	)	)	PUNCT
ejde-636	441	20	such	such	ADJ
ejde-636	441	21	that	that	SCONJ
ejde-636	441	22	ep	ep	PROPN
ejde-636	441	23	,	,	PUNCT
ejde-636	441	24	q(un	q(un	PROPN
ejde-636	441	25	)	)	PUNCT
ejde-636	441	26	→	→	SYM
ejde-636	441	27	mp	mp	PROPN
ejde-636	441	28	,	,	PUNCT
ejde-636	441	29	q(c	q(c	PROPN
ejde-636	441	30	)	)	PUNCT
ejde-636	441	31	as	as	ADP
ejde-636	441	32	n→	n→	ADV
ejde-636	441	33	+	+	PROPN
ejde-636	441	34	∞.	∞.	PROPN
ejde-636	441	35	(	(	PUNCT
ejde-636	441	36	4.9	4.9	NUM
ejde-636	441	37	)	)	PUNCT
ejde-636	441	38	by	by	ADP
ejde-636	441	39	lemma	lemma	PROPN
ejde-636	441	40	4.2(4	4.2(4	NUM
ejde-636	441	41	)	)	PUNCT
ejde-636	441	42	,	,	PUNCT
ejde-636	441	43	it	it	PRON
ejde-636	441	44	follows	follow	VERB
ejde-636	441	45	that	that	SCONJ
ejde-636	441	46	{	{	PUNCT
ejde-636	441	47	un	un	PROPN
ejde-636	441	48	}	}	PUNCT
ejde-636	441	49	is	be	AUX
ejde-636	441	50	bounded	bound	VERB
ejde-636	441	51	in	in	ADP
ejde-636	441	52	h2(rn	h2(rn	PROPN
ejde-636	441	53	)	)	PUNCT
ejde-636	441	54	.	.	PUNCT
ejde-636	442	1	we	we	PRON
ejde-636	442	2	claim	claim	VERB
ejde-636	442	3	that	that	SCONJ
ejde-636	442	4	{	{	PUNCT
ejde-636	442	5	un	un	PROPN
ejde-636	442	6	}	}	PUNCT
ejde-636	442	7	is	be	AUX
ejde-636	442	8	non	non	ADJ
ejde-636	442	9	-	-	ADJ
ejde-636	442	10	vanishing	vanishing	ADJ
ejde-636	442	11	.	.	PUNCT
ejde-636	443	1	indeed	indeed	ADV
ejde-636	443	2	,	,	PUNCT
ejde-636	443	3	if	if	SCONJ
ejde-636	443	4	{	{	PUNCT
ejde-636	443	5	un	un	ADJ
ejde-636	443	6	}	}	PUNCT
ejde-636	443	7	is	be	AUX
ejde-636	443	8	vanishing	vanish	VERB
ejde-636	443	9	,	,	PUNCT
ejde-636	443	10	then	then	ADV
ejde-636	443	11	it	it	PRON
ejde-636	443	12	follows	follow	VERB
ejde-636	443	13	from	from	ADP
ejde-636	443	14	lemma	lemma	PROPN
ejde-636	443	15	2.1	2.1	NUM
ejde-636	443	16	that∫	that∫	NOUN
ejde-636	443	17	rn	rn	PROPN
ejde-636	443	18	|un|rdx→	|un|rdx→	PROPN
ejde-636	443	19	0	0	NUM
ejde-636	443	20	,	,	PUNCT
ejde-636	443	21	for	for	ADP
ejde-636	443	22	r	r	PROPN
ejde-636	443	23	∈	∈	PROPN
ejde-636	443	24	(	(	PUNCT
ejde-636	443	25	2	2	NUM
ejde-636	443	26	,	,	PUNCT
ejde-636	443	27	4∗	4∗	NOUN
ejde-636	443	28	)	)	PUNCT
ejde-636	443	29	.	.	PUNCT
ejde-636	444	1	since	since	SCONJ
ejde-636	444	2	qp	qp	NOUN
ejde-636	444	3	,	,	PUNCT
ejde-636	444	4	q(un	q(un	PROPN
ejde-636	444	5	)	)	PUNCT
ejde-636	444	6	=	=	SYM
ejde-636	444	7	0	0	NUM
ejde-636	444	8	and	and	CCONJ
ejde-636	444	9	p	p	NOUN
ejde-636	444	10	≤	≤	NOUN
ejde-636	444	11	q	q	NOUN
ejde-636	444	12	<	<	X
ejde-636	444	13	p	p	X
ejde-636	444	14	<	<	X
ejde-636	444	15	4∗	4∗	NOUN
ejde-636	444	16	,	,	PUNCT
ejde-636	444	17	it	it	PRON
ejde-636	444	18	follows	follow	VERB
ejde-636	444	19	that	that	SCONJ
ejde-636	444	20	|∆un|2	|∆un|2	ADJ
ejde-636	444	21	+	+	CCONJ
ejde-636	444	22	1	1	NUM
ejde-636	444	23	2	2	NUM
ejde-636	444	24	|∇un|2	|∇un|2	NOUN
ejde-636	444	25	=	=	SYM
ejde-636	444	26	µγq∥un∥qq	µγq∥un∥qq	X
ejde-636	444	27	+	+	CCONJ
ejde-636	444	28	γp∥un∥pp	γp∥un∥pp	ADJ
ejde-636	444	29	→	→	SYM
ejde-636	444	30	0	0	NUM
ejde-636	444	31	,	,	PUNCT
ejde-636	444	32	as	as	ADP
ejde-636	444	33	n→	n→	ADV
ejde-636	444	34	∞	∞	PROPN
ejde-636	444	35	,	,	PUNCT
ejde-636	444	36	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	444	37	dispersion	dispersion	NOUN
ejde-636	444	38	nonlinear	nonlinear	NOUN
ejde-636	444	39	schrödinger	schrödinger	NOUN
ejde-636	444	40	equation	equation	NOUN
ejde-636	444	41	15	15	NUM
ejde-636	444	42	which	which	PRON
ejde-636	444	43	contradicts	contradict	VERB
ejde-636	444	44	lemma	lemma	PROPN
ejde-636	444	45	4.2(2	4.2(2	NUM
ejde-636	444	46	)	)	PUNCT
ejde-636	444	47	.	.	PUNCT
ejde-636	445	1	thus	thus	ADV
ejde-636	445	2	,	,	PUNCT
ejde-636	445	3	up	up	ADP
ejde-636	445	4	to	to	ADP
ejde-636	445	5	a	a	DET
ejde-636	445	6	subsequence	subsequence	NOUN
ejde-636	445	7	,	,	PUNCT
ejde-636	445	8	we	we	PRON
ejde-636	445	9	obtain	obtain	VERB
ejde-636	445	10	that	that	DET
ejde-636	445	11	un	un	PROPN
ejde-636	445	12	⇀	⇀	PROPN
ejde-636	445	13	u0	u0	PROPN
ejde-636	445	14	̸=	̸=	PROPN
ejde-636	445	15	0	0	NUM
ejde-636	445	16	in	in	ADP
ejde-636	445	17	h2(rn	h2(rn	PROPN
ejde-636	445	18	)	)	PUNCT
ejde-636	445	19	.	.	PUNCT
ejde-636	446	1	denote	denote	VERB
ejde-636	446	2	un,0	un,0	PROPN
ejde-636	446	3	=	=	PROPN
ejde-636	446	4	un	un	PROPN
ejde-636	446	5	−	−	PROPN
ejde-636	446	6	u0	u0	PROPN
ejde-636	446	7	.	.	PUNCT
ejde-636	447	1	it	it	PRON
ejde-636	447	2	is	be	AUX
ejde-636	447	3	easily	easily	ADV
ejde-636	447	4	seen	see	VERB
ejde-636	447	5	that	that	DET
ejde-636	447	6	∥un∥22	∥un∥22	PROPN
ejde-636	448	1	=	=	SYM
ejde-636	448	2	∥u0∥22	∥u0∥22	PROPN
ejde-636	449	1	+	+	CCONJ
ejde-636	449	2	∥un,0∥22	∥un,0∥22	ADJ
ejde-636	449	3	+	+	PUNCT
ejde-636	449	4	on(1	on(1	NOUN
ejde-636	449	5	)	)	PUNCT
ejde-636	449	6	,	,	PUNCT
ejde-636	449	7	∥∇un∥22	∥∇un∥22	PROPN
ejde-636	449	8	=	=	PRON
ejde-636	449	9	∥∇u0∥22	∥∇u0∥22	PROPN
ejde-636	449	10	+	+	NUM
ejde-636	449	11	∥∇un,0∥22	∥∇un,0∥22	PROPN
ejde-636	449	12	+	+	CCONJ
ejde-636	449	13	on(1	on(1	NOUN
ejde-636	449	14	)	)	PUNCT
ejde-636	449	15	,	,	PUNCT
ejde-636	449	16	∥∆un∥22	∥∆un∥22	PUNCT
ejde-636	449	17	=	=	PRON
ejde-636	449	18	∥∆u0∥22	∥∆u0∥22	PROPN
ejde-636	449	19	+	+	CCONJ
ejde-636	449	20	∥∆un,0∥22	∥∆un,0∥22	PROPN
ejde-636	449	21	+	+	X
ejde-636	449	22	on(1	on(1	NOUN
ejde-636	449	23	)	)	PUNCT
ejde-636	449	24	.	.	PUNCT
ejde-636	450	1	by	by	ADP
ejde-636	450	2	the	the	DET
ejde-636	450	3	splitting	splitting	NOUN
ejde-636	450	4	properties	property	NOUN
ejde-636	450	5	of	of	ADP
ejde-636	450	6	brezis	brezis	NOUN
ejde-636	450	7	-	-	PUNCT
ejde-636	450	8	lieb	lieb	NOUN
ejde-636	450	9	we	we	PRON
ejde-636	450	10	have	have	VERB
ejde-636	450	11	h(un	h(un	NOUN
ejde-636	450	12	)	)	PUNCT
ejde-636	450	13	=	=	SYM
ejde-636	450	14	h(u0	h(u0	NOUN
ejde-636	450	15	)	)	PUNCT
ejde-636	451	1	+	+	NOUN
ejde-636	451	2	h(un,0	h(un,0	NUM
ejde-636	451	3	)	)	PUNCT
ejde-636	452	1	+	+	CCONJ
ejde-636	452	2	on(1	on(1	NOUN
ejde-636	452	3	)	)	PUNCT
ejde-636	452	4	,	,	PUNCT
ejde-636	452	5	(	(	PUNCT
ejde-636	452	6	4.10	4.10	NUM
ejde-636	452	7	)	)	PUNCT
ejde-636	452	8	ep	ep	PROPN
ejde-636	452	9	,	,	PUNCT
ejde-636	452	10	q(un	q(un	PROPN
ejde-636	452	11	)	)	PUNCT
ejde-636	453	1	=	=	SYM
ejde-636	453	2	ep	ep	PROPN
ejde-636	453	3	,	,	PUNCT
ejde-636	453	4	q(u0	q(u0	PROPN
ejde-636	453	5	)	)	PUNCT
ejde-636	454	1	+	+	CCONJ
ejde-636	454	2	ep	ep	PROPN
ejde-636	454	3	,	,	PUNCT
ejde-636	454	4	q(un,0	q(un,0	NOUN
ejde-636	454	5	)	)	PUNCT
ejde-636	454	6	+	+	SYM
ejde-636	454	7	on(1	on(1	NOUN
ejde-636	454	8	)	)	PUNCT
ejde-636	454	9	,	,	PUNCT
ejde-636	454	10	(	(	PUNCT
ejde-636	454	11	4.11	4.11	NUM
ejde-636	454	12	)	)	PUNCT
ejde-636	454	13	qp	qp	PROPN
ejde-636	454	14	,	,	PUNCT
ejde-636	454	15	q(un	q(un	PROPN
ejde-636	454	16	)	)	PUNCT
ejde-636	454	17	=	=	SYM
ejde-636	454	18	qp	qp	PROPN
ejde-636	454	19	,	,	PUNCT
ejde-636	454	20	q(u0	q(u0	PROPN
ejde-636	454	21	)	)	PUNCT
ejde-636	455	1	+	+	NOUN
ejde-636	455	2	qp	qp	NOUN
ejde-636	455	3	,	,	PUNCT
ejde-636	455	4	q(un,0	q(un,0	NOUN
ejde-636	455	5	)	)	PUNCT
ejde-636	455	6	+	+	SYM
ejde-636	456	1	on(1	on(1	NOUN
ejde-636	456	2	)	)	PUNCT
ejde-636	456	3	.	.	PUNCT
ejde-636	457	1	(	(	PUNCT
ejde-636	457	2	4.12	4.12	NUM
ejde-636	457	3	)	)	PUNCT
ejde-636	457	4	we	we	PRON
ejde-636	457	5	claim	claim	VERB
ejde-636	457	6	that	that	SCONJ
ejde-636	457	7	qp	qp	ADP
ejde-636	457	8	,	,	PUNCT
ejde-636	457	9	q(u0	q(u0	PROPN
ejde-636	457	10	)	)	PUNCT
ejde-636	457	11	≤	≤	NUM
ejde-636	457	12	0	0	NUM
ejde-636	457	13	.	.	PUNCT
ejde-636	458	1	up	up	ADP
ejde-636	458	2	to	to	ADP
ejde-636	458	3	a	a	DET
ejde-636	458	4	subsequence	subsequence	NOUN
ejde-636	458	5	,	,	PUNCT
ejde-636	458	6	we	we	PRON
ejde-636	458	7	assume	assume	VERB
ejde-636	458	8	that	that	SCONJ
ejde-636	458	9	δn	δn	NOUN
ejde-636	458	10	:	:	PUNCT
ejde-636	458	11	=	=	NUM
ejde-636	458	12	∫	∫	PROPN
ejde-636	458	13	rn	rn	PROPN
ejde-636	458	14	|∆un,0|2dx+	|∆un,0|2dx+	NOUN
ejde-636	458	15	1	1	NUM
ejde-636	458	16	2	2	NUM
ejde-636	458	17	∫	∫	PROPN
ejde-636	458	18	rn	rn	PROPN
ejde-636	458	19	|∇un,0|2dx→	|∇un,0|2dx→	PROPN
ejde-636	458	20	δ0	δ0	PROPN
ejde-636	458	21	≥	≥	NOUN
ejde-636	458	22	0	0	NUM
ejde-636	458	23	.	.	PUNCT
ejde-636	459	1	now	now	ADV
ejde-636	459	2	we	we	PRON
ejde-636	459	3	need	need	VERB
ejde-636	459	4	to	to	PART
ejde-636	459	5	consider	consider	VERB
ejde-636	459	6	two	two	NUM
ejde-636	459	7	cases	case	NOUN
ejde-636	459	8	.	.	PUNCT
ejde-636	460	1	case	case	NOUN
ejde-636	460	2	1	1	NUM
ejde-636	460	3	.	.	PUNCT
ejde-636	460	4	δ0	δ0	NOUN
ejde-636	460	5	=	=	SYM
ejde-636	460	6	0	0	NUM
ejde-636	460	7	.	.	PUNCT
ejde-636	460	8	by	by	ADP
ejde-636	460	9	lemma	lemma	PROPN
ejde-636	460	10	2.3	2.3	NUM
ejde-636	460	11	,	,	PUNCT
ejde-636	460	12	for	for	ADP
ejde-636	460	13	any	any	DET
ejde-636	460	14	r	r	NOUN
ejde-636	460	15	∈	∈	NOUN
ejde-636	460	16	(	(	PUNCT
ejde-636	460	17	2	2	NUM
ejde-636	460	18	,	,	PUNCT
ejde-636	460	19	4∗	4∗	NOUN
ejde-636	460	20	)	)	PUNCT
ejde-636	460	21	,	,	PUNCT
ejde-636	460	22	we	we	PRON
ejde-636	460	23	have	have	VERB
ejde-636	460	24	∫	∫	PROPN
ejde-636	460	25	rn	rn	PROPN
ejde-636	460	26	|un,0|rdx	|un,0|rdx	PROPN
ejde-636	460	27	→	→	SYM
ejde-636	460	28	0	0	X
ejde-636	460	29	.	.	PUNCT
ejde-636	461	1	then	then	ADV
ejde-636	461	2	qp	qp	PROPN
ejde-636	461	3	,	,	PUNCT
ejde-636	461	4	q(un,0	q(un,0	NOUN
ejde-636	461	5	)	)	PUNCT
ejde-636	461	6	→	→	SYM
ejde-636	461	7	0	0	NUM
ejde-636	461	8	as	as	ADP
ejde-636	461	9	n→	n→	ADV
ejde-636	461	10	+	+	PROPN
ejde-636	461	11	∞.	∞.	PROPN
ejde-636	461	12	hence	hence	ADV
ejde-636	461	13	,	,	PUNCT
ejde-636	461	14	from	from	ADP
ejde-636	461	15	(	(	PUNCT
ejde-636	461	16	4.12	4.12	NUM
ejde-636	461	17	)	)	PUNCT
ejde-636	461	18	we	we	PRON
ejde-636	461	19	derive	derive	VERB
ejde-636	461	20	qp	qp	NOUN
ejde-636	461	21	,	,	PUNCT
ejde-636	461	22	q(u0	q(u0	PROPN
ejde-636	461	23	)	)	PUNCT
ejde-636	461	24	=	=	SYM
ejde-636	462	1	0	0	X
ejde-636	462	2	.	.	PUNCT
ejde-636	462	3	case	case	NOUN
ejde-636	462	4	2	2	NUM
ejde-636	462	5	.	.	PUNCT
ejde-636	463	1	δ0	δ0	VERB
ejde-636	463	2	>	>	X
ejde-636	463	3	0	0	X
ejde-636	463	4	.	.	PUNCT
ejde-636	464	1	by	by	ADP
ejde-636	464	2	contradiction	contradiction	NOUN
ejde-636	464	3	,	,	PUNCT
ejde-636	464	4	we	we	PRON
ejde-636	464	5	suppose	suppose	VERB
ejde-636	464	6	that	that	SCONJ
ejde-636	464	7	qp	qp	ADP
ejde-636	464	8	,	,	PUNCT
ejde-636	464	9	q(u0	q(u0	PROPN
ejde-636	464	10	)	)	PUNCT
ejde-636	464	11	>	>	X
ejde-636	464	12	0	0	X
ejde-636	464	13	.	.	PUNCT
ejde-636	465	1	from	from	ADP
ejde-636	465	2	(	(	PUNCT
ejde-636	465	3	4.12	4.12	NUM
ejde-636	465	4	)	)	PUNCT
ejde-636	465	5	it	it	PRON
ejde-636	465	6	follows	follow	VERB
ejde-636	465	7	that	that	SCONJ
ejde-636	465	8	qp	qp	ADP
ejde-636	465	9	,	,	PUNCT
ejde-636	465	10	q(un,0	q(un,0	NOUN
ejde-636	465	11	)	)	PUNCT
ejde-636	465	12	≤	≤	NOUN
ejde-636	465	13	0	0	NUM
ejde-636	465	14	.	.	PUNCT
ejde-636	466	1	according	accord	VERB
ejde-636	466	2	to	to	ADP
ejde-636	466	3	lemma	lemma	PROPN
ejde-636	466	4	4.1	4.1	NUM
ejde-636	466	5	,	,	PUNCT
ejde-636	466	6	there	there	PRON
ejde-636	466	7	exists	exist	VERB
ejde-636	466	8	sun,0	sun,0	NOUN
ejde-636	466	9	∈	∈	PROPN
ejde-636	466	10	(	(	PUNCT
ejde-636	466	11	0	0	NUM
ejde-636	466	12	,	,	PUNCT
ejde-636	466	13	1	1	NUM
ejde-636	466	14	]	]	PUNCT
ejde-636	466	15	such	such	ADJ
ejde-636	466	16	that	that	SCONJ
ejde-636	466	17	qp	qp	NOUN
ejde-636	466	18	,	,	PUNCT
ejde-636	466	19	q((un,0)sun,0	q((un,0)sun,0	PROPN
ejde-636	466	20	)	)	PUNCT
ejde-636	466	21	=	=	PUNCT
ejde-636	467	1	0	0	X
ejde-636	467	2	.	.	PUNCT
ejde-636	468	1	in	in	ADP
ejde-636	468	2	view	view	NOUN
ejde-636	468	3	of	of	ADP
ejde-636	468	4	the	the	DET
ejde-636	468	5	fact	fact	NOUN
ejde-636	468	6	that	that	SCONJ
ejde-636	468	7	h(s	h(	NOUN
ejde-636	468	8	)	)	PUNCT
ejde-636	468	9	|s|2	|s|2	NOUN
ejde-636	468	10	+	+	CCONJ
ejde-636	468	11	8	8	NUM
ejde-636	468	12	n	n	NOUN
ejde-636	468	13	is	be	AUX
ejde-636	468	14	strictly	strictly	ADV
ejde-636	468	15	increasing	increase	VERB
ejde-636	468	16	for	for	ADP
ejde-636	468	17	s	s	PROPN
ejde-636	468	18	∈	∈	PROPN
ejde-636	468	19	(	(	PUNCT
ejde-636	468	20	0,∞	0,∞	NOUN
ejde-636	468	21	)	)	PUNCT
ejde-636	468	22	,	,	PUNCT
ejde-636	468	23	we	we	PRON
ejde-636	468	24	deduce	deduce	VERB
ejde-636	468	25	ep	ep	PROPN
ejde-636	468	26	,	,	PUNCT
ejde-636	468	27	q(un,0)−	q(un,0)−	PROPN
ejde-636	468	28	ep	ep	PROPN
ejde-636	468	29	,	,	PUNCT
ejde-636	468	30	q((un,0)sun,0	q((un,0)sun,0	PROPN
ejde-636	468	31	)	)	PUNCT
ejde-636	469	1	=	=	PUNCT
ejde-636	470	1	1−	1−	NUM
ejde-636	470	2	s2un,0	s2un,0	NOUN
ejde-636	470	3	2	2	NUM
ejde-636	470	4	∫	∫	PROPN
ejde-636	470	5	rn	rn	PROPN
ejde-636	470	6	|∆un,0|2dx+	|∆un,0|2dx+	PROPN
ejde-636	470	7	1−	1−	NUM
ejde-636	470	8	sun,0	sun,0	NOUN
ejde-636	470	9	2	2	NUM
ejde-636	470	10	∫	∫	PROPN
ejde-636	470	11	rn	rn	PROPN
ejde-636	470	12	|∇un,0|2dx	|∇un,0|2dx	NUM
ejde-636	470	13	−	−	PROPN
ejde-636	470	14	∫	∫	PROPN
ejde-636	470	15	rn	rn	PROPN
ejde-636	470	16	f	f	PROPN
ejde-636	470	17	(	(	PUNCT
ejde-636	470	18	un,0)dx+	un,0)dx+	ADJ
ejde-636	470	19	s−n/2	s−n/2	PROPN
ejde-636	470	20	un,0	un,0	PROPN
ejde-636	470	21	∫	∫	PROPN
ejde-636	470	22	rn	rn	PROPN
ejde-636	470	23	f	f	PROPN
ejde-636	470	24	(	(	PUNCT
ejde-636	470	25	sn/4	sn/4	NOUN
ejde-636	470	26	un,0	un,0	PROPN
ejde-636	470	27	un,0)dx	un,0)dx	PROPN
ejde-636	470	28	=	=	SYM
ejde-636	470	29	1−	1−	NUM
ejde-636	470	30	s2un,0	s2un,0	PROPN
ejde-636	470	31	2	2	NUM
ejde-636	470	32	qp	qp	ADP
ejde-636	470	33	,	,	PUNCT
ejde-636	470	34	q(un,0	q(un,0	NOUN
ejde-636	470	35	)	)	PUNCT
ejde-636	470	36	+	+	CCONJ
ejde-636	470	37	(	(	PUNCT
ejde-636	470	38	1−	1−	NUM
ejde-636	470	39	sn,0	sn,0	NOUN
ejde-636	470	40	2	2	NUM
ejde-636	470	41	−	−	NOUN
ejde-636	470	42	1−	1−	NUM
ejde-636	470	43	s2n,0	s2n,0	NOUN
ejde-636	470	44	4	4	NUM
ejde-636	470	45	)	)	PUNCT
ejde-636	470	46	∫	∫	PROPN
ejde-636	470	47	rn	rn	PROPN
ejde-636	470	48	|∇un,0|2dx	|∇un,0|2dx	PRON
ejde-636	470	49	+	+	CCONJ
ejde-636	471	1	1−	1−	NUM
ejde-636	471	2	s2n,0	s2n,0	NOUN
ejde-636	471	3	2	2	NUM
ejde-636	471	4	n	n	NUM
ejde-636	471	5	4	4	NUM
ejde-636	471	6	∫	∫	PROPN
ejde-636	471	7	rn	rn	PROPN
ejde-636	471	8	(	(	PUNCT
ejde-636	471	9	f(un,0)un,0	f(un,0)un,0	PROPN
ejde-636	471	10	−	−	PROPN
ejde-636	471	11	2f	2f	NUM
ejde-636	471	12	(	(	PUNCT
ejde-636	471	13	un,0))dx	un,0))dx	INTJ
ejde-636	471	14	−	−	PROPN
ejde-636	471	15	∫	∫	PROPN
ejde-636	471	16	rn	rn	PROPN
ejde-636	471	17	f	f	PROPN
ejde-636	471	18	(	(	PUNCT
ejde-636	471	19	un,0)dx+	un,0)dx+	ADJ
ejde-636	471	20	s−n/2	s−n/2	PROPN
ejde-636	471	21	un,0	un,0	PROPN
ejde-636	471	22	∫	∫	PROPN
ejde-636	471	23	rn	rn	PROPN
ejde-636	471	24	f	f	PROPN
ejde-636	471	25	(	(	PUNCT
ejde-636	471	26	sn/4	sn/4	NOUN
ejde-636	471	27	un,0	un,0	PROPN
ejde-636	471	28	un,0)dx	un,0)dx	PROPN
ejde-636	471	29	≥	≥	NUM
ejde-636	471	30	1−	1−	NUM
ejde-636	471	31	s2n,0	s2n,0	NOUN
ejde-636	471	32	2	2	NUM
ejde-636	471	33	n	n	NUM
ejde-636	471	34	4	4	NUM
ejde-636	471	35	∫	∫	PROPN
ejde-636	471	36	rn	rn	PROPN
ejde-636	471	37	(	(	PUNCT
ejde-636	471	38	f(un,0)un,0	f(un,0)un,0	PROPN
ejde-636	471	39	−	−	PROPN
ejde-636	471	40	2f	2f	NUM
ejde-636	471	41	(	(	PUNCT
ejde-636	471	42	un,0))dx	un,0))dx	INTJ
ejde-636	471	43	−	−	PROPN
ejde-636	471	44	∫	∫	PROPN
ejde-636	471	45	rn	rn	PROPN
ejde-636	471	46	f	f	PROPN
ejde-636	471	47	(	(	PUNCT
ejde-636	471	48	un,0)dx+	un,0)dx+	ADJ
ejde-636	471	49	s−n/2	s−n/2	PROPN
ejde-636	471	50	un,0	un,0	PROPN
ejde-636	471	51	∫	∫	PROPN
ejde-636	471	52	rn	rn	PROPN
ejde-636	471	53	f	f	PROPN
ejde-636	471	54	(	(	PUNCT
ejde-636	471	55	sn/4	sn/4	NOUN
ejde-636	471	56	un,0	un,0	PROPN
ejde-636	471	57	un,0)dx+	un,0)dx+	VERB
ejde-636	471	58	1−	1−	NUM
ejde-636	471	59	s2un,0	s2un,0	PROPN
ejde-636	471	60	2	2	NUM
ejde-636	471	61	qp	qp	ADP
ejde-636	471	62	,	,	PUNCT
ejde-636	471	63	q(un,0	q(un,0	NOUN
ejde-636	471	64	)	)	PUNCT
ejde-636	471	65	=	=	SYM
ejde-636	472	1	∫	∫	PROPN
ejde-636	472	2	rn	rn	PROPN
ejde-636	472	3	∫	∫	PROPN
ejde-636	472	4	1	1	NUM
ejde-636	472	5	sn,0	sn,0	PROPN
ejde-636	472	6	n	n	CCONJ
ejde-636	472	7	4	4	NUM
ejde-636	472	8	s|un,0|2	s|un,0|2	ADJ
ejde-636	472	9	+	+	NUM
ejde-636	472	10	8	8	NUM
ejde-636	472	11	n	n	PRON
ejde-636	472	12	(	(	PUNCT
ejde-636	472	13	h(un,0	h(un,0	NOUN
ejde-636	472	14	)	)	PUNCT
ejde-636	473	1	|un,0|2	|un,0|2	X
ejde-636	473	2	+	+	NUM
ejde-636	473	3	8	8	NUM
ejde-636	473	4	n	n	NUM
ejde-636	473	5	−	−	PROPN
ejde-636	473	6	h(sn/4un,0	h(sn/4un,0	NOUN
ejde-636	473	7	)	)	PUNCT
ejde-636	473	8	|sn/4un,0|2	|sn/4un,0|2	ADV
ejde-636	473	9	+	+	NUM
ejde-636	473	10	8	8	NUM
ejde-636	473	11	n	n	NOUN
ejde-636	473	12	)	)	PUNCT
ejde-636	473	13	dsdx	dsdx	NOUN
ejde-636	473	14	+	+	CCONJ
ejde-636	473	15	1−	1−	NUM
ejde-636	473	16	s2un,0	s2un,0	PROPN
ejde-636	473	17	2	2	NUM
ejde-636	473	18	qp	qp	ADP
ejde-636	473	19	,	,	PUNCT
ejde-636	473	20	q(un,0	q(un,0	NOUN
ejde-636	473	21	)	)	PUNCT
ejde-636	473	22	≥	≥	NOUN
ejde-636	473	23	1−	1−	NUM
ejde-636	473	24	sun,0	sun,0	NOUN
ejde-636	473	25	2	2	NUM
ejde-636	473	26	qp	qp	NOUN
ejde-636	473	27	,	,	PUNCT
ejde-636	473	28	q(un,0	q(un,0	NOUN
ejde-636	473	29	)	)	PUNCT
ejde-636	473	30	.	.	PUNCT
ejde-636	474	1	16	16	NUM
ejde-636	474	2	z.	z.	PROPN
ejde-636	474	3	ma	ma	PROPN
ejde-636	474	4	,	,	PUNCT
ejde-636	474	5	x.	x.	PROPN
ejde-636	474	6	chang	chang	PROPN
ejde-636	474	7	,	,	PUNCT
ejde-636	474	8	z.	z.	PROPN
ejde-636	474	9	feng	feng	PROPN
ejde-636	474	10	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	474	11	we	we	PRON
ejde-636	474	12	denote	denote	VERB
ejde-636	474	13	cn,0	cn,0	PROPN
ejde-636	474	14	:	:	PUNCT
ejde-636	474	15	=	=	PUNCT
ejde-636	474	16	∥un,0∥22	∥un,0∥22	X
ejde-636	474	17	.	.	PUNCT
ejde-636	475	1	clearly	clearly	ADV
ejde-636	475	2	,	,	PUNCT
ejde-636	475	3	cn,0	cn,0	PROPN
ejde-636	475	4	≤	≤	PROPN
ejde-636	475	5	c.	c.	NOUN
ejde-636	475	6	from	from	ADP
ejde-636	475	7	lemma	lemma	PROPN
ejde-636	475	8	4.4	4.4	NUM
ejde-636	475	9	we	we	PRON
ejde-636	475	10	derive	derive	VERB
ejde-636	475	11	mp	mp	PROPN
ejde-636	475	12	,	,	PUNCT
ejde-636	475	13	q(c	q(c	PROPN
ejde-636	475	14	)	)	PUNCT
ejde-636	476	1	=	=	VERB
ejde-636	476	2	lim	lim	PROPN
ejde-636	476	3	n→+∞	n→+∞	PROPN
ejde-636	476	4	(	(	PUNCT
ejde-636	476	5	ep	ep	PROPN
ejde-636	476	6	,	,	PUNCT
ejde-636	476	7	q(un)−	q(un)−	VERB
ejde-636	476	8	1	1	NUM
ejde-636	476	9	2	2	NUM
ejde-636	476	10	qp	qp	NOUN
ejde-636	476	11	,	,	PUNCT
ejde-636	476	12	q(un	q(un	PROPN
ejde-636	476	13	)	)	PUNCT
ejde-636	476	14	)	)	PUNCT
ejde-636	477	1	=	=	VERB
ejde-636	477	2	lim	lim	PROPN
ejde-636	477	3	n→+∞	n→+∞	VERB
ejde-636	477	4	[	[	X
ejde-636	477	5	(	(	PUNCT
ejde-636	477	6	n	n	CCONJ
ejde-636	477	7	8	8	NUM
ejde-636	477	8	∫	∫	PROPN
ejde-636	477	9	rn	rn	PROPN
ejde-636	477	10	h(un)dx−	h(un)dx−	PROPN
ejde-636	477	11	∫	∫	PROPN
ejde-636	477	12	rn	rn	PROPN
ejde-636	477	13	f	f	PROPN
ejde-636	477	14	(	(	PUNCT
ejde-636	477	15	un)dx	un)dx	PUNCT
ejde-636	477	16	)	)	PUNCT
ejde-636	478	1	+	+	CCONJ
ejde-636	478	2	1	1	NUM
ejde-636	478	3	4	4	NUM
ejde-636	478	4	∥∇un∥22	∥∇un∥22	NUM
ejde-636	478	5	]	]	PUNCT
ejde-636	479	1	=	=	SYM
ejde-636	479	2	(	(	PUNCT
ejde-636	479	3	n	n	CCONJ
ejde-636	479	4	8	8	NUM
ejde-636	479	5	∫	∫	PROPN
ejde-636	479	6	rn	rn	PROPN
ejde-636	479	7	h(u0)dx−	h(u0)dx−	PROPN
ejde-636	479	8	∫	∫	PROPN
ejde-636	479	9	rn	rn	PROPN
ejde-636	479	10	f	f	PROPN
ejde-636	479	11	(	(	PUNCT
ejde-636	479	12	u0)dx+	u0)dx+	NUM
ejde-636	479	13	1	1	NUM
ejde-636	479	14	4	4	NUM
ejde-636	479	15	∥∇u0∥22	∥∇u0∥22	NUM
ejde-636	479	16	)	)	PUNCT
ejde-636	480	1	+	+	CCONJ
ejde-636	480	2	lim	lim	PROPN
ejde-636	480	3	n→+∞	n→+∞	PROPN
ejde-636	480	4	(	(	PUNCT
ejde-636	480	5	n	n	CCONJ
ejde-636	480	6	8	8	NUM
ejde-636	480	7	∫	∫	PROPN
ejde-636	480	8	rn	rn	PROPN
ejde-636	480	9	h(un,0)dx−	h(un,0)dx−	PROPN
ejde-636	480	10	∫	∫	PROPN
ejde-636	481	1	rn	rn	PROPN
ejde-636	481	2	f	f	PROPN
ejde-636	481	3	(	(	PUNCT
ejde-636	481	4	un,0)dx+	un,0)dx+	PROPN
ejde-636	481	5	1	1	NUM
ejde-636	481	6	4	4	NUM
ejde-636	481	7	∥∇un,0∥22	∥∇un,0∥22	PROPN
ejde-636	481	8	)	)	PUNCT
ejde-636	482	1	=	=	PUNCT
ejde-636	483	1	[	[	X
ejde-636	483	2	n	n	NUM
ejde-636	483	3	8	8	NUM
ejde-636	483	4	∫	∫	PROPN
ejde-636	483	5	rn	rn	PROPN
ejde-636	483	6	(	(	PUNCT
ejde-636	483	7	f(u0)u0	f(u0)u0	ADV
ejde-636	483	8	−	−	PROPN
ejde-636	483	9	(	(	PUNCT
ejde-636	483	10	2	2	NUM
ejde-636	483	11	+	+	SYM
ejde-636	483	12	8	8	NUM
ejde-636	483	13	n	n	NOUN
ejde-636	483	14	)	)	PUNCT
ejde-636	483	15	f	f	PROPN
ejde-636	483	16	(	(	PUNCT
ejde-636	483	17	u0	u0	ADJ
ejde-636	483	18	)	)	PUNCT
ejde-636	483	19	)	)	PUNCT
ejde-636	483	20	dx+	dx+	NOUN
ejde-636	483	21	1	1	NUM
ejde-636	483	22	4	4	NUM
ejde-636	483	23	∥∇u0∥22	∥∇u0∥22	NUM
ejde-636	483	24	]	]	PUNCT
ejde-636	484	1	+	+	CCONJ
ejde-636	484	2	lim	lim	PROPN
ejde-636	484	3	n→+∞	n→+∞	PROPN
ejde-636	484	4	(	(	PUNCT
ejde-636	484	5	ep	ep	PROPN
ejde-636	484	6	,	,	PUNCT
ejde-636	484	7	q(un,0)−	q(un,0)−	PROPN
ejde-636	484	8	1	1	NUM
ejde-636	484	9	2	2	NUM
ejde-636	484	10	qp	qp	NOUN
ejde-636	484	11	,	,	PUNCT
ejde-636	484	12	q(un,0	q(un,0	NOUN
ejde-636	484	13	)	)	PUNCT
ejde-636	484	14	)	)	PUNCT
ejde-636	484	15	≥	≥	PROPN
ejde-636	484	16	lim	lim	PROPN
ejde-636	484	17	n→+∞	n→+∞	PROPN
ejde-636	484	18	(	(	PUNCT
ejde-636	484	19	ep	ep	PROPN
ejde-636	484	20	,	,	PUNCT
ejde-636	484	21	q(un,0)−	q(un,0)−	PROPN
ejde-636	484	22	1	1	NUM
ejde-636	484	23	2	2	NUM
ejde-636	484	24	qp	qp	NOUN
ejde-636	484	25	,	,	PUNCT
ejde-636	484	26	q(un,0	q(un,0	NOUN
ejde-636	484	27	)	)	PUNCT
ejde-636	484	28	)	)	PUNCT
ejde-636	485	1	≥	≥	PROPN
ejde-636	485	2	lim	lim	PROPN
ejde-636	485	3	n→+∞	n→+∞	PROPN
ejde-636	485	4	(	(	PUNCT
ejde-636	485	5	ep	ep	PROPN
ejde-636	485	6	,	,	PUNCT
ejde-636	485	7	q((un,0)sun,0	q((un,0)sun,0	PROPN
ejde-636	485	8	)	)	PUNCT
ejde-636	485	9	−	−	PROPN
ejde-636	486	1	s2un,0	s2un,0	NOUN
ejde-636	486	2	2	2	NUM
ejde-636	486	3	qp	qp	NOUN
ejde-636	486	4	,	,	PUNCT
ejde-636	486	5	q(un,0	q(un,0	NOUN
ejde-636	486	6	)	)	PUNCT
ejde-636	486	7	)	)	PUNCT
ejde-636	487	1	≥	≥	PROPN
ejde-636	487	2	lim	lim	PROPN
ejde-636	487	3	n→+∞	n→+∞	VERB
ejde-636	487	4	ep	ep	PROPN
ejde-636	487	5	,	,	PUNCT
ejde-636	487	6	q((un,0)sun,0	q((un,0)sun,0	PROPN
ejde-636	487	7	)	)	PUNCT
ejde-636	487	8	≥	≥	PROPN
ejde-636	487	9	lim	lim	PROPN
ejde-636	487	10	n→+∞	n→+∞	PROPN
ejde-636	487	11	mp	mp	PROPN
ejde-636	487	12	,	,	PUNCT
ejde-636	487	13	q(cn,0	q(cn,0	PROPN
ejde-636	487	14	)	)	PUNCT
ejde-636	487	15	≥	≥	PROPN
ejde-636	487	16	mp	mp	PROPN
ejde-636	487	17	,	,	PUNCT
ejde-636	487	18	q(c	q(c	PROPN
ejde-636	487	19	)	)	PUNCT
ejde-636	487	20	.	.	PUNCT
ejde-636	488	1	this	this	PRON
ejde-636	488	2	indicates	indicate	VERB
ejde-636	488	3	that	that	SCONJ
ejde-636	488	4	limn→+∞qp	limn→+∞qp	NUM
ejde-636	488	5	,	,	PUNCT
ejde-636	488	6	q(un,0	q(un,0	NOUN
ejde-636	488	7	)	)	PUNCT
ejde-636	488	8	=	=	SYM
ejde-636	488	9	0	0	PUNCT
ejde-636	488	10	and	and	CCONJ
ejde-636	488	11	lim	lim	PROPN
ejde-636	488	12	n→+∞	n→+∞	VERB
ejde-636	488	13	ep	ep	PROPN
ejde-636	488	14	,	,	PUNCT
ejde-636	488	15	q(un,0	q(un,0	NOUN
ejde-636	488	16	)	)	PUNCT
ejde-636	489	1	=	=	VERB
ejde-636	489	2	lim	lim	PROPN
ejde-636	489	3	n→+∞	n→+∞	PROPN
ejde-636	489	4	mp	mp	PROPN
ejde-636	489	5	,	,	PUNCT
ejde-636	489	6	q(cn,0	q(cn,0	NOUN
ejde-636	489	7	)	)	PUNCT
ejde-636	490	1	=	=	SYM
ejde-636	490	2	mp	mp	PROPN
ejde-636	490	3	,	,	PUNCT
ejde-636	490	4	q(c	q(c	PROPN
ejde-636	490	5	)	)	PUNCT
ejde-636	490	6	.	.	PUNCT
ejde-636	491	1	(	(	PUNCT
ejde-636	491	2	4.13	4.13	NUM
ejde-636	491	3	)	)	PUNCT
ejde-636	491	4	on	on	ADP
ejde-636	491	5	the	the	DET
ejde-636	491	6	other	other	ADJ
ejde-636	491	7	hand	hand	NOUN
ejde-636	491	8	,	,	PUNCT
ejde-636	491	9	combining	combine	VERB
ejde-636	491	10	(	(	PUNCT
ejde-636	491	11	4.9	4.9	NUM
ejde-636	491	12	)	)	PUNCT
ejde-636	491	13	and	and	CCONJ
ejde-636	491	14	(	(	PUNCT
ejde-636	491	15	4.11	4.11	NUM
ejde-636	491	16	)	)	PUNCT
ejde-636	491	17	yields	yield	NOUN
ejde-636	491	18	mp	mp	PROPN
ejde-636	491	19	,	,	PUNCT
ejde-636	491	20	q(c	q(c	PROPN
ejde-636	491	21	)	)	PUNCT
ejde-636	491	22	=	=	SYM
ejde-636	491	23	ep	ep	PROPN
ejde-636	491	24	,	,	PUNCT
ejde-636	491	25	q(un	q(un	PROPN
ejde-636	491	26	)	)	PUNCT
ejde-636	491	27	+	+	SYM
ejde-636	491	28	on(1	on(1	NOUN
ejde-636	491	29	)	)	PUNCT
ejde-636	491	30	=	=	SYM
ejde-636	491	31	ep	ep	PROPN
ejde-636	491	32	,	,	PUNCT
ejde-636	491	33	q(u0	q(u0	PROPN
ejde-636	491	34	)	)	PUNCT
ejde-636	492	1	+	+	CCONJ
ejde-636	492	2	ep	ep	PROPN
ejde-636	492	3	,	,	PUNCT
ejde-636	492	4	q(un,0	q(un,0	NOUN
ejde-636	492	5	)	)	PUNCT
ejde-636	492	6	+	+	SYM
ejde-636	492	7	on(1	on(1	NOUN
ejde-636	492	8	)	)	PUNCT
ejde-636	492	9	.	.	PUNCT
ejde-636	493	1	in	in	ADP
ejde-636	493	2	view	view	NOUN
ejde-636	493	3	of	of	ADP
ejde-636	493	4	ep	ep	PROPN
ejde-636	493	5	,	,	PUNCT
ejde-636	493	6	q(u0	q(u0	PROPN
ejde-636	493	7	)	)	PUNCT
ejde-636	493	8	>	>	X
ejde-636	493	9	0	0	NUM
ejde-636	493	10	,	,	PUNCT
ejde-636	493	11	from	from	ADP
ejde-636	493	12	(	(	PUNCT
ejde-636	493	13	4.13	4.13	NUM
ejde-636	493	14	)	)	PUNCT
ejde-636	493	15	it	it	PRON
ejde-636	493	16	follows	follow	VERB
ejde-636	493	17	that	that	SCONJ
ejde-636	493	18	mp	mp	NOUN
ejde-636	493	19	,	,	PUNCT
ejde-636	493	20	q(c	q(c	PROPN
ejde-636	493	21	)	)	PUNCT
ejde-636	493	22	>	>	X
ejde-636	493	23	mp	mp	PROPN
ejde-636	493	24	,	,	PUNCT
ejde-636	493	25	q(c)−	q(c)−	PROPN
ejde-636	493	26	ep	ep	PROPN
ejde-636	493	27	,	,	PUNCT
ejde-636	493	28	q(u0	q(u0	PROPN
ejde-636	493	29	)	)	PUNCT
ejde-636	494	1	=	=	VERB
ejde-636	494	2	lim	lim	PROPN
ejde-636	494	3	n→+∞	n→+∞	VERB
ejde-636	494	4	ep	ep	PROPN
ejde-636	494	5	,	,	PUNCT
ejde-636	494	6	q(un,0	q(un,0	NOUN
ejde-636	494	7	)	)	PUNCT
ejde-636	495	1	=	=	VERB
ejde-636	495	2	lim	lim	PROPN
ejde-636	495	3	n→+∞	n→+∞	PROPN
ejde-636	495	4	mp	mp	PROPN
ejde-636	495	5	,	,	PUNCT
ejde-636	495	6	q(cn,0	q(cn,0	NOUN
ejde-636	495	7	)	)	PUNCT
ejde-636	496	1	=	=	SYM
ejde-636	496	2	mp	mp	PROPN
ejde-636	496	3	,	,	PUNCT
ejde-636	496	4	q(c	q(c	PROPN
ejde-636	496	5	)	)	PUNCT
ejde-636	496	6	.	.	PUNCT
ejde-636	497	1	this	this	PRON
ejde-636	497	2	yields	yield	VERB
ejde-636	497	3	a	a	DET
ejde-636	497	4	contradiction	contradiction	NOUN
ejde-636	497	5	.	.	PUNCT
ejde-636	498	1	using	use	VERB
ejde-636	498	2	qp	qp	NOUN
ejde-636	498	3	,	,	PUNCT
ejde-636	498	4	q(u0	q(u0	PROPN
ejde-636	498	5	)	)	PUNCT
ejde-636	498	6	≤	≤	NOUN
ejde-636	498	7	0	0	NUM
ejde-636	499	1	and	and	CCONJ
ejde-636	499	2	similar	similar	ADJ
ejde-636	499	3	arguments	argument	NOUN
ejde-636	499	4	as	as	ADP
ejde-636	499	5	above	above	ADV
ejde-636	499	6	,	,	PUNCT
ejde-636	499	7	there	there	PRON
ejde-636	499	8	exists	exist	VERB
ejde-636	499	9	s0	s0	PROPN
ejde-636	499	10	∈	∈	PROPN
ejde-636	499	11	(	(	PUNCT
ejde-636	499	12	0	0	NUM
ejde-636	499	13	,	,	PUNCT
ejde-636	499	14	1	1	NUM
ejde-636	499	15	]	]	PUNCT
ejde-636	499	16	such	such	ADJ
ejde-636	499	17	that	that	PRON
ejde-636	499	18	(	(	PUNCT
ejde-636	499	19	u0)s0	u0)s0	PROPN
ejde-636	499	20	∈	∈	PROPN
ejde-636	499	21	qp	qp	PROPN
ejde-636	499	22	,	,	PUNCT
ejde-636	499	23	q(c0	q(c0	NOUN
ejde-636	499	24	)	)	PUNCT
ejde-636	499	25	and	and	CCONJ
ejde-636	499	26	ep	ep	PROPN
ejde-636	499	27	,	,	PUNCT
ejde-636	499	28	q(u0)−	q(u0)−	PROPN
ejde-636	499	29	ep	ep	PROPN
ejde-636	499	30	,	,	PUNCT
ejde-636	499	31	q((u0)s0	q((u0)s0	PROPN
ejde-636	499	32	)	)	PUNCT
ejde-636	499	33	≥	≥	PROPN
ejde-636	499	34	1−	1−	NUM
ejde-636	499	35	s20	s20	PROPN
ejde-636	499	36	2	2	NUM
ejde-636	499	37	qp	qp	NOUN
ejde-636	499	38	,	,	PUNCT
ejde-636	499	39	q(u0	q(u0	PROPN
ejde-636	499	40	)	)	PUNCT
ejde-636	499	41	.	.	PUNCT
ejde-636	500	1	(	(	PUNCT
ejde-636	500	2	4.14	4.14	NUM
ejde-636	500	3	)	)	PUNCT
ejde-636	500	4	we	we	PRON
ejde-636	500	5	denote	denote	VERB
ejde-636	500	6	c0	c0	PROPN
ejde-636	500	7	=	=	PUNCT
ejde-636	501	1	∥u0∥22	∥u0∥22	PROPN
ejde-636	501	2	.	.	PUNCT
ejde-636	502	1	clearly	clearly	ADV
ejde-636	502	2	,	,	PUNCT
ejde-636	502	3	c0	c0	PROPN
ejde-636	502	4	∈	∈	PROPN
ejde-636	502	5	(	(	PUNCT
ejde-636	502	6	0	0	NUM
ejde-636	502	7	,	,	PUNCT
ejde-636	502	8	c	c	NOUN
ejde-636	502	9	]	]	PUNCT
ejde-636	502	10	.	.	PUNCT
ejde-636	503	1	by	by	ADP
ejde-636	503	2	(	(	PUNCT
ejde-636	503	3	4.14	4.14	NUM
ejde-636	503	4	)	)	PUNCT
ejde-636	503	5	and	and	CCONJ
ejde-636	503	6	lemma	lemma	PROPN
ejde-636	503	7	4.4	4.4	NUM
ejde-636	503	8	we	we	PRON
ejde-636	503	9	have	have	VERB
ejde-636	503	10	mp	mp	NOUN
ejde-636	503	11	,	,	PUNCT
ejde-636	503	12	q(c	q(c	PROPN
ejde-636	503	13	)	)	PUNCT
ejde-636	504	1	=	=	VERB
ejde-636	504	2	lim	lim	PROPN
ejde-636	504	3	n→+∞	n→+∞	PROPN
ejde-636	504	4	(	(	PUNCT
ejde-636	504	5	ep	ep	PROPN
ejde-636	504	6	,	,	PUNCT
ejde-636	504	7	q(un)−	q(un)−	VERB
ejde-636	504	8	1	1	NUM
ejde-636	504	9	2	2	NUM
ejde-636	504	10	qp	qp	NOUN
ejde-636	504	11	,	,	PUNCT
ejde-636	504	12	q(un	q(un	PROPN
ejde-636	504	13	)	)	PUNCT
ejde-636	504	14	)	)	PUNCT
ejde-636	505	1	=	=	VERB
ejde-636	505	2	lim	lim	PROPN
ejde-636	505	3	n→+∞	n→+∞	VERB
ejde-636	505	4	[	[	X
ejde-636	505	5	(	(	PUNCT
ejde-636	505	6	n	n	CCONJ
ejde-636	505	7	8	8	NUM
ejde-636	505	8	∫	∫	PROPN
ejde-636	505	9	rn	rn	PROPN
ejde-636	505	10	h(un)dx−	h(un)dx−	PROPN
ejde-636	505	11	∫	∫	PROPN
ejde-636	505	12	rn	rn	PROPN
ejde-636	505	13	f	f	PROPN
ejde-636	505	14	(	(	PUNCT
ejde-636	505	15	un)dx	un)dx	PUNCT
ejde-636	505	16	)	)	PUNCT
ejde-636	506	1	+	+	CCONJ
ejde-636	506	2	1	1	NUM
ejde-636	506	3	4	4	NUM
ejde-636	506	4	∥∇un∥22	∥∇un∥22	NUM
ejde-636	506	5	]	]	PUNCT
ejde-636	507	1	=	=	PUNCT
ejde-636	507	2	lim	lim	PROPN
ejde-636	507	3	n→+∞	n→+∞	VERB
ejde-636	508	1	[	[	X
ejde-636	508	2	n	n	NUM
ejde-636	508	3	8	8	NUM
ejde-636	508	4	∫	∫	PROPN
ejde-636	508	5	rn	rn	PROPN
ejde-636	508	6	(	(	PUNCT
ejde-636	508	7	f(un,0)un,0	f(un,0)un,0	PROPN
ejde-636	508	8	−	−	PROPN
ejde-636	508	9	(	(	PUNCT
ejde-636	508	10	2	2	NUM
ejde-636	508	11	+	+	SYM
ejde-636	508	12	8	8	NUM
ejde-636	508	13	n	n	NOUN
ejde-636	508	14	)	)	PUNCT
ejde-636	508	15	f	f	PROPN
ejde-636	508	16	(	(	PUNCT
ejde-636	508	17	un,0	un,0	PROPN
ejde-636	508	18	)	)	PUNCT
ejde-636	508	19	)	)	PUNCT
ejde-636	508	20	dx	dx	PROPN
ejde-636	509	1	+	+	CCONJ
ejde-636	509	2	1	1	NUM
ejde-636	509	3	4	4	NUM
ejde-636	509	4	∥∇un,0∥22	∥∇un,0∥22	PROPN
ejde-636	509	5	]	]	PUNCT
ejde-636	510	1	+	+	CCONJ
ejde-636	510	2	(	(	PUNCT
ejde-636	510	3	ep	ep	PROPN
ejde-636	510	4	,	,	PUNCT
ejde-636	510	5	q(u0)−	q(u0)−	PROPN
ejde-636	510	6	1	1	NUM
ejde-636	510	7	2	2	NUM
ejde-636	510	8	qp	qp	NOUN
ejde-636	510	9	,	,	PUNCT
ejde-636	510	10	q(u0	q(u0	PROPN
ejde-636	510	11	)	)	PUNCT
ejde-636	510	12	)	)	PUNCT
ejde-636	510	13	≥	≥	PROPN
ejde-636	510	14	ep	ep	PROPN
ejde-636	510	15	,	,	PUNCT
ejde-636	510	16	q((u0)s0)−	q((u0)s0)−	NOUN
ejde-636	510	17	s20	s20	PROPN
ejde-636	510	18	2	2	NUM
ejde-636	510	19	qp	qp	PROPN
ejde-636	510	20	,	,	PUNCT
ejde-636	510	21	q(u0	q(u0	NOUN
ejde-636	510	22	)	)	PUNCT
ejde-636	510	23	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	510	24	dispersion	dispersion	NOUN
ejde-636	510	25	nonlinear	nonlinear	NOUN
ejde-636	510	26	schrödinger	schrödinger	NOUN
ejde-636	510	27	equation	equation	NOUN
ejde-636	510	28	17	17	NUM
ejde-636	510	29	≥	≥	NOUN
ejde-636	510	30	mp	mp	PROPN
ejde-636	510	31	,	,	PUNCT
ejde-636	510	32	q(c0	q(c0	NOUN
ejde-636	510	33	)	)	PUNCT
ejde-636	510	34	≥	≥	PROPN
ejde-636	510	35	mp	mp	PROPN
ejde-636	510	36	,	,	PUNCT
ejde-636	510	37	q(c	q(c	PROPN
ejde-636	510	38	)	)	PUNCT
ejde-636	510	39	,	,	PUNCT
ejde-636	510	40	which	which	PRON
ejde-636	510	41	implies	imply	VERB
ejde-636	510	42	mp	mp	PROPN
ejde-636	510	43	,	,	PUNCT
ejde-636	510	44	q(c0	q(c0	NOUN
ejde-636	510	45	)	)	PUNCT
ejde-636	510	46	=	=	SYM
ejde-636	510	47	mp	mp	PROPN
ejde-636	510	48	,	,	PUNCT
ejde-636	510	49	q(c	q(c	PROPN
ejde-636	510	50	)	)	PUNCT
ejde-636	510	51	and	and	CCONJ
ejde-636	510	52	qp	qp	PROPN
ejde-636	510	53	,	,	PUNCT
ejde-636	510	54	q(u0	q(u0	PROPN
ejde-636	510	55	)	)	PUNCT
ejde-636	510	56	=	=	SYM
ejde-636	511	1	0	0	NUM
ejde-636	511	2	,	,	PUNCT
ejde-636	511	3	that	that	ADV
ejde-636	511	4	is	is	ADV
ejde-636	511	5	,	,	PUNCT
ejde-636	511	6	s0	s0	PROPN
ejde-636	511	7	=	=	SYM
ejde-636	511	8	1	1	X
ejde-636	511	9	.	.	PUNCT
ejde-636	512	1	thus	thus	ADV
ejde-636	512	2	we	we	PRON
ejde-636	512	3	have	have	VERB
ejde-636	512	4	u0	u0	PROPN
ejde-636	512	5	∈	∈	PROPN
ejde-636	512	6	qp	qp	PROPN
ejde-636	512	7	,	,	PUNCT
ejde-636	512	8	q(c0	q(c0	NOUN
ejde-636	512	9	)	)	PUNCT
ejde-636	512	10	and	and	CCONJ
ejde-636	512	11	ep	ep	PROPN
ejde-636	512	12	,	,	PUNCT
ejde-636	512	13	q(u0	q(u0	PROPN
ejde-636	512	14	)	)	PUNCT
ejde-636	512	15	=	=	SYM
ejde-636	512	16	mp	mp	PROPN
ejde-636	512	17	,	,	PUNCT
ejde-636	512	18	q(c0	q(c0	NOUN
ejde-636	512	19	)	)	PUNCT
ejde-636	512	20	.	.	PUNCT
ejde-636	513	1	using	use	VERB
ejde-636	513	2	lemma	lemma	PROPN
ejde-636	513	3	4.6	4.6	NUM
ejde-636	513	4	at	at	ADP
ejde-636	513	5	c0	c0	PROPN
ejde-636	513	6	and	and	CCONJ
ejde-636	513	7	mp	mp	PROPN
ejde-636	513	8	,	,	PUNCT
ejde-636	513	9	q(c0	q(c0	NOUN
ejde-636	513	10	)	)	PUNCT
ejde-636	513	11	=	=	SYM
ejde-636	513	12	mp	mp	PROPN
ejde-636	513	13	,	,	PUNCT
ejde-636	513	14	q(c	q(c	PROPN
ejde-636	513	15	)	)	PUNCT
ejde-636	513	16	,	,	PUNCT
ejde-636	513	17	we	we	PRON
ejde-636	513	18	obtain	obtain	VERB
ejde-636	513	19	c0	c0	NOUN
ejde-636	513	20	=	=	PROPN
ejde-636	513	21	c	c	PROPN
ejde-636	513	22	and	and	CCONJ
ejde-636	513	23	thus	thus	ADV
ejde-636	513	24	ep	ep	PROPN
ejde-636	513	25	,	,	PUNCT
ejde-636	513	26	q(u0	q(u0	PROPN
ejde-636	513	27	)	)	PUNCT
ejde-636	513	28	=	=	SYM
ejde-636	513	29	mp	mp	PROPN
ejde-636	513	30	,	,	PUNCT
ejde-636	513	31	q(c	q(c	PROPN
ejde-636	513	32	)	)	PUNCT
ejde-636	513	33	.	.	PUNCT
ejde-636	514	1	□	□	PUNCT
ejde-636	514	2	proof	proof	NOUN
ejde-636	514	3	of	of	ADP
ejde-636	514	4	theorem	theorem	NOUN
ejde-636	514	5	1.3	1.3	NUM
ejde-636	514	6	.	.	PUNCT
ejde-636	514	7	consider	consider	VERB
ejde-636	514	8	the	the	DET
ejde-636	514	9	functional	functional	ADJ
ejde-636	514	10	ψ(u	ψ(u	PROPN
ejde-636	514	11	)	)	PUNCT
ejde-636	514	12	:	:	PUNCT
ejde-636	515	1	s(c	s(c	X
ejde-636	515	2	)	)	PUNCT
ejde-636	515	3	→	→	SYM
ejde-636	515	4	r	r	NOUN
ejde-636	515	5	defined	define	VERB
ejde-636	515	6	by	by	ADP
ejde-636	515	7	ψ(u	ψ(u	PROPN
ejde-636	515	8	)	)	PUNCT
ejde-636	515	9	:	:	PUNCT
ejde-636	515	10	=	=	SYM
ejde-636	515	11	ep	ep	PROPN
ejde-636	515	12	,	,	PUNCT
ejde-636	515	13	q(usu	q(usu	PROPN
ejde-636	515	14	)	)	PUNCT
ejde-636	515	15	=	=	SYM
ejde-636	515	16	1	1	NUM
ejde-636	515	17	2	2	NUM
ejde-636	515	18	s2u∥∆u∥22	s2u∥∆u∥22	ADJ
ejde-636	515	19	+	+	CCONJ
ejde-636	515	20	su	su	PROPN
ejde-636	515	21	2	2	NUM
ejde-636	515	22	∥∇u∥22	∥∇u∥22	PROPN
ejde-636	515	23	−	−	PROPN
ejde-636	515	24	µ	µ	PRON
ejde-636	515	25	q	q	X
ejde-636	515	26	s	s	PROPN
ejde-636	515	27	n(q−2	n(q−2	X
ejde-636	515	28	)	)	PUNCT
ejde-636	515	29	4	4	NUM
ejde-636	515	30	u	u	NOUN
ejde-636	515	31	∥u∥qq	∥u∥qq	ADP
ejde-636	515	32	−	−	PROPN
ejde-636	515	33	1	1	NUM
ejde-636	515	34	p	p	NOUN
ejde-636	515	35	s	s	X
ejde-636	515	36	n(p−2	n(p−2	PROPN
ejde-636	515	37	)	)	PUNCT
ejde-636	515	38	4	4	NUM
ejde-636	515	39	u	u	NOUN
ejde-636	515	40	∥u∥pp	∥u∥pp	PROPN
ejde-636	515	41	,	,	PUNCT
ejde-636	515	42	where	where	SCONJ
ejde-636	515	43	su	su	PROPN
ejde-636	515	44	is	be	AUX
ejde-636	515	45	given	give	VERB
ejde-636	515	46	in	in	ADP
ejde-636	515	47	lemma	lemma	PROPN
ejde-636	515	48	4.1	4.1	NUM
ejde-636	515	49	and	and	CCONJ
ejde-636	515	50	usu	usu	PROPN
ejde-636	515	51	∈	∈	PROPN
ejde-636	515	52	qp	qp	PROPN
ejde-636	515	53	,	,	PUNCT
ejde-636	515	54	q(c	q(c	PROPN
ejde-636	515	55	)	)	PUNCT
ejde-636	515	56	.	.	PUNCT
ejde-636	516	1	according	accord	VERB
ejde-636	516	2	to	to	ADP
ejde-636	516	3	lemma	lemma	PROPN
ejde-636	516	4	4.7	4.7	NUM
ejde-636	516	5	,	,	PUNCT
ejde-636	516	6	we	we	PRON
ejde-636	516	7	find	find	VERB
ejde-636	516	8	u0	u0	PROPN
ejde-636	516	9	∈	∈	PROPN
ejde-636	516	10	qp	qp	NOUN
ejde-636	516	11	,	,	PUNCT
ejde-636	516	12	q(c	q(c	PROPN
ejde-636	516	13	)	)	PUNCT
ejde-636	516	14	such	such	ADJ
ejde-636	516	15	that	that	SCONJ
ejde-636	516	16	ep	ep	PROPN
ejde-636	516	17	,	,	PUNCT
ejde-636	516	18	q(u0	q(u0	PROPN
ejde-636	516	19	)	)	PUNCT
ejde-636	516	20	=	=	SYM
ejde-636	516	21	mp	mp	PROPN
ejde-636	516	22	,	,	PUNCT
ejde-636	516	23	q(c	q(c	PROPN
ejde-636	516	24	)	)	PUNCT
ejde-636	516	25	.	.	PUNCT
ejde-636	517	1	then	then	ADV
ejde-636	517	2	there	there	PRON
ejde-636	517	3	exists	exist	VERB
ejde-636	517	4	v0	v0	PROPN
ejde-636	517	5	∈	∈	PROPN
ejde-636	517	6	s(c	s(c	PROPN
ejde-636	517	7	)	)	PUNCT
ejde-636	517	8	such	such	ADJ
ejde-636	517	9	that	that	SCONJ
ejde-636	517	10	(	(	PUNCT
ejde-636	517	11	v0)sv0	v0)sv0	X
ejde-636	517	12	=	=	SYM
ejde-636	517	13	u0	u0	ADJ
ejde-636	517	14	and	and	CCONJ
ejde-636	517	15	ψ(v0	ψ(v0	ADJ
ejde-636	517	16	)	)	PUNCT
ejde-636	517	17	=	=	SYM
ejde-636	518	1	ep	ep	PROPN
ejde-636	518	2	,	,	PUNCT
ejde-636	518	3	q((v0)sv0	q((v0)sv0	X
ejde-636	518	4	)	)	PUNCT
ejde-636	519	1	=	=	SYM
ejde-636	519	2	ep	ep	PROPN
ejde-636	519	3	,	,	PUNCT
ejde-636	519	4	q(u0	q(u0	PROPN
ejde-636	519	5	)	)	PUNCT
ejde-636	519	6	=	=	SYM
ejde-636	519	7	mp	mp	PROPN
ejde-636	519	8	,	,	PUNCT
ejde-636	519	9	q(c	q(c	PROPN
ejde-636	519	10	)	)	PUNCT
ejde-636	519	11	.	.	PUNCT
ejde-636	520	1	this	this	PRON
ejde-636	520	2	implies	imply	VERB
ejde-636	520	3	that	that	SCONJ
ejde-636	520	4	v0	v0	NOUN
ejde-636	520	5	is	be	AUX
ejde-636	520	6	a	a	DET
ejde-636	520	7	minimizer	minimizer	NOUN
ejde-636	520	8	of	of	ADP
ejde-636	520	9	ep	ep	PROPN
ejde-636	520	10	,	,	PUNCT
ejde-636	520	11	q	q	PUNCT
ejde-636	520	12	restricted	restrict	VERB
ejde-636	520	13	on	on	ADP
ejde-636	520	14	s(c	s(c	NUM
ejde-636	520	15	)	)	PUNCT
ejde-636	520	16	.	.	PUNCT
ejde-636	521	1	we	we	PRON
ejde-636	521	2	claim	claim	VERB
ejde-636	521	3	that	that	SCONJ
ejde-636	521	4	ψ	ψ	NOUN
ejde-636	521	5	is	be	AUX
ejde-636	521	6	of	of	ADP
ejde-636	521	7	class	class	NOUN
ejde-636	521	8	c1	c1	NOUN
ejde-636	521	9	and	and	CCONJ
ejde-636	521	10	dψ(u)[φ	dψ(u)[φ	ADJ
ejde-636	521	11	]	]	X
ejde-636	521	12	=	=	SYM
ejde-636	521	13	dep	dep	NOUN
ejde-636	521	14	,	,	PUNCT
ejde-636	521	15	q(usu)[φsu	q(usu)[φsu	ADP
ejde-636	521	16	]	]	PUNCT
ejde-636	521	17	(	(	PUNCT
ejde-636	521	18	4.15	4.15	NUM
ejde-636	521	19	)	)	PUNCT
ejde-636	521	20	for	for	ADP
ejde-636	521	21	any	any	DET
ejde-636	521	22	u	u	PROPN
ejde-636	521	23	∈	∈	PROPN
ejde-636	521	24	s(c	s(c	PROPN
ejde-636	521	25	)	)	PUNCT
ejde-636	521	26	and	and	CCONJ
ejde-636	521	27	φ	φ	NUM
ejde-636	521	28	∈	∈	PROPN
ejde-636	521	29	tus(c	tus(c	NOUN
ejde-636	521	30	)	)	PUNCT
ejde-636	521	31	.	.	PUNCT
ejde-636	522	1	in	in	ADP
ejde-636	522	2	fact	fact	NOUN
ejde-636	522	3	,	,	PUNCT
ejde-636	522	4	by	by	ADP
ejde-636	522	5	the	the	DET
ejde-636	522	6	definition	definition	NOUN
ejde-636	522	7	of	of	ADP
ejde-636	522	8	ψ	ψ	PRON
ejde-636	522	9	we	we	PRON
ejde-636	522	10	have	have	VERB
ejde-636	522	11	ψ(u+	ψ(u+	NOUN
ejde-636	522	12	tφ)−ψ(u	tφ)−ψ(u	NOUN
ejde-636	522	13	)	)	PUNCT
ejde-636	523	1	=	=	SYM
ejde-636	523	2	ep	ep	NOUN
ejde-636	523	3	,	,	PUNCT
ejde-636	523	4	q((u+	q((u+	VERB
ejde-636	523	5	tφ)st)−	tφ)st)−	PROPN
ejde-636	523	6	ep	ep	PROPN
ejde-636	523	7	,	,	PUNCT
ejde-636	523	8	q(us0	q(us0	PROPN
ejde-636	523	9	)	)	PUNCT
ejde-636	523	10	,	,	PUNCT
ejde-636	523	11	where	where	SCONJ
ejde-636	523	12	|t|	|t|	NOUN
ejde-636	523	13	is	be	AUX
ejde-636	523	14	small	small	ADJ
ejde-636	523	15	enough	enough	ADV
ejde-636	523	16	,	,	PUNCT
ejde-636	523	17	st	st	PROPN
ejde-636	523	18	=	=	PUNCT
ejde-636	523	19	su+tφ	su+tφ	PROPN
ejde-636	523	20	and	and	CCONJ
ejde-636	523	21	s0	s0	PROPN
ejde-636	523	22	=	=	SYM
ejde-636	523	23	su	su	PROPN
ejde-636	523	24	is	be	AUX
ejde-636	523	25	the	the	DET
ejde-636	523	26	unique	unique	ADJ
ejde-636	523	27	maximum	maximum	ADJ
ejde-636	523	28	point	point	NOUN
ejde-636	523	29	of	of	ADP
ejde-636	523	30	the	the	DET
ejde-636	523	31	functional	functional	ADJ
ejde-636	523	32	ep	ep	PROPN
ejde-636	523	33	,	,	PUNCT
ejde-636	523	34	q(us	q(us	PROPN
ejde-636	523	35	)	)	PUNCT
ejde-636	523	36	.	.	PUNCT
ejde-636	524	1	by	by	ADP
ejde-636	524	2	the	the	DET
ejde-636	524	3	mean	mean	ADJ
ejde-636	524	4	value	value	NOUN
ejde-636	524	5	theorem	theorem	VERB
ejde-636	524	6	we	we	PRON
ejde-636	524	7	obtain	obtain	VERB
ejde-636	524	8	ep	ep	NOUN
ejde-636	524	9	,	,	PUNCT
ejde-636	524	10	q((u+	q((u+	VERB
ejde-636	524	11	tφ)st)−	tφ)st)−	PROPN
ejde-636	524	12	ep	ep	PROPN
ejde-636	524	13	,	,	PUNCT
ejde-636	524	14	q(us0	q(us0	PROPN
ejde-636	524	15	)	)	PUNCT
ejde-636	524	16	≤	≤	NOUN
ejde-636	524	17	ep	ep	PROPN
ejde-636	524	18	,	,	PUNCT
ejde-636	524	19	q((u+	q((u+	VERB
ejde-636	524	20	tφ)st)−	tφ)st)−	PROPN
ejde-636	524	21	ep	ep	PROPN
ejde-636	524	22	,	,	PUNCT
ejde-636	524	23	q(ust	q(ust	ADV
ejde-636	524	24	)	)	PUNCT
ejde-636	524	25	=	=	SYM
ejde-636	524	26	s2	s2	PROPN
ejde-636	524	27	t	t	PROPN
ejde-636	524	28	2	2	NUM
ejde-636	524	29	(	(	PUNCT
ejde-636	524	30	∫	∫	PROPN
ejde-636	524	31	rn	rn	PROPN
ejde-636	524	32	2t∆u	2t∆u	NUM
ejde-636	524	33	·	·	SYM
ejde-636	524	34	∆φ+	∆φ+	PROPN
ejde-636	524	35	t2|∆φ|2dx	t2|∆φ|2dx	PROPN
ejde-636	524	36	)	)	PUNCT
ejde-636	525	1	+	+	CCONJ
ejde-636	525	2	st	st	PROPN
ejde-636	525	3	2	2	NUM
ejde-636	525	4	(	(	PUNCT
ejde-636	525	5	∫	∫	PROPN
ejde-636	525	6	rn	rn	PROPN
ejde-636	525	7	2t∇u	2t∇u	PROPN
ejde-636	525	8	·	·	PUNCT
ejde-636	525	9	∇φ+	∇φ+	PROPN
ejde-636	525	10	t2|∇φ|2dx	t2|∇φ|2dx	PUNCT
ejde-636	525	11	)	)	PUNCT
ejde-636	525	12	−	−	PRON
ejde-636	525	13	µs	µs	NOUN
ejde-636	525	14	n(q−2	n(q−2	X
ejde-636	525	15	)	)	PUNCT
ejde-636	525	16	4	4	NUM
ejde-636	525	17	t	t	PROPN
ejde-636	525	18	∫	∫	PROPN
ejde-636	525	19	rn	rn	PROPN
ejde-636	525	20	(	(	PUNCT
ejde-636	525	21	∫	∫	PROPN
ejde-636	525	22	1	1	NUM
ejde-636	525	23	0	0	NUM
ejde-636	525	24	|u+	|u+	X
ejde-636	525	25	sηtφ|q−2(u+	sηtφ|q−2(u+	PROPN
ejde-636	525	26	tηtφ)tφdt	tηtφ)tφdt	PROPN
ejde-636	525	27	)	)	PUNCT
ejde-636	525	28	dx	dx	PROPN
ejde-636	526	1	−	−	PROPN
ejde-636	526	2	s	s	PROPN
ejde-636	526	3	n(p−2	n(p−2	PROPN
ejde-636	526	4	)	)	PUNCT
ejde-636	526	5	4	4	NUM
ejde-636	526	6	t	t	PROPN
ejde-636	526	7	∫	∫	PROPN
ejde-636	526	8	rn	rn	PROPN
ejde-636	526	9	(	(	PUNCT
ejde-636	526	10	∫	∫	PROPN
ejde-636	526	11	1	1	NUM
ejde-636	526	12	0	0	NUM
ejde-636	526	13	|u+	|u+	X
ejde-636	526	14	tηtφ|p−2(u+	tηtφ|p−2(u+	SYM
ejde-636	526	15	tηtφ)tφdt	tηtφ)tφdt	PROPN
ejde-636	526	16	)	)	PUNCT
ejde-636	526	17	dx	dx	PROPN
ejde-636	526	18	,	,	PUNCT
ejde-636	526	19	(	(	PUNCT
ejde-636	526	20	4.16	4.16	NUM
ejde-636	526	21	)	)	PUNCT
ejde-636	526	22	where	where	SCONJ
ejde-636	526	23	ηt	ηt	ADP
ejde-636	526	24	∈	∈	PROPN
ejde-636	526	25	(	(	PUNCT
ejde-636	526	26	0	0	NUM
ejde-636	526	27	,	,	PUNCT
ejde-636	526	28	1	1	NUM
ejde-636	526	29	)	)	PUNCT
ejde-636	526	30	.	.	PUNCT
ejde-636	527	1	similarly	similarly	ADV
ejde-636	527	2	,	,	PUNCT
ejde-636	527	3	we	we	PRON
ejde-636	527	4	derive	derive	VERB
ejde-636	527	5	ep	ep	NOUN
ejde-636	527	6	,	,	PUNCT
ejde-636	527	7	q((u+	q((u+	VERB
ejde-636	527	8	tφ)st)−	tφ)st)−	PROPN
ejde-636	527	9	ep	ep	PROPN
ejde-636	527	10	,	,	PUNCT
ejde-636	527	11	q(us0	q(us0	PROPN
ejde-636	527	12	)	)	PUNCT
ejde-636	527	13	≥	≥	NOUN
ejde-636	527	14	ep	ep	PROPN
ejde-636	527	15	,	,	PUNCT
ejde-636	527	16	q((u+	q((u+	VERB
ejde-636	527	17	tφ)s0)−	tφ)s0)−	X
ejde-636	527	18	ep	ep	NOUN
ejde-636	527	19	,	,	PUNCT
ejde-636	527	20	q(us0	q(us0	PROPN
ejde-636	527	21	)	)	PUNCT
ejde-636	527	22	=	=	NOUN
ejde-636	527	23	s20	s20	X
ejde-636	527	24	2	2	NUM
ejde-636	527	25	(	(	PUNCT
ejde-636	527	26	∫	∫	PROPN
ejde-636	527	27	rn	rn	PROPN
ejde-636	527	28	2t∆u	2t∆u	NUM
ejde-636	527	29	·	·	SYM
ejde-636	527	30	∆φ+	∆φ+	PROPN
ejde-636	527	31	t2|∆φ|2dx	t2|∆φ|2dx	PROPN
ejde-636	527	32	)	)	PUNCT
ejde-636	528	1	+	+	CCONJ
ejde-636	528	2	s0	s0	PROPN
ejde-636	528	3	2	2	NUM
ejde-636	528	4	(	(	PUNCT
ejde-636	528	5	∫	∫	PROPN
ejde-636	528	6	rn	rn	PROPN
ejde-636	528	7	2t∇u	2t∇u	PROPN
ejde-636	528	8	·	·	PUNCT
ejde-636	528	9	∇φ+	∇φ+	PROPN
ejde-636	528	10	t2|∇φ|2dx	t2|∇φ|2dx	PUNCT
ejde-636	528	11	)	)	PUNCT
ejde-636	528	12	−	−	PRON
ejde-636	528	13	µs	µs	NOUN
ejde-636	528	14	n(q−2	n(q−2	X
ejde-636	528	15	)	)	PUNCT
ejde-636	528	16	4	4	NUM
ejde-636	528	17	0	0	NUM
ejde-636	528	18	∫	∫	PROPN
ejde-636	528	19	rn	rn	PROPN
ejde-636	528	20	(	(	PUNCT
ejde-636	528	21	∫	∫	PROPN
ejde-636	528	22	1	1	NUM
ejde-636	528	23	0	0	NUM
ejde-636	528	24	|u+	|u+	PROPN
ejde-636	528	25	tθtφ|q−2(u+	tθtφ|q−2(u+	PUNCT
ejde-636	528	26	tθtφ)tφdt	tθtφ)tφdt	NOUN
ejde-636	528	27	)	)	PUNCT
ejde-636	528	28	dx	dx	PROPN
ejde-636	528	29	−	−	PROPN
ejde-636	528	30	s	s	PROPN
ejde-636	528	31	n(p−2	n(p−2	PROPN
ejde-636	528	32	)	)	PUNCT
ejde-636	528	33	4	4	NUM
ejde-636	528	34	0	0	NUM
ejde-636	528	35	∫	∫	PROPN
ejde-636	528	36	rn	rn	PROPN
ejde-636	528	37	(	(	PUNCT
ejde-636	528	38	∫	∫	PROPN
ejde-636	528	39	1	1	NUM
ejde-636	528	40	0	0	NUM
ejde-636	528	41	|u+	|u+	NOUN
ejde-636	528	42	tθtφ|p−2(u+	tθtφ|p−2(u+	X
ejde-636	528	43	tθtφ)tφdt	tθtφ)tφdt	NOUN
ejde-636	528	44	)	)	PUNCT
ejde-636	528	45	dx	dx	PROPN
ejde-636	528	46	,	,	PUNCT
ejde-636	528	47	(	(	PUNCT
ejde-636	528	48	4.17	4.17	NUM
ejde-636	528	49	)	)	PUNCT
ejde-636	528	50	where	where	SCONJ
ejde-636	528	51	θt	θt	PROPN
ejde-636	528	52	∈	∈	PROPN
ejde-636	528	53	(	(	PUNCT
ejde-636	528	54	0	0	NUM
ejde-636	528	55	,	,	PUNCT
ejde-636	528	56	1	1	NUM
ejde-636	528	57	)	)	PUNCT
ejde-636	528	58	.	.	PUNCT
ejde-636	529	1	since	since	SCONJ
ejde-636	529	2	the	the	DET
ejde-636	529	3	map	map	NOUN
ejde-636	529	4	u	u	PROPN
ejde-636	529	5	7→	7→	NUM
ejde-636	529	6	su	su	NOUN
ejde-636	529	7	is	be	AUX
ejde-636	529	8	of	of	ADP
ejde-636	529	9	class	class	NOUN
ejde-636	529	10	c1	c1	NOUN
ejde-636	529	11	,	,	PUNCT
ejde-636	529	12	from	from	ADP
ejde-636	529	13	(	(	PUNCT
ejde-636	529	14	4.16	4.16	NUM
ejde-636	529	15	)	)	PUNCT
ejde-636	529	16	and	and	CCONJ
ejde-636	529	17	(	(	PUNCT
ejde-636	529	18	4.17	4.17	X
ejde-636	529	19	)	)	PUNCT
ejde-636	529	20	it	it	PRON
ejde-636	529	21	follows	follow	VERB
ejde-636	529	22	that	that	SCONJ
ejde-636	529	23	lim	lim	PROPN
ejde-636	529	24	t→0	t→0	PUNCT
ejde-636	529	25	ψ(u+	ψ(u+	PROPN
ejde-636	529	26	tφ)−ψ(u	tφ)−ψ(u	NOUN
ejde-636	529	27	)	)	PUNCT
ejde-636	529	28	t	t	NOUN
ejde-636	530	1	=	=	SYM
ejde-636	530	2	s2u	s2u	PROPN
ejde-636	530	3	∫	∫	PROPN
ejde-636	530	4	rn	rn	PROPN
ejde-636	530	5	∆u	∆u	PROPN
ejde-636	530	6	·	·	PUNCT
ejde-636	530	7	∆φdx+	∆φdx+	ADJ
ejde-636	530	8	su	su	PROPN
ejde-636	530	9	∫	∫	PROPN
ejde-636	530	10	rn	rn	PROPN
ejde-636	530	11	∇u	∇u	PROPN
ejde-636	530	12	·	·	PUNCT
ejde-636	531	1	∇φdx	∇φdx	NOUN
ejde-636	531	2	18	18	NUM
ejde-636	531	3	z.	z.	PROPN
ejde-636	531	4	ma	ma	PROPN
ejde-636	531	5	,	,	PUNCT
ejde-636	531	6	x.	x.	PROPN
ejde-636	531	7	chang	chang	PROPN
ejde-636	531	8	,	,	PUNCT
ejde-636	531	9	z.	z.	PROPN
ejde-636	531	10	feng	feng	PROPN
ejde-636	531	11	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	531	12	−	−	PROPN
ejde-636	531	13	s	s	PART
ejde-636	531	14	n(q−2	n(q−2	X
ejde-636	531	15	)	)	PUNCT
ejde-636	531	16	4	4	NUM
ejde-636	531	17	u	u	NOUN
ejde-636	531	18	µ	µ	X
ejde-636	531	19	∫	∫	PROPN
ejde-636	531	20	rn	rn	PROPN
ejde-636	531	21	|u|q−2u	|u|q−2u	PROPN
ejde-636	531	22	·	·	PUNCT
ejde-636	532	1	φdx−	φdx−	PROPN
ejde-636	532	2	s	s	PROPN
ejde-636	532	3	n(p−2	n(p−2	PROPN
ejde-636	532	4	)	)	PUNCT
ejde-636	532	5	4	4	NUM
ejde-636	532	6	u	u	NOUN
ejde-636	532	7	µ	µ	X
ejde-636	532	8	∫	∫	PROPN
ejde-636	532	9	rn	rn	PROPN
ejde-636	532	10	|u|p−2u	|u|p−2u	PROPN
ejde-636	532	11	·	·	PUNCT
ejde-636	532	12	φdx	φdx	PROPN
ejde-636	532	13	.	.	PUNCT
ejde-636	533	1	so	so	ADV
ejde-636	533	2	the	the	DET
ejde-636	533	3	gâteaux	gâteaux	ADJ
ejde-636	533	4	derivative	derivative	NOUN
ejde-636	533	5	of	of	ADP
ejde-636	533	6	ψ	ψ	NOUN
ejde-636	533	7	is	be	AUX
ejde-636	533	8	bounded	bound	VERB
ejde-636	533	9	linear	linear	PROPN
ejde-636	533	10	in	in	ADP
ejde-636	533	11	φ	φ	PROPN
ejde-636	533	12	and	and	CCONJ
ejde-636	533	13	continuous	continuous	ADJ
ejde-636	533	14	in	in	ADP
ejde-636	533	15	u.	u.	PROPN
ejde-636	533	16	therefore	therefore	ADV
ejde-636	533	17	,	,	PUNCT
ejde-636	533	18	ψ	ψ	X
ejde-636	533	19	is	be	AUX
ejde-636	533	20	of	of	ADP
ejde-636	533	21	class	class	NOUN
ejde-636	533	22	c1	c1	NOUN
ejde-636	533	23	.	.	PUNCT
ejde-636	534	1	in	in	ADP
ejde-636	534	2	particular	particular	ADJ
ejde-636	534	3	,	,	PUNCT
ejde-636	534	4	by	by	ADP
ejde-636	534	5	changing	change	VERB
ejde-636	534	6	variables	variable	NOUN
ejde-636	534	7	in	in	ADP
ejde-636	534	8	the	the	DET
ejde-636	534	9	integrals	integral	NOUN
ejde-636	534	10	,	,	PUNCT
ejde-636	534	11	we	we	PRON
ejde-636	534	12	have	have	VERB
ejde-636	534	13	dψ(u)[φ	dψ(u)[φ	ADV
ejde-636	534	14	]	]	X
ejde-636	534	15	=	=	SYM
ejde-636	534	16	s2u	s2u	PROPN
ejde-636	534	17	∫	∫	PROPN
ejde-636	534	18	rn	rn	PROPN
ejde-636	534	19	∆u	∆u	PROPN
ejde-636	534	20	·	·	PUNCT
ejde-636	534	21	∆φdx+	∆φdx+	ADJ
ejde-636	534	22	su	su	PROPN
ejde-636	534	23	∫	∫	PROPN
ejde-636	534	24	rn	rn	PROPN
ejde-636	534	25	∇u	∇u	PROPN
ejde-636	534	26	·	·	PUNCT
ejde-636	535	1	∇φdx	∇φdx	NOUN
ejde-636	535	2	−	−	PROPN
ejde-636	535	3	s	s	PART
ejde-636	535	4	n(q−2	n(q−2	X
ejde-636	535	5	)	)	PUNCT
ejde-636	535	6	4	4	NUM
ejde-636	535	7	u	u	NOUN
ejde-636	535	8	µ	µ	X
ejde-636	535	9	∫	∫	PROPN
ejde-636	535	10	rn	rn	PROPN
ejde-636	535	11	|u|q−2u	|u|q−2u	PROPN
ejde-636	535	12	·	·	PUNCT
ejde-636	536	1	φdx−	φdx−	PROPN
ejde-636	536	2	s	s	PROPN
ejde-636	536	3	n(p−2	n(p−2	PROPN
ejde-636	536	4	)	)	PUNCT
ejde-636	536	5	4	4	NUM
ejde-636	536	6	u	u	NOUN
ejde-636	536	7	∫	∫	PROPN
ejde-636	536	8	rn	rn	PROPN
ejde-636	536	9	|u|p−2u	|u|p−2u	PROPN
ejde-636	536	10	·	·	PUNCT
ejde-636	536	11	φdx	φdx	NOUN
ejde-636	536	12	=	=	SYM
ejde-636	536	13	∫	∫	PROPN
ejde-636	536	14	rn	rn	PROPN
ejde-636	536	15	∆usu	∆usu	X
ejde-636	536	16	·	·	PUNCT
ejde-636	536	17	∆φsudx+	∆φsudx+	NUM
ejde-636	536	18	∫	∫	PROPN
ejde-636	536	19	rn	rn	PROPN
ejde-636	536	20	∇usu	∇usu	PROPN
ejde-636	536	21	·	·	PUNCT
ejde-636	536	22	∇φsudx	∇φsudx	NOUN
ejde-636	536	23	−	−	X
ejde-636	536	24	µ	µ	PROPN
ejde-636	536	25	∫	∫	PROPN
ejde-636	536	26	rn	rn	PROPN
ejde-636	536	27	|usu	|usu	PROPN
ejde-636	536	28	|q−2usu	|q−2usu	X
ejde-636	536	29	·	·	PUNCT
ejde-636	536	30	φsudx−	φsudx−	NOUN
ejde-636	536	31	∫	∫	PROPN
ejde-636	536	32	rn	rn	PROPN
ejde-636	536	33	|usu	|usu	PROPN
ejde-636	536	34	|p−2usu	|p−2usu	X
ejde-636	536	35	·	·	PUNCT
ejde-636	536	36	φsudx	φsudx	NOUN
ejde-636	536	37	.	.	PUNCT
ejde-636	537	1	=	=	PUNCT
ejde-636	537	2	dep	dep	NOUN
ejde-636	537	3	,	,	PUNCT
ejde-636	537	4	q(usu)[φsu	q(usu)[φsu	ADP
ejde-636	537	5	]	]	PUNCT
ejde-636	537	6	.	.	PUNCT
ejde-636	538	1	so	so	ADV
ejde-636	538	2	the	the	DET
ejde-636	538	3	claim	claim	NOUN
ejde-636	538	4	(	(	PUNCT
ejde-636	538	5	4.15	4.15	NUM
ejde-636	538	6	)	)	PUNCT
ejde-636	538	7	is	be	AUX
ejde-636	538	8	true	true	ADJ
ejde-636	538	9	,	,	PUNCT
ejde-636	538	10	from	from	ADP
ejde-636	538	11	which	which	PRON
ejde-636	538	12	we	we	PRON
ejde-636	538	13	deduce	deduce	VERB
ejde-636	538	14	∥dep	∥dep	PROPN
ejde-636	538	15	,	,	PUNCT
ejde-636	538	16	q(u0)∥(tu0s(c))∗	q(u0)∥(tu0s(c))∗	NOUN
ejde-636	538	17	=	=	PUNCT
ejde-636	538	18	sup	sup	PROPN
ejde-636	538	19	φ∈tu0	φ∈tu0	ADJ
ejde-636	538	20	s(c),∥φ∥≤1	s(c),∥φ∥≤1	VERB
ejde-636	538	21	|dep	|dep	ADJ
ejde-636	538	22	,	,	PUNCT
ejde-636	538	23	q(u0)[φ]|	q(u0)[φ]|	ADJ
ejde-636	538	24	=	=	PUNCT
ejde-636	538	25	sup	sup	NOUN
ejde-636	538	26	φ∈tu0	φ∈tu0	ADJ
ejde-636	538	27	s(c),∥φ∥≤1	s(c),∥φ∥≤1	VERB
ejde-636	538	28	|dep	|dep	ADJ
ejde-636	538	29	,	,	PUNCT
ejde-636	538	30	q((v0)sv0	q((v0)sv0	NUM
ejde-636	538	31	)	)	PUNCT
ejde-636	539	1	[	[	X
ejde-636	539	2	(	(	PUNCT
ejde-636	539	3	φs−1	φs−1	PROPN
ejde-636	539	4	v0	v0	NOUN
ejde-636	539	5	)	)	PUNCT
ejde-636	539	6	sv0	sv0	ADJ
ejde-636	539	7	]	]	X
ejde-636	539	8	|	|	NOUN
ejde-636	539	9	=	=	SYM
ejde-636	539	10	sup	sup	PROPN
ejde-636	539	11	φ∈tu0	φ∈tu0	PUNCT
ejde-636	539	12	s(c),∥φ∥≤1	s(c),∥φ∥≤1	VERB
ejde-636	539	13	|dψ(v0)[φs−1	|dψ(v0)[φs−1	NOUN
ejde-636	539	14	v0	v0	NOUN
ejde-636	539	15	]	]	PUNCT
ejde-636	539	16	|	|	ADV
ejde-636	539	17	≤	≤	ADJ
ejde-636	539	18	∥dψ(v0)∥(tv0s(c))∗	∥dψ(v0)∥(tv0s(c))∗	PROPN
ejde-636	539	19	·	·	PUNCT
ejde-636	539	20	sup	sup	PROPN
ejde-636	539	21	φ∈tu0	φ∈tu0	ADJ
ejde-636	539	22	s(c),∥φ∥≤1	s(c),∥φ∥≤1	VERB
ejde-636	539	23	∥φs−1	∥φs−1	NUM
ejde-636	539	24	v0	v0	NOUN
ejde-636	539	25	∥	∥	PUNCT
ejde-636	539	26	≤	≤	NUM
ejde-636	539	27	max{s−1	max{s−1	ADJ
ejde-636	539	28	v0	v0	NOUN
ejde-636	539	29	,	,	PUNCT
ejde-636	539	30	1}∥dep	1}∥dep	NUM
ejde-636	539	31	,	,	PUNCT
ejde-636	539	32	q(v0)∥(tv0s(c))∗	q(v0)∥(tv0s(c))∗	NOUN
ejde-636	539	33	=	=	NOUN
ejde-636	539	34	0	0	PROPN
ejde-636	539	35	.	.	PUNCT
ejde-636	540	1	it	it	PRON
ejde-636	540	2	follows	follow	VERB
ejde-636	540	3	that	that	SCONJ
ejde-636	540	4	u0	u0	ADJ
ejde-636	540	5	is	be	AUX
ejde-636	540	6	a	a	DET
ejde-636	540	7	critical	critical	ADJ
ejde-636	540	8	point	point	NOUN
ejde-636	540	9	of	of	ADP
ejde-636	540	10	ep	ep	PROPN
ejde-636	540	11	,	,	PUNCT
ejde-636	540	12	q	q	PUNCT
ejde-636	540	13	restricted	restrict	VERB
ejde-636	540	14	on	on	ADP
ejde-636	540	15	s(c	s(c	NUM
ejde-636	540	16	)	)	PUNCT
ejde-636	540	17	.	.	PUNCT
ejde-636	541	1	by	by	ADP
ejde-636	541	2	lemma	lemma	PROPN
ejde-636	541	3	4.5	4.5	NUM
ejde-636	541	4	for	for	ADP
ejde-636	541	5	some	some	DET
ejde-636	541	6	ω	ω	PROPN
ejde-636	541	7	>	>	X
ejde-636	541	8	0	0	PROPN
ejde-636	541	9	,	,	PUNCT
ejde-636	541	10	u0	u0	ADJ
ejde-636	541	11	weakly	weakly	ADJ
ejde-636	541	12	solves	solve	NOUN
ejde-636	541	13	(	(	PUNCT
ejde-636	541	14	1.2	1.2	NUM
ejde-636	541	15	)	)	PUNCT
ejde-636	541	16	.	.	PUNCT
ejde-636	542	1	in	in	ADP
ejde-636	542	2	view	view	NOUN
ejde-636	542	3	of	of	ADP
ejde-636	542	4	ep	ep	PROPN
ejde-636	542	5	,	,	PUNCT
ejde-636	542	6	q(u0	q(u0	PROPN
ejde-636	542	7	)	)	PUNCT
ejde-636	542	8	=	=	SYM
ejde-636	542	9	mp	mp	PROPN
ejde-636	542	10	,	,	PUNCT
ejde-636	542	11	q(c	q(c	PROPN
ejde-636	542	12	)	)	PUNCT
ejde-636	542	13	,	,	PUNCT
ejde-636	542	14	we	we	PRON
ejde-636	542	15	infer	infer	VERB
ejde-636	542	16	that	that	SCONJ
ejde-636	542	17	u0	u0	PROPN
ejde-636	542	18	is	be	AUX
ejde-636	542	19	a	a	DET
ejde-636	542	20	normalized	normalize	VERB
ejde-636	542	21	ground	ground	NOUN
ejde-636	542	22	state	state	NOUN
ejde-636	542	23	solution	solution	NOUN
ejde-636	542	24	of	of	ADP
ejde-636	542	25	problem	problem	NOUN
ejde-636	542	26	(	(	PUNCT
ejde-636	542	27	1.2	1.2	NUM
ejde-636	542	28	)	)	PUNCT
ejde-636	542	29	.	.	PUNCT
ejde-636	543	1	□	□	PUNCT
ejde-636	543	2	acknowledgments	acknowledgment	NOUN
ejde-636	543	3	.	.	PUNCT
ejde-636	544	1	this	this	DET
ejde-636	544	2	work	work	NOUN
ejde-636	544	3	is	be	AUX
ejde-636	544	4	supported	support	VERB
ejde-636	544	5	by	by	ADP
ejde-636	544	6	national	national	ADJ
ejde-636	544	7	natural	natural	PROPN
ejde-636	544	8	science	science	PROPN
ejde-636	544	9	foundation	foundation	PROPN
ejde-636	544	10	of	of	ADP
ejde-636	544	11	china	china	PROPN
ejde-636	544	12	no	no	INTJ
ejde-636	544	13	.	.	PROPN
ejde-636	544	14	11971095	11971095	NUM
ejde-636	544	15	.	.	PUNCT
ejde-636	545	1	references	reference	NOUN
ejde-636	545	2	[	[	X
ejde-636	545	3	1	1	X
ejde-636	545	4	]	]	PUNCT
ejde-636	545	5	t.	t.	PROPN
ejde-636	545	6	bartsch	bartsch	PROPN
ejde-636	545	7	,	,	PUNCT
ejde-636	545	8	n.	n.	PROPN
ejde-636	545	9	soave	soave	PROPN
ejde-636	545	10	;	;	PUNCT
ejde-636	545	11	a	a	DET
ejde-636	545	12	natural	natural	ADJ
ejde-636	545	13	constraint	constraint	NOUN
ejde-636	545	14	approach	approach	NOUN
ejde-636	545	15	to	to	ADP
ejde-636	545	16	normalized	normalize	VERB
ejde-636	545	17	solutions	solution	NOUN
ejde-636	545	18	of	of	ADP
ejde-636	545	19	nonlinear	nonlinear	ADJ
ejde-636	545	20	schrödinger	schrödinger	NOUN
ejde-636	545	21	equations	equation	NOUN
ejde-636	545	22	and	and	CCONJ
ejde-636	545	23	systems	system	NOUN
ejde-636	545	24	,	,	PUNCT
ejde-636	545	25	j.	j.	PROPN
ejde-636	545	26	funct	funct	PROPN
ejde-636	545	27	.	.	PUNCT
ejde-636	546	1	anal	anal	PROPN
ejde-636	546	2	.	.	PROPN
ejde-636	546	3	,	,	PUNCT
ejde-636	546	4	272	272	NUM
ejde-636	546	5	(	(	PUNCT
ejde-636	546	6	2017	2017	NUM
ejde-636	546	7	)	)	PUNCT
ejde-636	546	8	,	,	PUNCT
ejde-636	546	9	4998	4998	NUM
ejde-636	546	10	-	-	SYM
ejde-636	546	11	5037	5037	NUM
ejde-636	546	12	.	.	PUNCT
ejde-636	547	1	[	[	X
ejde-636	547	2	2	2	X
ejde-636	547	3	]	]	PUNCT
ejde-636	547	4	j.	j.	PROPN
ejde-636	547	5	bellazzini	bellazzini	PROPN
ejde-636	547	6	,	,	PUNCT
ejde-636	547	7	l.	l.	PROPN
ejde-636	547	8	jeanjean	jeanjean	PROPN
ejde-636	547	9	,	,	PUNCT
ejde-636	547	10	t.	t.	PROPN
ejde-636	547	11	luo	luo	PROPN
ejde-636	547	12	;	;	PUNCT
ejde-636	547	13	existence	existence	NOUN
ejde-636	547	14	and	and	CCONJ
ejde-636	547	15	instability	instability	NOUN
ejde-636	547	16	of	of	ADP
ejde-636	547	17	standing	standing	ADJ
ejde-636	547	18	waves	wave	NOUN
ejde-636	547	19	with	with	ADP
ejde-636	547	20	prescribed	prescribed	ADJ
ejde-636	547	21	norm	norm	NOUN
ejde-636	547	22	for	for	ADP
ejde-636	547	23	a	a	DET
ejde-636	547	24	class	class	NOUN
ejde-636	547	25	of	of	ADP
ejde-636	547	26	schrödinger	schrödinger	NOUN
ejde-636	547	27	-	-	PUNCT
ejde-636	547	28	poisson	poisson	NOUN
ejde-636	547	29	equations	equation	NOUN
ejde-636	547	30	,	,	PUNCT
ejde-636	547	31	proc	proc	NOUN
ejde-636	547	32	.	.	PUNCT
ejde-636	548	1	lond	lond	PROPN
ejde-636	548	2	.	.	PUNCT
ejde-636	549	1	math	math	NOUN
ejde-636	549	2	.	.	PUNCT
ejde-636	550	1	soc	soc	PROPN
ejde-636	550	2	.	.	PUNCT
ejde-636	551	1	(	(	PUNCT
ejde-636	551	2	3	3	NUM
ejde-636	551	3	)	)	PUNCT
ejde-636	551	4	,	,	PUNCT
ejde-636	551	5	107	107	NUM
ejde-636	551	6	(	(	PUNCT
ejde-636	551	7	2013	2013	NUM
ejde-636	551	8	)	)	PUNCT
ejde-636	551	9	,	,	PUNCT
ejde-636	551	10	303	303	NUM
ejde-636	551	11	-	-	SYM
ejde-636	551	12	339	339	NUM
ejde-636	551	13	.	.	PUNCT
ejde-636	552	1	[	[	X
ejde-636	552	2	3	3	X
ejde-636	552	3	]	]	PUNCT
ejde-636	552	4	d.	d.	NOUN
ejde-636	552	5	bonheure	bonheure	PROPN
ejde-636	552	6	,	,	PUNCT
ejde-636	552	7	j.-b	j.-b	PROPN
ejde-636	552	8	.	.	PUNCT
ejde-636	552	9	casteras	casteras	PROPN
ejde-636	552	10	,	,	PUNCT
ejde-636	552	11	t.	t.	PROPN
ejde-636	552	12	gou	gou	PROPN
ejde-636	552	13	,	,	PUNCT
ejde-636	552	14	l.	l.	PROPN
ejde-636	552	15	jeanjean	jeanjean	PROPN
ejde-636	552	16	;	;	PUNCT
ejde-636	552	17	normalized	normalize	VERB
ejde-636	552	18	solutions	solution	NOUN
ejde-636	552	19	to	to	ADP
ejde-636	552	20	the	the	DET
ejde-636	552	21	mixed	mixed	ADJ
ejde-636	552	22	dispersion	dispersion	NOUN
ejde-636	552	23	nonlinear	nonlinear	NOUN
ejde-636	552	24	schrödinger	schrödinger	NOUN
ejde-636	552	25	equation	equation	NOUN
ejde-636	552	26	in	in	ADP
ejde-636	552	27	the	the	DET
ejde-636	552	28	mass	mass	NOUN
ejde-636	552	29	critical	critical	ADJ
ejde-636	552	30	and	and	CCONJ
ejde-636	552	31	subcritical	subcritical	ADJ
ejde-636	552	32	regime	regime	NOUN
ejde-636	552	33	,	,	PUNCT
ejde-636	552	34	trans	trans	PROPN
ejde-636	552	35	.	.	PROPN
ejde-636	553	1	amer	amer	PROPN
ejde-636	553	2	.	.	PUNCT
ejde-636	553	3	math	math	PROPN
ejde-636	553	4	.	.	PUNCT
ejde-636	554	1	soc	soc	PROPN
ejde-636	554	2	.	.	PUNCT
ejde-636	554	3	,	,	PUNCT
ejde-636	554	4	372	372	NUM
ejde-636	554	5	(	(	PUNCT
ejde-636	554	6	2019	2019	NUM
ejde-636	554	7	)	)	PUNCT
ejde-636	554	8	,	,	PUNCT
ejde-636	554	9	2167	2167	NUM
ejde-636	554	10	-	-	SYM
ejde-636	554	11	2212	2212	NUM
ejde-636	554	12	.	.	PUNCT
ejde-636	555	1	[	[	X
ejde-636	555	2	4	4	X
ejde-636	555	3	]	]	PUNCT
ejde-636	555	4	d.	d.	NOUN
ejde-636	555	5	bonheure	bonheure	PROPN
ejde-636	555	6	,	,	PUNCT
ejde-636	555	7	j.-b	j.-b	PROPN
ejde-636	555	8	.	.	PUNCT
ejde-636	555	9	casteras	casteras	PROPN
ejde-636	555	10	,	,	PUNCT
ejde-636	555	11	t.	t.	PROPN
ejde-636	555	12	gou	gou	PROPN
ejde-636	555	13	,	,	PUNCT
ejde-636	555	14	l.	l.	PROPN
ejde-636	555	15	jeanjean	jeanjean	PROPN
ejde-636	555	16	;	;	PUNCT
ejde-636	555	17	strong	strong	ADJ
ejde-636	555	18	instability	instability	NOUN
ejde-636	555	19	of	of	ADP
ejde-636	555	20	ground	ground	NOUN
ejde-636	555	21	states	state	NOUN
ejde-636	555	22	to	to	ADP
ejde-636	555	23	a	a	DET
ejde-636	555	24	fourth	fourth	ADJ
ejde-636	555	25	order	order	NOUN
ejde-636	555	26	schrödinger	schrödinger	NOUN
ejde-636	555	27	equation	equation	NOUN
ejde-636	555	28	,	,	PUNCT
ejde-636	555	29	int	int	NOUN
ejde-636	555	30	.	.	PUNCT
ejde-636	555	31	math	math	NOUN
ejde-636	555	32	.	.	PUNCT
ejde-636	556	1	res	re	NOUN
ejde-636	556	2	.	.	PUNCT
ejde-636	557	1	not	not	PART
ejde-636	557	2	.	.	PUNCT
ejde-636	558	1	imrn	imrn	PROPN
ejde-636	558	2	,	,	PUNCT
ejde-636	558	3	(	(	PUNCT
ejde-636	558	4	2019	2019	NUM
ejde-636	558	5	)	)	PUNCT
ejde-636	558	6	,	,	PUNCT
ejde-636	558	7	5299	5299	NUM
ejde-636	558	8	-	-	SYM
ejde-636	558	9	5315	5315	NUM
ejde-636	558	10	.	.	PUNCT
ejde-636	559	1	[	[	X
ejde-636	559	2	5	5	NUM
ejde-636	559	3	]	]	PUNCT
ejde-636	559	4	d.	d.	NOUN
ejde-636	559	5	bonheure	bonheure	PROPN
ejde-636	559	6	,	,	PUNCT
ejde-636	559	7	j.-b	j.-b	PROPN
ejde-636	559	8	.	.	PUNCT
ejde-636	559	9	casteras	casteras	PROPN
ejde-636	559	10	,	,	PUNCT
ejde-636	559	11	e.	e.	PROPN
ejde-636	559	12	moreira	moreira	PROPN
ejde-636	559	13	dos	dos	PROPN
ejde-636	559	14	santos	santos	PROPN
ejde-636	559	15	,	,	PUNCT
ejde-636	559	16	r.	r.	PROPN
ejde-636	559	17	nascimento	nascimento	PROPN
ejde-636	559	18	;	;	PUNCT
ejde-636	559	19	orbitally	orbitally	ADV
ejde-636	559	20	stable	stable	ADJ
ejde-636	559	21	standing	standing	ADJ
ejde-636	559	22	waves	wave	NOUN
ejde-636	559	23	of	of	ADP
ejde-636	559	24	a	a	DET
ejde-636	559	25	mixed	mixed	ADJ
ejde-636	559	26	dispersion	dispersion	NOUN
ejde-636	559	27	nonlinear	nonlinear	NOUN
ejde-636	559	28	schrödinger	schrödinger	NOUN
ejde-636	559	29	equation	equation	NOUN
ejde-636	559	30	,	,	PUNCT
ejde-636	559	31	siam	siam	PROPN
ejde-636	559	32	j.	j.	PROPN
ejde-636	559	33	math	math	PROPN
ejde-636	559	34	.	.	PUNCT
ejde-636	560	1	anal	anal	PROPN
ejde-636	560	2	.	.	PROPN
ejde-636	560	3	,	,	PUNCT
ejde-636	560	4	50	50	NUM
ejde-636	560	5	(	(	PUNCT
ejde-636	560	6	2018	2018	NUM
ejde-636	560	7	)	)	PUNCT
ejde-636	560	8	,	,	PUNCT
ejde-636	560	9	5027	5027	NUM
ejde-636	560	10	-	-	SYM
ejde-636	560	11	5071	5071	NUM
ejde-636	560	12	.	.	PUNCT
ejde-636	561	1	[	[	X
ejde-636	561	2	6	6	NUM
ejde-636	561	3	]	]	PUNCT
ejde-636	561	4	d.	d.	NOUN
ejde-636	561	5	bonheure	bonheure	PROPN
ejde-636	561	6	,	,	PUNCT
ejde-636	561	7	r.	r.	PROPN
ejde-636	561	8	nascimento	nascimento	PROPN
ejde-636	561	9	;	;	PUNCT
ejde-636	561	10	waveguide	waveguide	VERB
ejde-636	561	11	solutions	solution	NOUN
ejde-636	561	12	for	for	ADP
ejde-636	561	13	a	a	DET
ejde-636	561	14	nonlinear	nonlinear	ADJ
ejde-636	561	15	schrödinger	schrödinger	NOUN
ejde-636	561	16	equations	equation	NOUN
ejde-636	561	17	with	with	ADP
ejde-636	561	18	mixed	mixed	ADJ
ejde-636	561	19	dispersion	dispersion	NOUN
ejde-636	561	20	,	,	PUNCT
ejde-636	561	21	in	in	ADP
ejde-636	561	22	:	:	PUNCT
ejde-636	561	23	contributions	contribution	NOUN
ejde-636	561	24	to	to	ADP
ejde-636	561	25	nonlinear	nonlinear	ADJ
ejde-636	561	26	elliptic	elliptic	ADJ
ejde-636	561	27	equations	equation	NOUN
ejde-636	561	28	and	and	CCONJ
ejde-636	561	29	systems	system	NOUN
ejde-636	561	30	,	,	PUNCT
ejde-636	561	31	progr	progr	NOUN
ejde-636	561	32	.	.	PUNCT
ejde-636	562	1	nonlinear	nonlinear	ADJ
ejde-636	562	2	differential	differential	ADJ
ejde-636	562	3	equations	equation	NOUN
ejde-636	562	4	appl	appl	PROPN
ejde-636	562	5	.	.	PROPN
ejde-636	562	6	,	,	PUNCT
ejde-636	562	7	86	86	NUM
ejde-636	562	8	,	,	PUNCT
ejde-636	562	9	birkhäuser	birkhäuser	NOUN
ejde-636	562	10	/	/	SYM
ejde-636	562	11	springer	springer	NOUN
ejde-636	562	12	,	,	PUNCT
ejde-636	562	13	cham	cham	PROPN
ejde-636	562	14	,	,	PUNCT
ejde-636	562	15	(	(	PUNCT
ejde-636	562	16	2015	2015	NUM
ejde-636	562	17	)	)	PUNCT
ejde-636	562	18	,	,	PUNCT
ejde-636	562	19	31	31	NUM
ejde-636	562	20	-	-	SYM
ejde-636	562	21	53	53	NUM
ejde-636	562	22	.	.	PUNCT
ejde-636	562	23	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	562	24	dispersion	dispersion	NOUN
ejde-636	562	25	nonlinear	nonlinear	NOUN
ejde-636	562	26	schrödinger	schrödinger	NOUN
ejde-636	562	27	equation	equation	NOUN
ejde-636	562	28	19	19	NUM
ejde-636	563	1	[	[	X
ejde-636	563	2	7	7	X
ejde-636	563	3	]	]	X
ejde-636	563	4	j.	j.	PROPN
ejde-636	563	5	borthwick	borthwick	PROPN
ejde-636	563	6	,	,	PUNCT
ejde-636	563	7	x.	x.	PROPN
ejde-636	563	8	chang	chang	PROPN
ejde-636	563	9	,	,	PUNCT
ejde-636	563	10	l.	l.	PROPN
ejde-636	563	11	jeanjean	jeanjean	PROPN
ejde-636	563	12	,	,	PUNCT
ejde-636	563	13	n.	n.	PROPN
ejde-636	563	14	soave	soave	PROPN
ejde-636	563	15	;	;	PUNCT
ejde-636	563	16	normalized	normalize	VERB
ejde-636	563	17	solutions	solution	NOUN
ejde-636	563	18	of	of	ADP
ejde-636	563	19	l2	l2	NOUN
ejde-636	563	20	-	-	PUNCT
ejde-636	563	21	supercritical	supercritical	ADJ
ejde-636	563	22	nls	nls	NOUN
ejde-636	563	23	equations	equation	NOUN
ejde-636	563	24	on	on	ADP
ejde-636	563	25	noncompact	noncompact	ADJ
ejde-636	563	26	metric	metric	ADJ
ejde-636	563	27	graphs	graph	NOUN
ejde-636	563	28	with	with	ADP
ejde-636	563	29	localized	localized	ADJ
ejde-636	563	30	nonlinearities	nonlinearitie	NOUN
ejde-636	563	31	,	,	PUNCT
ejde-636	563	32	nonlinearity	nonlinearity	NOUN
ejde-636	563	33	,	,	PUNCT
ejde-636	563	34	36	36	NUM
ejde-636	563	35	,	,	PUNCT
ejde-636	563	36	(	(	PUNCT
ejde-636	563	37	2023	2023	NUM
ejde-636	563	38	)	)	PUNCT
ejde-636	563	39	,	,	PUNCT
ejde-636	563	40	3776	3776	NUM
ejde-636	563	41	-	-	SYM
ejde-636	563	42	3795	3795	NUM
ejde-636	563	43	.	.	PUNCT
ejde-636	564	1	[	[	X
ejde-636	564	2	8	8	X
ejde-636	564	3	]	]	PUNCT
ejde-636	564	4	t.	t.	NOUN
ejde-636	564	5	boulenger	boulenger	NOUN
ejde-636	564	6	,	,	PUNCT
ejde-636	564	7	e.	e.	PROPN
ejde-636	564	8	lenzman	lenzman	PROPN
ejde-636	564	9	;	;	PUNCT
ejde-636	564	10	blowup	blowup	ADJ
ejde-636	564	11	for	for	ADP
ejde-636	564	12	biharmonic	biharmonic	ADJ
ejde-636	564	13	nls	nls	NOUN
ejde-636	564	14	,	,	PUNCT
ejde-636	564	15	ann	ann	PROPN
ejde-636	564	16	.	.	PUNCT
ejde-636	565	1	sci	sci	PROPN
ejde-636	565	2	.	.	PUNCT
ejde-636	566	1	éc	éc	PROPN
ejde-636	566	2	.	.	PUNCT
ejde-636	567	1	norm	norm	NOUN
ejde-636	567	2	.	.	PUNCT
ejde-636	568	1	supér	supér	PROPN
ejde-636	568	2	.	.	PUNCT
ejde-636	569	1	(	(	PUNCT
ejde-636	569	2	4	4	NUM
ejde-636	569	3	)	)	PUNCT
ejde-636	569	4	,	,	PUNCT
ejde-636	569	5	50	50	NUM
ejde-636	569	6	(	(	PUNCT
ejde-636	569	7	2017	2017	NUM
ejde-636	569	8	)	)	PUNCT
ejde-636	569	9	,	,	PUNCT
ejde-636	569	10	503	503	NUM
ejde-636	569	11	-	-	SYM
ejde-636	569	12	544	544	NUM
ejde-636	569	13	.	.	PUNCT
ejde-636	570	1	[	[	X
ejde-636	570	2	9	9	NUM
ejde-636	570	3	]	]	X
ejde-636	570	4	n.	n.	NOUN
ejde-636	570	5	boussäıd	boussäıd	PROPN
ejde-636	570	6	,	,	PUNCT
ejde-636	570	7	a.	a.	PROPN
ejde-636	570	8	j.	j.	PROPN
ejde-636	570	9	fernández	fernández	PROPN
ejde-636	570	10	,	,	PUNCT
ejde-636	570	11	l.	l.	PROPN
ejde-636	570	12	jeanjean	jeanjean	PROPN
ejde-636	570	13	;	;	PUNCT
ejde-636	570	14	some	some	DET
ejde-636	570	15	remarks	remark	NOUN
ejde-636	570	16	on	on	ADP
ejde-636	570	17	a	a	DET
ejde-636	570	18	minimization	minimization	NOUN
ejde-636	570	19	problem	problem	NOUN
ejde-636	570	20	associated	associate	VERB
ejde-636	570	21	to	to	ADP
ejde-636	570	22	a	a	DET
ejde-636	570	23	fourth	fourth	ADJ
ejde-636	570	24	order	order	NOUN
ejde-636	570	25	nonlinear	nonlinear	NOUN
ejde-636	570	26	scrhödinger	scrhödinger	NUM
ejde-636	570	27	equation	equation	NOUN
ejde-636	570	28	,	,	PUNCT
ejde-636	570	29	arxiv.1910.13177	arxiv.1910.13177	ADJ
ejde-636	570	30	.	.	PUNCT
ejde-636	571	1	[	[	X
ejde-636	571	2	10	10	NUM
ejde-636	571	3	]	]	PUNCT
ejde-636	571	4	t.	t.	PROPN
ejde-636	571	5	cazenave	cazenave	PROPN
ejde-636	571	6	,	,	PUNCT
ejde-636	571	7	p.-l	p.-l	PROPN
ejde-636	571	8	.	.	PUNCT
ejde-636	572	1	lions	lion	NOUN
ejde-636	572	2	;	;	PUNCT
ejde-636	572	3	orbital	orbital	ADJ
ejde-636	572	4	stability	stability	NOUN
ejde-636	572	5	of	of	ADP
ejde-636	572	6	standing	stand	VERB
ejde-636	572	7	waves	wave	NOUN
ejde-636	572	8	for	for	ADP
ejde-636	572	9	some	some	DET
ejde-636	572	10	nonlinear	nonlinear	ADJ
ejde-636	572	11	schrödinger	schrödinger	NOUN
ejde-636	572	12	equations	equation	NOUN
ejde-636	572	13	,	,	PUNCT
ejde-636	572	14	comm	comm	NOUN
ejde-636	572	15	.	.	PUNCT
ejde-636	572	16	math	math	NOUN
ejde-636	572	17	.	.	PUNCT
ejde-636	573	1	phys	phy	NOUN
ejde-636	573	2	.	.	PUNCT
ejde-636	573	3	,	,	PUNCT
ejde-636	573	4	85	85	NUM
ejde-636	573	5	(	(	PUNCT
ejde-636	573	6	1982	1982	NUM
ejde-636	573	7	)	)	PUNCT
ejde-636	573	8	,	,	PUNCT
ejde-636	573	9	549	549	NUM
ejde-636	573	10	-	-	SYM
ejde-636	573	11	561	561	NUM
ejde-636	573	12	.	.	PUNCT
ejde-636	574	1	[	[	X
ejde-636	574	2	11	11	NUM
ejde-636	574	3	]	]	PUNCT
ejde-636	574	4	l.	l.	PROPN
ejde-636	574	5	cely	cely	ADV
ejde-636	574	6	;	;	PUNCT
ejde-636	574	7	stability	stability	NOUN
ejde-636	574	8	of	of	ADP
ejde-636	574	9	ground	ground	NOUN
ejde-636	574	10	states	state	NOUN
ejde-636	574	11	of	of	ADP
ejde-636	574	12	nonlinear	nonlinear	ADJ
ejde-636	574	13	schrodinger	schrodinger	PROPN
ejde-636	574	14	systems	systems	PROPN
ejde-636	574	15	,	,	PUNCT
ejde-636	574	16	electron	electron	NOUN
ejde-636	574	17	.	.	PUNCT
ejde-636	575	1	j.	j.	PROPN
ejde-636	575	2	differential	differential	PROPN
ejde-636	575	3	equations	equation	NOUN
ejde-636	575	4	,	,	PUNCT
ejde-636	575	5	2023	2023	NUM
ejde-636	575	6	(	(	PUNCT
ejde-636	575	7	2023	2023	NUM
ejde-636	575	8	)	)	PUNCT
ejde-636	575	9	,	,	PUNCT
ejde-636	575	10	no	no	INTJ
ejde-636	575	11	.	.	PROPN
ejde-636	575	12	76	76	NUM
ejde-636	575	13	,	,	PUNCT
ejde-636	575	14	1	1	NUM
ejde-636	575	15	-	-	SYM
ejde-636	575	16	20	20	NUM
ejde-636	575	17	.	.	PUNCT
ejde-636	576	1	[	[	X
ejde-636	576	2	12	12	NUM
ejde-636	576	3	]	]	PUNCT
ejde-636	576	4	x.	x.	PROPN
ejde-636	576	5	chang	chang	PROPN
ejde-636	576	6	,	,	PUNCT
ejde-636	576	7	l.	l.	PROPN
ejde-636	576	8	jeanjean	jeanjean	PROPN
ejde-636	576	9	,	,	PUNCT
ejde-636	576	10	n.	n.	PROPN
ejde-636	576	11	soave	soave	PROPN
ejde-636	576	12	;	;	PUNCT
ejde-636	576	13	normalized	normalize	VERB
ejde-636	576	14	solutions	solution	NOUN
ejde-636	576	15	of	of	ADP
ejde-636	576	16	l2	l2	NOUN
ejde-636	576	17	-	-	PUNCT
ejde-636	576	18	supercritical	supercritical	ADJ
ejde-636	576	19	nls	nls	NOUN
ejde-636	576	20	equations	equation	NOUN
ejde-636	576	21	on	on	ADP
ejde-636	576	22	compact	compact	ADJ
ejde-636	576	23	metric	metric	ADJ
ejde-636	576	24	graphs	graph	NOUN
ejde-636	576	25	,	,	PUNCT
ejde-636	576	26	ann	ann	PROPN
ejde-636	576	27	.	.	PROPN
ejde-636	576	28	inst	inst	PROPN
ejde-636	576	29	.	.	PUNCT
ejde-636	577	1	h.	h.	PROPN
ejde-636	577	2	poincaré	poincaré	PROPN
ejde-636	577	3	c	c	PROPN
ejde-636	577	4	anal	anal	PROPN
ejde-636	577	5	.	.	PUNCT
ejde-636	578	1	non	non	PROPN
ejde-636	578	2	linéaire	linéaire	PROPN
ejde-636	578	3	,	,	PUNCT
ejde-636	578	4	doi	doi	NOUN
ejde-636	578	5	:	:	PUNCT
ejde-636	578	6	10.4171	10.4171	NUM
ejde-636	578	7	/	/	SYM
ejde-636	578	8	aihpc/8	aihpc/8	NOUN
ejde-636	578	9	.	.	PUNCT
ejde-636	579	1	[	[	X
ejde-636	579	2	13	13	NUM
ejde-636	579	3	]	]	PUNCT
ejde-636	579	4	x.	x.	PROPN
ejde-636	579	5	chang	chang	PROPN
ejde-636	579	6	,	,	PUNCT
ejde-636	579	7	m.	m.	PROPN
ejde-636	579	8	liu	liu	PROPN
ejde-636	579	9	,	,	PUNCT
ejde-636	579	10	d.	d.	PROPN
ejde-636	579	11	yan	yan	PROPN
ejde-636	579	12	;	;	PUNCT
ejde-636	579	13	normalized	normalize	VERB
ejde-636	579	14	ground	ground	NOUN
ejde-636	579	15	state	state	NOUN
ejde-636	579	16	solutions	solution	NOUN
ejde-636	579	17	of	of	ADP
ejde-636	579	18	nonlinear	nonlinear	ADJ
ejde-636	579	19	schrödinger	schrödinger	NOUN
ejde-636	579	20	equations	equation	NOUN
ejde-636	579	21	involving	involve	VERB
ejde-636	579	22	exponential	exponential	ADJ
ejde-636	579	23	critical	critical	ADJ
ejde-636	579	24	growth	growth	NOUN
ejde-636	579	25	,	,	PUNCT
ejde-636	579	26	j.	j.	PROPN
ejde-636	579	27	geom	geom	PROPN
ejde-636	579	28	.	.	PUNCT
ejde-636	580	1	anal	anal	PROPN
ejde-636	580	2	.	.	PROPN
ejde-636	580	3	,	,	PUNCT
ejde-636	580	4	33	33	NUM
ejde-636	580	5	(	(	PUNCT
ejde-636	580	6	2023	2023	NUM
ejde-636	580	7	)	)	PUNCT
ejde-636	580	8	,	,	PUNCT
ejde-636	580	9	paper	paper	NOUN
ejde-636	580	10	no	no	INTJ
ejde-636	580	11	.	.	PROPN
ejde-636	580	12	83	83	NUM
ejde-636	580	13	,	,	PUNCT
ejde-636	580	14	20pp	20pp	NOUN
ejde-636	580	15	.	.	PUNCT
ejde-636	581	1	[	[	X
ejde-636	581	2	14	14	NUM
ejde-636	581	3	]	]	PUNCT
ejde-636	581	4	a.	a.	NOUN
ejde-636	581	5	fernández	fernández	PROPN
ejde-636	581	6	,	,	PUNCT
ejde-636	581	7	l.	l.	PROPN
ejde-636	581	8	jeanjean	jeanjean	PROPN
ejde-636	581	9	,	,	PUNCT
ejde-636	581	10	r.	r.	PROPN
ejde-636	581	11	mandel	mandel	PROPN
ejde-636	581	12	,	,	PUNCT
ejde-636	581	13	m.	m.	PROPN
ejde-636	581	14	maris	maris	PROPN
ejde-636	581	15	;	;	PUNCT
ejde-636	581	16	non	non	ADJ
ejde-636	581	17	-	-	ADJ
ejde-636	581	18	homogeneous	homogeneous	ADJ
ejde-636	581	19	gagliardo	gagliardo	NOUN
ejde-636	581	20	-	-	PUNCT
ejde-636	581	21	nirenberg	nirenberg	PROPN
ejde-636	581	22	inequalities	inequality	NOUN
ejde-636	581	23	in	in	ADP
ejde-636	581	24	rn	rn	PROPN
ejde-636	581	25	and	and	CCONJ
ejde-636	581	26	application	application	NOUN
ejde-636	581	27	to	to	ADP
ejde-636	581	28	a	a	DET
ejde-636	581	29	biharmonic	biharmonic	ADJ
ejde-636	581	30	non	non	ADJ
ejde-636	581	31	-	-	ADJ
ejde-636	581	32	linear	linear	ADJ
ejde-636	581	33	schrödinger	schrödinger	NOUN
ejde-636	581	34	equation	equation	NOUN
ejde-636	581	35	,	,	PUNCT
ejde-636	581	36	j.	j.	PROPN
ejde-636	581	37	differential	differential	PROPN
ejde-636	581	38	equations	equation	NOUN
ejde-636	581	39	,	,	PUNCT
ejde-636	581	40	330	330	NUM
ejde-636	581	41	(	(	PUNCT
ejde-636	581	42	2022	2022	NUM
ejde-636	581	43	)	)	PUNCT
ejde-636	581	44	,	,	PUNCT
ejde-636	581	45	1	1	NUM
ejde-636	581	46	-	-	SYM
ejde-636	581	47	65	65	NUM
ejde-636	581	48	.	.	PUNCT
ejde-636	582	1	[	[	X
ejde-636	582	2	15	15	NUM
ejde-636	582	3	]	]	X
ejde-636	582	4	g.	g.	PROPN
ejde-636	582	5	fibich	fibich	PROPN
ejde-636	582	6	,	,	PUNCT
ejde-636	582	7	b.	b.	PROPN
ejde-636	582	8	ilan	ilan	PROPN
ejde-636	582	9	,	,	PUNCT
ejde-636	582	10	g.	g.	PROPN
ejde-636	582	11	papanicolaou	papanicolaou	PROPN
ejde-636	582	12	;	;	PUNCT
ejde-636	582	13	self	self	NOUN
ejde-636	582	14	-	-	PUNCT
ejde-636	582	15	focusing	focus	VERB
ejde-636	582	16	with	with	ADP
ejde-636	582	17	fourth	fourth	ADJ
ejde-636	582	18	-	-	PUNCT
ejde-636	582	19	order	order	NOUN
ejde-636	582	20	dispersion	dispersion	NOUN
ejde-636	582	21	,	,	PUNCT
ejde-636	582	22	siam	siam	PROPN
ejde-636	582	23	j.	j.	PROPN
ejde-636	582	24	appl	appl	PROPN
ejde-636	582	25	.	.	PROPN
ejde-636	582	26	math	math	PROPN
ejde-636	582	27	.	.	PUNCT
ejde-636	582	28	,	,	PUNCT
ejde-636	582	29	62	62	NUM
ejde-636	582	30	(	(	PUNCT
ejde-636	582	31	2002	2002	NUM
ejde-636	582	32	)	)	PUNCT
ejde-636	582	33	,	,	PUNCT
ejde-636	582	34	1437	1437	NUM
ejde-636	582	35	-	-	SYM
ejde-636	582	36	1462	1462	NUM
ejde-636	582	37	.	.	PUNCT
ejde-636	583	1	[	[	X
ejde-636	583	2	16	16	NUM
ejde-636	583	3	]	]	X
ejde-636	583	4	b.	b.	PROPN
ejde-636	583	5	a.	a.	PROPN
ejde-636	583	6	ivanov	ivanov	PROPN
ejde-636	583	7	,	,	PUNCT
ejde-636	583	8	a.	a.	NOUN
ejde-636	583	9	m.	m.	NOUN
ejde-636	583	10	kosevich	kosevich	PROPN
ejde-636	583	11	;	;	PUNCT
ejde-636	583	12	stable	stable	ADJ
ejde-636	583	13	three	three	NUM
ejde-636	583	14	-	-	PUNCT
ejde-636	583	15	dimensional	dimensional	ADJ
ejde-636	583	16	small	small	ADJ
ejde-636	583	17	-	-	PUNCT
ejde-636	583	18	amplitude	amplitude	NOUN
ejde-636	583	19	soliton	soliton	NOUN
ejde-636	583	20	in	in	ADP
ejde-636	583	21	magnetic	magnetic	ADJ
ejde-636	583	22	materials	material	NOUN
ejde-636	583	23	,	,	PUNCT
ejde-636	583	24	so	so	ADV
ejde-636	583	25	.	.	PUNCT
ejde-636	584	1	j.	j.	PROPN
ejde-636	584	2	low	low	PROPN
ejde-636	584	3	temp	temp	PROPN
ejde-636	584	4	.	.	PUNCT
ejde-636	585	1	phys	phy	NOUN
ejde-636	585	2	.	.	PUNCT
ejde-636	585	3	,	,	PUNCT
ejde-636	585	4	9	9	NUM
ejde-636	585	5	(	(	PUNCT
ejde-636	585	6	1983	1983	NUM
ejde-636	585	7	)	)	PUNCT
ejde-636	585	8	,	,	PUNCT
ejde-636	585	9	439	439	NUM
ejde-636	585	10	-	-	SYM
ejde-636	585	11	442	442	NUM
ejde-636	585	12	.	.	PUNCT
ejde-636	586	1	[	[	X
ejde-636	586	2	17	17	NUM
ejde-636	586	3	]	]	PUNCT
ejde-636	586	4	l.	l.	PROPN
ejde-636	586	5	jeanjean	jeanjean	PROPN
ejde-636	586	6	;	;	PUNCT
ejde-636	586	7	existence	existence	NOUN
ejde-636	586	8	of	of	ADP
ejde-636	586	9	solutions	solution	NOUN
ejde-636	586	10	with	with	ADP
ejde-636	586	11	prescribed	prescribed	ADJ
ejde-636	586	12	norm	norm	NOUN
ejde-636	586	13	for	for	ADP
ejde-636	586	14	semilinear	semilinear	PROPN
ejde-636	586	15	elliptic	elliptic	ADJ
ejde-636	586	16	equations	equation	NOUN
ejde-636	586	17	,	,	PUNCT
ejde-636	586	18	nonlinear	nonlinear	ADJ
ejde-636	586	19	anal	anal	NOUN
ejde-636	586	20	.	.	PUNCT
ejde-636	586	21	,	,	PUNCT
ejde-636	586	22	28	28	NUM
ejde-636	586	23	(	(	PUNCT
ejde-636	586	24	1997	1997	NUM
ejde-636	586	25	)	)	PUNCT
ejde-636	586	26	,	,	PUNCT
ejde-636	586	27	1633	1633	NUM
ejde-636	586	28	-	-	SYM
ejde-636	586	29	1659	1659	NUM
ejde-636	586	30	.	.	PUNCT
ejde-636	587	1	[	[	X
ejde-636	587	2	18	18	NUM
ejde-636	587	3	]	]	X
ejde-636	587	4	l.	l.	PROPN
ejde-636	587	5	jeanjean	jeanjean	PROPN
ejde-636	587	6	,	,	PUNCT
ejde-636	587	7	j.	j.	PROPN
ejde-636	587	8	jendrej	jendrej	PROPN
ejde-636	587	9	,	,	PUNCT
ejde-636	587	10	t.	t.	PROPN
ejde-636	587	11	t.	t.	PROPN
ejde-636	587	12	le	le	PROPN
ejde-636	587	13	,	,	PUNCT
ejde-636	587	14	n.	n.	PROPN
ejde-636	587	15	visciglia	visciglia	PROPN
ejde-636	587	16	;	;	PUNCT
ejde-636	587	17	orbital	orbital	ADJ
ejde-636	587	18	stability	stability	NOUN
ejde-636	587	19	of	of	ADP
ejde-636	587	20	ground	ground	NOUN
ejde-636	587	21	states	state	NOUN
ejde-636	587	22	for	for	ADP
ejde-636	587	23	a	a	DET
ejde-636	587	24	sobolev	sobolev	NOUN
ejde-636	587	25	critical	critical	ADJ
ejde-636	587	26	schrödinger	schrödinger	NOUN
ejde-636	587	27	equation	equation	NOUN
ejde-636	587	28	,	,	PUNCT
ejde-636	587	29	j.	j.	PROPN
ejde-636	587	30	math	math	PROPN
ejde-636	587	31	.	.	PUNCT
ejde-636	588	1	pures	pure	NOUN
ejde-636	588	2	appl	appl	PROPN
ejde-636	588	3	.	.	PUNCT
ejde-636	589	1	(	(	PUNCT
ejde-636	589	2	9	9	NUM
ejde-636	589	3	)	)	PUNCT
ejde-636	589	4	,	,	PUNCT
ejde-636	589	5	164	164	NUM
ejde-636	589	6	(	(	PUNCT
ejde-636	589	7	2022	2022	NUM
ejde-636	589	8	)	)	PUNCT
ejde-636	589	9	,	,	PUNCT
ejde-636	589	10	158	158	NUM
ejde-636	589	11	-	-	SYM
ejde-636	589	12	179	179	NUM
ejde-636	589	13	.	.	PUNCT
ejde-636	590	1	[	[	X
ejde-636	590	2	19	19	NUM
ejde-636	590	3	]	]	PUNCT
ejde-636	590	4	l.	l.	PROPN
ejde-636	590	5	jeanjean	jeanjean	PROPN
ejde-636	590	6	,	,	PUNCT
ejde-636	590	7	t.	t.	PROPN
ejde-636	590	8	t.	t.	PROPN
ejde-636	590	9	le	le	PROPN
ejde-636	590	10	;	;	PUNCT
ejde-636	590	11	multiple	multiple	ADJ
ejde-636	590	12	normalized	normalize	VERB
ejde-636	590	13	solutions	solution	NOUN
ejde-636	590	14	for	for	ADP
ejde-636	590	15	a	a	DET
ejde-636	590	16	sobolev	sobolev	NOUN
ejde-636	590	17	critical	critical	ADJ
ejde-636	590	18	schrödinger	schrödinger	NOUN
ejde-636	590	19	equation	equation	NOUN
ejde-636	590	20	,	,	PUNCT
ejde-636	590	21	math	math	NOUN
ejde-636	590	22	.	.	PUNCT
ejde-636	591	1	ann	ann	PROPN
ejde-636	591	2	.	.	PROPN
ejde-636	591	3	,	,	PUNCT
ejde-636	591	4	384	384	NUM
ejde-636	591	5	(	(	PUNCT
ejde-636	591	6	2022	2022	NUM
ejde-636	591	7	)	)	PUNCT
ejde-636	591	8	,	,	PUNCT
ejde-636	591	9	101	101	NUM
ejde-636	591	10	-	-	SYM
ejde-636	591	11	134	134	NUM
ejde-636	591	12	.	.	PUNCT
ejde-636	592	1	[	[	X
ejde-636	592	2	20	20	NUM
ejde-636	592	3	]	]	PUNCT
ejde-636	592	4	v.	v.	PROPN
ejde-636	592	5	i.	i.	PROPN
ejde-636	592	6	karpman	karpman	PROPN
ejde-636	592	7	;	;	PUNCT
ejde-636	592	8	stabilization	stabilization	NOUN
ejde-636	592	9	of	of	ADP
ejde-636	592	10	soliton	soliton	NOUN
ejde-636	592	11	instabilities	instability	NOUN
ejde-636	592	12	by	by	ADP
ejde-636	592	13	higher	high	ADJ
ejde-636	592	14	-	-	PUNCT
ejde-636	592	15	order	order	NOUN
ejde-636	592	16	dispersion	dispersion	NOUN
ejde-636	592	17	:	:	PUNCT
ejde-636	592	18	fourth	fourth	ADJ
ejde-636	592	19	-	-	PUNCT
ejde-636	592	20	order	order	NOUN
ejde-636	592	21	nonlinear	nonlinear	ADJ
ejde-636	592	22	schrödinger	schrödinger	NOUN
ejde-636	592	23	-	-	PUNCT
ejde-636	592	24	type	type	NOUN
ejde-636	592	25	equations	equation	NOUN
ejde-636	592	26	,	,	PUNCT
ejde-636	592	27	phys	phy	NOUN
ejde-636	592	28	.	.	PUNCT
ejde-636	593	1	rev	rev	PROPN
ejde-636	593	2	.	.	PROPN
ejde-636	594	1	e	e	X
ejde-636	594	2	,	,	PUNCT
ejde-636	594	3	53	53	NUM
ejde-636	594	4	(	(	PUNCT
ejde-636	594	5	1996	1996	NUM
ejde-636	594	6	)	)	PUNCT
ejde-636	594	7	,	,	PUNCT
ejde-636	594	8	1336	1336	NUM
ejde-636	594	9	-	-	SYM
ejde-636	594	10	1339	1339	NUM
ejde-636	594	11	.	.	PUNCT
ejde-636	595	1	[	[	X
ejde-636	595	2	21	21	NUM
ejde-636	595	3	]	]	X
ejde-636	595	4	v.	v.	PROPN
ejde-636	595	5	i.	i.	PROPN
ejde-636	595	6	karpman	karpman	PROPN
ejde-636	595	7	,	,	PUNCT
ejde-636	595	8	a.	a.	PROPN
ejde-636	595	9	g.	g.	PROPN
ejde-636	595	10	shagalov	shagalov	PROPN
ejde-636	595	11	;	;	PUNCT
ejde-636	595	12	stability	stability	NOUN
ejde-636	595	13	of	of	ADP
ejde-636	595	14	solitons	soliton	NOUN
ejde-636	595	15	described	describe	VERB
ejde-636	595	16	by	by	ADP
ejde-636	595	17	nonlinear	nonlinear	ADJ
ejde-636	595	18	schrödinger	schrödinger	NOUN
ejde-636	595	19	-	-	PUNCT
ejde-636	595	20	type	type	NOUN
ejde-636	595	21	equations	equation	NOUN
ejde-636	595	22	with	with	ADP
ejde-636	595	23	higher	high	ADJ
ejde-636	595	24	-	-	PUNCT
ejde-636	595	25	order	order	NOUN
ejde-636	595	26	dispersion	dispersion	NOUN
ejde-636	595	27	,	,	PUNCT
ejde-636	595	28	phys	phy	NOUN
ejde-636	595	29	d	d	NOUN
ejde-636	595	30	,	,	PUNCT
ejde-636	595	31	144	144	NUM
ejde-636	595	32	(	(	PUNCT
ejde-636	595	33	2000	2000	NUM
ejde-636	595	34	)	)	PUNCT
ejde-636	595	35	,	,	PUNCT
ejde-636	595	36	194	194	NUM
ejde-636	595	37	-	-	SYM
ejde-636	595	38	210	210	NUM
ejde-636	595	39	.	.	PUNCT
ejde-636	596	1	[	[	X
ejde-636	596	2	22	22	NUM
ejde-636	596	3	]	]	X
ejde-636	596	4	y.	y.	PROPN
ejde-636	596	5	li	li	PROPN
ejde-636	596	6	,	,	PUNCT
ejde-636	596	7	x.	x.	PROPN
ejde-636	596	8	chang	chang	PROPN
ejde-636	596	9	,	,	PUNCT
ejde-636	596	10	z.	z.	PROPN
ejde-636	596	11	feng	feng	PROPN
ejde-636	596	12	;	;	PUNCT
ejde-636	596	13	normalized	normalize	VERB
ejde-636	596	14	solutions	solution	NOUN
ejde-636	596	15	for	for	ADP
ejde-636	596	16	sobolev	sobolev	NOUN
ejde-636	596	17	critical	critical	ADJ
ejde-636	596	18	schrödinger	schrödinger	NOUN
ejde-636	596	19	-	-	PUNCT
ejde-636	596	20	bopppodolsky	bopppodolsky	ADJ
ejde-636	596	21	systems	system	NOUN
ejde-636	596	22	,	,	PUNCT
ejde-636	596	23	electron	electron	NOUN
ejde-636	596	24	.	.	PUNCT
ejde-636	597	1	j.	j.	PROPN
ejde-636	597	2	differential	differential	PROPN
ejde-636	597	3	equations	equation	NOUN
ejde-636	597	4	,	,	PUNCT
ejde-636	597	5	2023	2023	NUM
ejde-636	597	6	(	(	PUNCT
ejde-636	597	7	2023	2023	NUM
ejde-636	597	8	)	)	PUNCT
ejde-636	597	9	,	,	PUNCT
ejde-636	597	10	no	no	INTJ
ejde-636	597	11	.	.	NOUN
ejde-636	597	12	56	56	NUM
ejde-636	597	13	,	,	PUNCT
ejde-636	597	14	1	1	NUM
ejde-636	597	15	-	-	SYM
ejde-636	597	16	19	19	NUM
ejde-636	597	17	.	.	PUNCT
ejde-636	598	1	[	[	X
ejde-636	598	2	23	23	NUM
ejde-636	598	3	]	]	PUNCT
ejde-636	598	4	m.	m.	NOUN
ejde-636	598	5	liu	liu	PROPN
ejde-636	598	6	,	,	PUNCT
ejde-636	598	7	x.	x.	PROPN
ejde-636	598	8	chang	chang	PROPN
ejde-636	598	9	;	;	PUNCT
ejde-636	598	10	normalized	normalize	VERB
ejde-636	598	11	ground	ground	NOUN
ejde-636	598	12	state	state	NOUN
ejde-636	598	13	solutions	solution	NOUN
ejde-636	598	14	for	for	ADP
ejde-636	598	15	nonlinear	nonlinear	ADJ
ejde-636	598	16	schrödinger	schrödinger	NOUN
ejde-636	598	17	equations	equation	NOUN
ejde-636	598	18	with	with	ADP
ejde-636	598	19	general	general	ADJ
ejde-636	598	20	sobolev	sobolev	PROPN
ejde-636	598	21	critical	critical	ADJ
ejde-636	598	22	nonlinearities	nonlinearitie	NOUN
ejde-636	598	23	,	,	PUNCT
ejde-636	598	24	discrete	discrete	ADJ
ejde-636	598	25	contin	contin	NOUN
ejde-636	598	26	.	.	PUNCT
ejde-636	599	1	dyn	dyn	NOUN
ejde-636	599	2	.	.	PUNCT
ejde-636	600	1	syst	syst	PROPN
ejde-636	600	2	.	.	PUNCT
ejde-636	601	1	ser	ser	PROPN
ejde-636	601	2	.	.	PUNCT
ejde-636	602	1	s.	s.	PROPN
ejde-636	602	2	,	,	PUNCT
ejde-636	602	3	doi	doi	NOUN
ejde-636	602	4	:	:	PUNCT
ejde-636	602	5	10.3934	10.3934	NUM
ejde-636	602	6	/	/	SYM
ejde-636	602	7	dcdss.2024035	dcdss.2024035	PROPN
ejde-636	602	8	.	.	PUNCT
ejde-636	603	1	[	[	X
ejde-636	603	2	24	24	NUM
ejde-636	603	3	]	]	PUNCT
ejde-636	603	4	t.	t.	PROPN
ejde-636	603	5	luo	luo	PROPN
ejde-636	603	6	,	,	PUNCT
ejde-636	603	7	s.	s.	PROPN
ejde-636	603	8	zheng	zheng	PROPN
ejde-636	603	9	,	,	PUNCT
ejde-636	603	10	s.	s.	PROPN
ejde-636	603	11	zhu	zhu	PROPN
ejde-636	603	12	;	;	PUNCT
ejde-636	603	13	the	the	DET
ejde-636	603	14	existence	existence	NOUN
ejde-636	603	15	and	and	CCONJ
ejde-636	603	16	stability	stability	NOUN
ejde-636	603	17	of	of	ADP
ejde-636	603	18	normalized	normalize	VERB
ejde-636	603	19	solutions	solution	NOUN
ejde-636	603	20	for	for	ADP
ejde-636	603	21	a	a	DET
ejde-636	603	22	bi	bi	ADJ
ejde-636	603	23	-	-	ADJ
ejde-636	603	24	harmonic	harmonic	ADJ
ejde-636	603	25	nonlinear	nonlinear	NOUN
ejde-636	603	26	schrödinger	schrödinger	NOUN
ejde-636	603	27	equation	equation	NOUN
ejde-636	603	28	with	with	ADP
ejde-636	603	29	mixed	mixed	ADJ
ejde-636	603	30	dispersion	dispersion	NOUN
ejde-636	603	31	,	,	PUNCT
ejde-636	603	32	acta	acta	PROPN
ejde-636	603	33	math	math	PROPN
ejde-636	603	34	.	.	PUNCT
ejde-636	604	1	sci	sci	PROPN
ejde-636	604	2	.	.	PUNCT
ejde-636	604	3	ser	ser	PROPN
ejde-636	604	4	.	.	PUNCT
ejde-636	605	1	b	b	PROPN
ejde-636	605	2	(	(	PUNCT
ejde-636	605	3	engl	engl	PROPN
ejde-636	605	4	.	.	PUNCT
ejde-636	606	1	ed	ed	NOUN
ejde-636	606	2	.	.	PUNCT
ejde-636	606	3	)	)	PUNCT
ejde-636	606	4	,	,	PUNCT
ejde-636	606	5	43	43	NUM
ejde-636	606	6	(	(	PUNCT
ejde-636	606	7	2023	2023	NUM
ejde-636	606	8	)	)	PUNCT
ejde-636	606	9	,	,	PUNCT
ejde-636	606	10	539	539	NUM
ejde-636	606	11	-	-	SYM
ejde-636	606	12	563	563	NUM
ejde-636	606	13	.	.	PUNCT
ejde-636	607	1	[	[	X
ejde-636	607	2	25	25	NUM
ejde-636	607	3	]	]	PUNCT
ejde-636	607	4	x.	x.	NOUN
ejde-636	607	5	luo	luo	PROPN
ejde-636	607	6	,	,	PUNCT
ejde-636	607	7	t.	t.	PROPN
ejde-636	607	8	yang	yang	PROPN
ejde-636	607	9	;	;	PUNCT
ejde-636	607	10	normalized	normalize	VERB
ejde-636	607	11	solutions	solution	NOUN
ejde-636	607	12	for	for	ADP
ejde-636	607	13	a	a	DET
ejde-636	607	14	fourth	fourth	ADJ
ejde-636	607	15	-	-	PUNCT
ejde-636	607	16	order	order	NOUN
ejde-636	607	17	schrödinger	schrödinger	NOUN
ejde-636	607	18	equation	equation	NOUN
ejde-636	607	19	with	with	ADP
ejde-636	607	20	a	a	DET
ejde-636	607	21	positive	positive	ADJ
ejde-636	607	22	second	second	ADJ
ejde-636	607	23	-	-	PUNCT
ejde-636	607	24	order	order	NOUN
ejde-636	607	25	dispersion	dispersion	NOUN
ejde-636	607	26	coefficient	coefficient	NOUN
ejde-636	607	27	,	,	PUNCT
ejde-636	607	28	sci	sci	PROPN
ejde-636	607	29	.	.	PUNCT
ejde-636	608	1	china	china	PROPN
ejde-636	608	2	math	math	PROPN
ejde-636	608	3	.	.	PUNCT
ejde-636	609	1	,	,	PUNCT
ejde-636	609	2	66	66	NUM
ejde-636	609	3	(	(	PUNCT
ejde-636	609	4	2023	2023	NUM
ejde-636	609	5	)	)	PUNCT
ejde-636	609	6	,	,	PUNCT
ejde-636	609	7	1237	1237	NUM
ejde-636	609	8	-	-	SYM
ejde-636	609	9	1262	1262	NUM
ejde-636	609	10	.	.	PUNCT
ejde-636	610	1	[	[	X
ejde-636	610	2	26	26	NUM
ejde-636	610	3	]	]	X
ejde-636	610	4	h.	h.	PROPN
ejde-636	610	5	lv	lv	PROPN
ejde-636	610	6	,	,	PUNCT
ejde-636	610	7	s.	s.	PROPN
ejde-636	610	8	zheng	zheng	PROPN
ejde-636	610	9	,	,	PUNCT
ejde-636	610	10	z.	z.	PROPN
ejde-636	610	11	feng	feng	PROPN
ejde-636	610	12	,	,	PUNCT
ejde-636	610	13	existence	existence	NOUN
ejde-636	610	14	results	result	VERB
ejde-636	610	15	for	for	ADP
ejde-636	610	16	nonlinear	nonlinear	ADJ
ejde-636	610	17	schrödinger	schrödinger	NOUN
ejde-636	610	18	equations	equation	NOUN
ejde-636	610	19	involving	involve	VERB
ejde-636	610	20	the	the	DET
ejde-636	610	21	fractional	fractional	ADJ
ejde-636	610	22	(	(	PUNCT
ejde-636	610	23	p	p	NOUN
ejde-636	610	24	,	,	PUNCT
ejde-636	610	25	q)-laplacian	q)-laplacian	PUNCT
ejde-636	610	26	and	and	CCONJ
ejde-636	610	27	critical	critical	ADJ
ejde-636	610	28	nonlinearities	nonlinearitie	NOUN
ejde-636	610	29	,	,	PUNCT
ejde-636	610	30	electron	electron	NOUN
ejde-636	610	31	.	.	PUNCT
ejde-636	611	1	j.	j.	PROPN
ejde-636	611	2	differential	differential	PROPN
ejde-636	611	3	equations	equation	NOUN
ejde-636	611	4	,	,	PUNCT
ejde-636	611	5	2021	2021	NUM
ejde-636	611	6	(	(	PUNCT
ejde-636	611	7	2021	2021	NUM
ejde-636	611	8	)	)	PUNCT
ejde-636	611	9	,	,	PUNCT
ejde-636	611	10	no	no	INTJ
ejde-636	611	11	.	.	NOUN
ejde-636	611	12	100	100	NUM
ejde-636	611	13	,	,	PUNCT
ejde-636	611	14	1	1	NUM
ejde-636	611	15	-	-	SYM
ejde-636	611	16	24	24	NUM
ejde-636	611	17	.	.	PUNCT
ejde-636	612	1	[	[	X
ejde-636	612	2	27	27	NUM
ejde-636	612	3	]	]	PUNCT
ejde-636	612	4	z.	z.	PROPN
ejde-636	612	5	ma	ma	PROPN
ejde-636	612	6	,	,	PUNCT
ejde-636	612	7	x.	x.	PROPN
ejde-636	612	8	chang	chang	PROPN
ejde-636	612	9	;	;	PUNCT
ejde-636	612	10	normalized	normalize	VERB
ejde-636	612	11	ground	ground	NOUN
ejde-636	612	12	states	state	NOUN
ejde-636	612	13	of	of	ADP
ejde-636	612	14	nonlinear	nonlinear	ADJ
ejde-636	612	15	biharmonic	biharmonic	NOUN
ejde-636	612	16	schrödinger	schrödinger	NOUN
ejde-636	612	17	equations	equation	NOUN
ejde-636	612	18	with	with	ADP
ejde-636	612	19	sobolev	sobolev	PROPN
ejde-636	612	20	critical	critical	ADJ
ejde-636	612	21	growth	growth	NOUN
ejde-636	612	22	and	and	CCONJ
ejde-636	612	23	combined	combined	ADJ
ejde-636	612	24	nonlinearities	nonlinearitie	NOUN
ejde-636	612	25	,	,	PUNCT
ejde-636	612	26	appl	appl	PROPN
ejde-636	612	27	.	.	PROPN
ejde-636	612	28	math	math	PROPN
ejde-636	612	29	.	.	PUNCT
ejde-636	613	1	lett	lett	PROPN
ejde-636	613	2	.	.	PROPN
ejde-636	613	3	,	,	PUNCT
ejde-636	613	4	135	135	NUM
ejde-636	613	5	(	(	PUNCT
ejde-636	613	6	2023	2023	NUM
ejde-636	613	7	)	)	PUNCT
ejde-636	613	8	,	,	PUNCT
ejde-636	613	9	paper	paper	NOUN
ejde-636	613	10	108388	108388	NUM
ejde-636	613	11	,	,	PUNCT
ejde-636	613	12	7pp	7pp	NOUN
ejde-636	613	13	.	.	PUNCT
ejde-636	614	1	[	[	X
ejde-636	614	2	28	28	NUM
ejde-636	614	3	]	]	X
ejde-636	614	4	z.	z.	PROPN
ejde-636	614	5	ma	ma	PROPN
ejde-636	614	6	,	,	PUNCT
ejde-636	614	7	x.	x.	PROPN
ejde-636	614	8	chang	chang	PROPN
ejde-636	614	9	,	,	PUNCT
ejde-636	614	10	h.	h.	PROPN
ejde-636	614	11	hajaiej	hajaiej	PROPN
ejde-636	614	12	,	,	PUNCT
ejde-636	614	13	l.	l.	PROPN
ejde-636	614	14	song	song	PROPN
ejde-636	614	15	;	;	PUNCT
ejde-636	614	16	existence	existence	NOUN
ejde-636	614	17	and	and	CCONJ
ejde-636	614	18	instability	instability	NOUN
ejde-636	614	19	of	of	ADP
ejde-636	614	20	standing	standing	ADJ
ejde-636	614	21	waves	wave	NOUN
ejde-636	614	22	for	for	ADP
ejde-636	614	23	the	the	DET
ejde-636	614	24	biharmonic	biharmonic	PROPN
ejde-636	614	25	nonlinear	nonlinear	PROPN
ejde-636	614	26	schrödinger	schrödinger	NOUN
ejde-636	614	27	equation	equation	NOUN
ejde-636	614	28	with	with	ADP
ejde-636	614	29	combined	combined	ADJ
ejde-636	614	30	nonlinearities	nonlinearitie	NOUN
ejde-636	614	31	,	,	PUNCT
ejde-636	614	32	arxiv.2305.00327	arxiv.2305.00327	NOUN
ejde-636	614	33	.	.	PUNCT
ejde-636	615	1	[	[	X
ejde-636	615	2	29	29	NUM
ejde-636	615	3	]	]	X
ejde-636	615	4	c.	c.	PROPN
ejde-636	615	5	miao	miao	PROPN
ejde-636	615	6	,	,	PUNCT
ejde-636	615	7	g.	g.	PROPN
ejde-636	615	8	xu	xu	PROPN
ejde-636	615	9	,	,	PUNCT
ejde-636	615	10	l.	l.	PROPN
ejde-636	615	11	zhao	zhao	PROPN
ejde-636	615	12	;	;	PUNCT
ejde-636	615	13	global	global	ADJ
ejde-636	615	14	well	well	ADJ
ejde-636	615	15	-	-	PUNCT
ejde-636	615	16	posedness	posedness	NOUN
ejde-636	615	17	and	and	CCONJ
ejde-636	615	18	scattering	scatter	VERB
ejde-636	615	19	for	for	ADP
ejde-636	615	20	the	the	DET
ejde-636	615	21	focusing	focus	VERB
ejde-636	615	22	energy	energy	NOUN
ejde-636	615	23	-	-	PUNCT
ejde-636	615	24	critical	critical	ADJ
ejde-636	615	25	nonlinear	nonlinear	NOUN
ejde-636	615	26	schrödinger	schrödinger	NOUN
ejde-636	615	27	equations	equation	NOUN
ejde-636	615	28	of	of	ADP
ejde-636	615	29	fourth	fourth	ADJ
ejde-636	615	30	order	order	NOUN
ejde-636	615	31	in	in	ADP
ejde-636	615	32	the	the	DET
ejde-636	615	33	radial	radial	ADJ
ejde-636	615	34	case	case	NOUN
ejde-636	615	35	,	,	PUNCT
ejde-636	615	36	j.	j.	PROPN
ejde-636	615	37	differential	differential	PROPN
ejde-636	615	38	equations	equations	PROPN
ejde-636	615	39	,	,	PUNCT
ejde-636	615	40	246	246	NUM
ejde-636	615	41	(	(	PUNCT
ejde-636	615	42	2009	2009	NUM
ejde-636	615	43	)	)	PUNCT
ejde-636	615	44	,	,	PUNCT
ejde-636	615	45	3715	3715	NUM
ejde-636	615	46	-	-	SYM
ejde-636	615	47	3749	3749	NUM
ejde-636	615	48	.	.	PUNCT
ejde-636	616	1	[	[	X
ejde-636	616	2	30	30	NUM
ejde-636	616	3	]	]	X
ejde-636	616	4	f.	f.	PROPN
ejde-636	616	5	natali	natali	PROPN
ejde-636	616	6	,	,	PUNCT
ejde-636	616	7	a.	a.	NOUN
ejde-636	616	8	pastor	pastor	NOUN
ejde-636	616	9	;	;	PUNCT
ejde-636	616	10	the	the	DET
ejde-636	616	11	fourth	fourth	ADJ
ejde-636	616	12	-	-	PUNCT
ejde-636	616	13	order	order	NOUN
ejde-636	616	14	dispersive	dispersive	ADJ
ejde-636	616	15	nonlinear	nonlinear	NOUN
ejde-636	616	16	schrödinger	schrödinger	NOUN
ejde-636	616	17	equation	equation	NOUN
ejde-636	616	18	:	:	PUNCT
ejde-636	616	19	orbital	orbital	ADJ
ejde-636	616	20	stability	stability	NOUN
ejde-636	616	21	of	of	ADP
ejde-636	616	22	a	a	DET
ejde-636	616	23	standing	standing	ADJ
ejde-636	616	24	wave	wave	NOUN
ejde-636	616	25	,	,	PUNCT
ejde-636	616	26	siam	siam	PROPN
ejde-636	616	27	j.	j.	PROPN
ejde-636	616	28	appl	appl	PROPN
ejde-636	616	29	.	.	PUNCT
ejde-636	616	30	dyn	dyn	PROPN
ejde-636	616	31	.	.	PUNCT
ejde-636	617	1	syst	syst	PROPN
ejde-636	617	2	.	.	PROPN
ejde-636	617	3	,	,	PUNCT
ejde-636	617	4	14	14	NUM
ejde-636	617	5	(	(	PUNCT
ejde-636	617	6	2015	2015	NUM
ejde-636	617	7	)	)	PUNCT
ejde-636	617	8	,	,	PUNCT
ejde-636	617	9	1326	1326	NUM
ejde-636	617	10	-	-	SYM
ejde-636	617	11	1347	1347	NUM
ejde-636	617	12	.	.	PUNCT
ejde-636	618	1	[	[	X
ejde-636	618	2	31	31	NUM
ejde-636	618	3	]	]	PUNCT
ejde-636	618	4	l.	l.	PROPN
ejde-636	618	5	nirenberg	nirenberg	PROPN
ejde-636	618	6	;	;	PUNCT
ejde-636	618	7	on	on	ADP
ejde-636	618	8	elliptic	elliptic	ADJ
ejde-636	618	9	partial	partial	ADJ
ejde-636	618	10	differential	differential	NOUN
ejde-636	618	11	equations	equation	NOUN
ejde-636	618	12	,	,	PUNCT
ejde-636	618	13	ann	ann	PROPN
ejde-636	618	14	.	.	PROPN
ejde-636	618	15	scuola	scuola	PROPN
ejde-636	618	16	norm	norm	NOUN
ejde-636	618	17	.	.	PUNCT
ejde-636	619	1	sup	sup	NOUN
ejde-636	619	2	.	.	PUNCT
ejde-636	619	3	pisa	pisa	PROPN
ejde-636	619	4	cl	cl	PROPN
ejde-636	619	5	.	.	PUNCT
ejde-636	620	1	sci	sci	PROPN
ejde-636	620	2	.	.	PUNCT
ejde-636	621	1	(	(	PUNCT
ejde-636	621	2	3	3	NUM
ejde-636	621	3	)	)	PUNCT
ejde-636	621	4	,	,	PUNCT
ejde-636	621	5	13	13	NUM
ejde-636	621	6	(	(	PUNCT
ejde-636	621	7	1959	1959	NUM
ejde-636	621	8	)	)	PUNCT
ejde-636	621	9	,	,	PUNCT
ejde-636	621	10	115	115	NUM
ejde-636	621	11	-	-	SYM
ejde-636	621	12	162	162	NUM
ejde-636	621	13	.	.	PUNCT
ejde-636	622	1	20	20	NUM
ejde-636	622	2	z.	z.	PROPN
ejde-636	622	3	ma	ma	PROPN
ejde-636	622	4	,	,	PUNCT
ejde-636	622	5	x.	x.	PROPN
ejde-636	622	6	chang	chang	PROPN
ejde-636	622	7	,	,	PUNCT
ejde-636	622	8	z.	z.	PROPN
ejde-636	622	9	feng	feng	PROPN
ejde-636	623	1	ejde-2024/29	ejde-2024/29	NOUN
ejde-636	623	2	[	[	X
ejde-636	623	3	32	32	NUM
ejde-636	623	4	]	]	PUNCT
ejde-636	623	5	b.	b.	PROPN
ejde-636	623	6	pausader	pausader	PROPN
ejde-636	623	7	,	,	PUNCT
ejde-636	623	8	s.	s.	PROPN
ejde-636	623	9	xia	xia	PROPN
ejde-636	623	10	;	;	PUNCT
ejde-636	623	11	scattering	scatter	VERB
ejde-636	623	12	theory	theory	NOUN
ejde-636	623	13	for	for	ADP
ejde-636	623	14	the	the	DET
ejde-636	623	15	fourth	fourth	ADJ
ejde-636	623	16	-	-	PUNCT
ejde-636	623	17	order	order	NOUN
ejde-636	623	18	schrödinger	schrödinger	NOUN
ejde-636	623	19	equation	equation	NOUN
ejde-636	623	20	in	in	ADP
ejde-636	623	21	low	low	ADJ
ejde-636	623	22	dimensions	dimension	NOUN
ejde-636	623	23	,	,	PUNCT
ejde-636	623	24	nonlinearity	nonlinearity	NOUN
ejde-636	623	25	,	,	PUNCT
ejde-636	623	26	26	26	NUM
ejde-636	623	27	(	(	PUNCT
ejde-636	623	28	2013	2013	NUM
ejde-636	623	29	)	)	PUNCT
ejde-636	623	30	,	,	PUNCT
ejde-636	623	31	2175	2175	NUM
ejde-636	623	32	-	-	SYM
ejde-636	623	33	2191	2191	NUM
ejde-636	623	34	.	.	PUNCT
ejde-636	624	1	[	[	X
ejde-636	624	2	33	33	NUM
ejde-636	624	3	]	]	PUNCT
ejde-636	624	4	t.	t.	PROPN
ejde-636	624	5	v.	v.	PROPN
ejde-636	624	6	phan	phan	PROPN
ejde-636	624	7	;	;	PUNCT
ejde-636	624	8	blowup	blowup	ADJ
ejde-636	624	9	for	for	ADP
ejde-636	624	10	biharmonic	biharmonic	NOUN
ejde-636	624	11	schrödinger	schrödinger	NOUN
ejde-636	624	12	equation	equation	NOUN
ejde-636	624	13	with	with	ADP
ejde-636	624	14	critical	critical	ADJ
ejde-636	624	15	nonlinearity	nonlinearity	NOUN
ejde-636	624	16	,	,	PUNCT
ejde-636	624	17	z.	z.	PROPN
ejde-636	624	18	angew	angew	PROPN
ejde-636	624	19	.	.	PUNCT
ejde-636	625	1	math	math	NOUN
ejde-636	625	2	.	.	PUNCT
ejde-636	626	1	phys	phy	NOUN
ejde-636	626	2	.	.	PUNCT
ejde-636	626	3	,	,	PUNCT
ejde-636	626	4	69	69	NUM
ejde-636	626	5	(	(	PUNCT
ejde-636	626	6	2018	2018	NUM
ejde-636	626	7	)	)	PUNCT
ejde-636	626	8	,	,	PUNCT
ejde-636	626	9	paper	paper	NOUN
ejde-636	626	10	no	no	NOUN
ejde-636	626	11	.	.	PROPN
ejde-636	626	12	31	31	NUM
ejde-636	626	13	,	,	PUNCT
ejde-636	626	14	11pp	11pp	NUM
ejde-636	626	15	.	.	PUNCT
ejde-636	627	1	[	[	X
ejde-636	627	2	34	34	NUM
ejde-636	627	3	]	]	X
ejde-636	627	4	n.	n.	PROPN
ejde-636	627	5	soave	soave	PROPN
ejde-636	627	6	;	;	PUNCT
ejde-636	627	7	normalized	normalize	VERB
ejde-636	627	8	ground	ground	NOUN
ejde-636	627	9	states	state	NOUN
ejde-636	627	10	for	for	ADP
ejde-636	627	11	the	the	DET
ejde-636	627	12	nls	nls	NOUN
ejde-636	627	13	equation	equation	NOUN
ejde-636	627	14	with	with	ADP
ejde-636	627	15	combined	combined	ADJ
ejde-636	627	16	nonlinearities	nonlinearitie	NOUN
ejde-636	627	17	,	,	PUNCT
ejde-636	627	18	j.	j.	PROPN
ejde-636	627	19	differential	differential	PROPN
ejde-636	627	20	equations	equations	PROPN
ejde-636	627	21	,	,	PUNCT
ejde-636	627	22	269	269	NUM
ejde-636	627	23	(	(	PUNCT
ejde-636	627	24	2020	2020	NUM
ejde-636	627	25	)	)	PUNCT
ejde-636	627	26	,	,	PUNCT
ejde-636	627	27	6941	6941	NUM
ejde-636	627	28	-	-	SYM
ejde-636	627	29	6987	6987	NUM
ejde-636	627	30	.	.	PUNCT
ejde-636	628	1	[	[	X
ejde-636	628	2	35	35	NUM
ejde-636	628	3	]	]	X
ejde-636	628	4	n.	n.	PROPN
ejde-636	628	5	soave	soave	PROPN
ejde-636	628	6	;	;	PUNCT
ejde-636	628	7	normalized	normalize	VERB
ejde-636	628	8	ground	ground	NOUN
ejde-636	628	9	states	state	NOUN
ejde-636	628	10	for	for	ADP
ejde-636	628	11	the	the	DET
ejde-636	628	12	nls	nls	NOUN
ejde-636	628	13	equation	equation	NOUN
ejde-636	628	14	with	with	ADP
ejde-636	628	15	combined	combined	ADJ
ejde-636	628	16	nonlinearities	nonlinearitie	NOUN
ejde-636	628	17	:	:	PUNCT
ejde-636	628	18	the	the	DET
ejde-636	628	19	sobolev	sobolev	ADJ
ejde-636	628	20	critical	critical	ADJ
ejde-636	628	21	case	case	NOUN
ejde-636	628	22	,	,	PUNCT
ejde-636	628	23	j.	j.	PROPN
ejde-636	628	24	funct	funct	PROPN
ejde-636	628	25	.	.	PUNCT
ejde-636	629	1	anal	anal	PROPN
ejde-636	629	2	.	.	PROPN
ejde-636	629	3	,	,	PUNCT
ejde-636	629	4	279	279	NUM
ejde-636	629	5	(	(	PUNCT
ejde-636	629	6	2020	2020	NUM
ejde-636	629	7	)	)	PUNCT
ejde-636	629	8	,	,	PUNCT
ejde-636	629	9	108610	108610	NUM
ejde-636	629	10	,	,	PUNCT
ejde-636	629	11	43pp	43pp	NOUN
ejde-636	629	12	.	.	PUNCT
ejde-636	630	1	[	[	X
ejde-636	630	2	36	36	NUM
ejde-636	630	3	]	]	X
ejde-636	630	4	c.	c.	PROPN
ejde-636	630	5	a.	a.	PROPN
ejde-636	630	6	swanson	swanson	PROPN
ejde-636	630	7	;	;	PUNCT
ejde-636	630	8	the	the	DET
ejde-636	630	9	best	good	ADJ
ejde-636	630	10	sobolev	sobolev	NOUN
ejde-636	630	11	constant	constant	ADJ
ejde-636	630	12	,	,	PUNCT
ejde-636	630	13	appl	appl	PROPN
ejde-636	630	14	.	.	PROPN
ejde-636	631	1	anal	anal	PROPN
ejde-636	631	2	.	.	PROPN
ejde-636	631	3	,	,	PUNCT
ejde-636	631	4	47	47	NUM
ejde-636	631	5	(	(	PUNCT
ejde-636	631	6	1992	1992	NUM
ejde-636	631	7	)	)	PUNCT
ejde-636	631	8	,	,	PUNCT
ejde-636	631	9	227	227	NUM
ejde-636	631	10	-	-	SYM
ejde-636	631	11	239	239	NUM
ejde-636	631	12	.	.	PUNCT
ejde-636	632	1	[	[	X
ejde-636	632	2	37	37	NUM
ejde-636	632	3	]	]	PUNCT
ejde-636	632	4	s.	s.	PROPN
ejde-636	632	5	k.	k.	PROPN
ejde-636	632	6	turitsyn	turitsyn	PROPN
ejde-636	632	7	;	;	PUNCT
ejde-636	632	8	three	three	NUM
ejde-636	632	9	-	-	PUNCT
ejde-636	632	10	dimensional	dimensional	ADJ
ejde-636	632	11	dispersion	dispersion	NOUN
ejde-636	632	12	of	of	ADP
ejde-636	632	13	nonlinearity	nonlinearity	NOUN
ejde-636	632	14	and	and	CCONJ
ejde-636	632	15	stability	stability	NOUN
ejde-636	632	16	of	of	ADP
ejde-636	632	17	multidimensional	multidimensional	ADJ
ejde-636	632	18	solitons	soliton	NOUN
ejde-636	632	19	,	,	PUNCT
ejde-636	632	20	teoret	teoret	NOUN
ejde-636	632	21	.	.	PUNCT
ejde-636	633	1	mat	mat	PROPN
ejde-636	633	2	.	.	PUNCT
ejde-636	633	3	fiz	fiz	PROPN
ejde-636	633	4	.	.	PROPN
ejde-636	633	5	,	,	PUNCT
ejde-636	633	6	64	64	NUM
ejde-636	633	7	(	(	PUNCT
ejde-636	633	8	1985	1985	NUM
ejde-636	633	9	)	)	PUNCT
ejde-636	633	10	,	,	PUNCT
ejde-636	633	11	226	226	NUM
ejde-636	633	12	-	-	SYM
ejde-636	633	13	232	232	NUM
ejde-636	633	14	.	.	PUNCT
ejde-636	634	1	(	(	PUNCT
ejde-636	634	2	in	in	ADP
ejde-636	634	3	russian	russian	NOUN
ejde-636	634	4	)	)	PUNCT
ejde-636	635	1	[	[	X
ejde-636	635	2	38	38	NUM
ejde-636	635	3	]	]	PUNCT
ejde-636	635	4	j.	j.	PROPN
ejde-636	635	5	wei	wei	PROPN
ejde-636	635	6	,	,	PUNCT
ejde-636	635	7	y.	y.	PROPN
ejde-636	635	8	wu	wu	PROPN
ejde-636	635	9	;	;	PUNCT
ejde-636	635	10	normalized	normalize	VERB
ejde-636	635	11	solutions	solution	NOUN
ejde-636	635	12	for	for	ADP
ejde-636	635	13	schrödinger	schrödinger	NOUN
ejde-636	635	14	equations	equation	NOUN
ejde-636	635	15	with	with	ADP
ejde-636	635	16	critical	critical	ADJ
ejde-636	635	17	sobolev	sobolev	NOUN
ejde-636	635	18	exponent	exponent	NOUN
ejde-636	635	19	and	and	CCONJ
ejde-636	635	20	mixed	mixed	ADJ
ejde-636	635	21	nonlinearities	nonlinearitie	NOUN
ejde-636	635	22	,	,	PUNCT
ejde-636	635	23	j.	j.	PROPN
ejde-636	635	24	funct	funct	PROPN
ejde-636	635	25	.	.	PUNCT
ejde-636	636	1	anal	anal	PROPN
ejde-636	636	2	.	.	PROPN
ejde-636	636	3	,	,	PUNCT
ejde-636	636	4	283	283	NUM
ejde-636	636	5	(	(	PUNCT
ejde-636	636	6	2022	2022	NUM
ejde-636	636	7	)	)	PUNCT
ejde-636	636	8	,	,	PUNCT
ejde-636	636	9	paper	paper	NOUN
ejde-636	636	10	no	no	NOUN
ejde-636	636	11	.	.	PROPN
ejde-636	637	1	109574	109574	NUM
ejde-636	637	2	,	,	PUNCT
ejde-636	637	3	46pp	46pp	ADJ
ejde-636	637	4	.	.	PUNCT
ejde-636	638	1	[	[	X
ejde-636	638	2	39	39	NUM
ejde-636	638	3	]	]	PUNCT
ejde-636	638	4	m.	m.	NOUN
ejde-636	638	5	willem	willem	PROPN
ejde-636	638	6	;	;	PUNCT
ejde-636	638	7	minimax	minimax	NOUN
ejde-636	638	8	theorems	theorem	NOUN
ejde-636	638	9	,	,	PUNCT
ejde-636	638	10	progr	progr	NOUN
ejde-636	638	11	.	.	PUNCT
ejde-636	639	1	nonlinear	nonlinear	ADJ
ejde-636	639	2	differential	differential	ADJ
ejde-636	639	3	equations	equation	NOUN
ejde-636	639	4	appl	appl	PROPN
ejde-636	639	5	.	.	PROPN
ejde-636	639	6	,	,	PUNCT
ejde-636	639	7	24	24	NUM
ejde-636	639	8	.	.	PUNCT
ejde-636	639	9	birkhäuser	birkhäuser	X
ejde-636	639	10	boston	boston	PROPN
ejde-636	639	11	,	,	PUNCT
ejde-636	639	12	inc	inc	PROPN
ejde-636	639	13	.	.	PROPN
ejde-636	639	14	,	,	PUNCT
ejde-636	639	15	boston	boston	PROPN
ejde-636	639	16	,	,	PUNCT
ejde-636	639	17	ma	ma	PROPN
ejde-636	639	18	,	,	PUNCT
ejde-636	639	19	1996	1996	NUM
ejde-636	639	20	.	.	PUNCT
ejde-636	640	1	zhouji	zhouji	PROPN
ejde-636	640	2	ma	ma	PROPN
ejde-636	640	3	school	school	PROPN
ejde-636	640	4	of	of	ADP
ejde-636	640	5	mathematics	mathematic	NOUN
ejde-636	640	6	and	and	CCONJ
ejde-636	640	7	statistics	statistic	NOUN
ejde-636	640	8	,	,	PUNCT
ejde-636	640	9	northeast	northeast	ADJ
ejde-636	640	10	normal	normal	ADJ
ejde-636	640	11	university	university	NOUN
ejde-636	640	12	,	,	PUNCT
ejde-636	640	13	changchun	changchun	PROPN
ejde-636	640	14	,	,	PUNCT
ejde-636	640	15	jilin	jilin	PROPN
ejde-636	640	16	130024	130024	NUM
ejde-636	640	17	,	,	PUNCT
ejde-636	640	18	china	china	PROPN
ejde-636	640	19	email	email	NOUN
ejde-636	640	20	address	address	NOUN
ejde-636	640	21	:	:	PUNCT
ejde-636	640	22	mazj588@nenu.edu.cn	mazj588@nenu.edu.cn	X
ejde-636	641	1	xiaojun	xiaojun	PROPN
ejde-636	641	2	chang	chang	PROPN
ejde-636	641	3	(	(	PUNCT
ejde-636	641	4	corresponding	corresponding	ADJ
ejde-636	641	5	author	author	NOUN
ejde-636	641	6	)	)	PUNCT
ejde-636	641	7	school	school	NOUN
ejde-636	641	8	of	of	ADP
ejde-636	641	9	mathematics	mathematic	NOUN
ejde-636	641	10	and	and	CCONJ
ejde-636	641	11	statistics	statistic	NOUN
ejde-636	641	12	&	&	CCONJ
ejde-636	641	13	center	center	PROPN
ejde-636	641	14	for	for	ADP
ejde-636	641	15	mathematics	mathematic	NOUN
ejde-636	641	16	and	and	CCONJ
ejde-636	641	17	interdisciplinary	interdisciplinary	ADJ
ejde-636	641	18	sciences	science	NOUN
ejde-636	641	19	,	,	PUNCT
ejde-636	641	20	northeast	northeast	ADJ
ejde-636	641	21	normal	normal	ADJ
ejde-636	641	22	university	university	NOUN
ejde-636	641	23	,	,	PUNCT
ejde-636	641	24	changchun	changchun	PROPN
ejde-636	641	25	,	,	PUNCT
ejde-636	641	26	jilin	jilin	PROPN
ejde-636	641	27	130024	130024	NUM
ejde-636	641	28	,	,	PUNCT
ejde-636	641	29	china	china	PROPN
ejde-636	641	30	email	email	NOUN
ejde-636	641	31	address	address	NOUN
ejde-636	641	32	:	:	PUNCT
ejde-636	641	33	changxj100@nenu.edu.cn	changxj100@nenu.edu.cn	X
ejde-636	641	34	zhaosheng	zhaosheng	PROPN
ejde-636	641	35	feng	feng	PROPN
ejde-636	641	36	school	school	PROPN
ejde-636	641	37	of	of	ADP
ejde-636	641	38	mathematical	mathematical	ADJ
ejde-636	641	39	and	and	CCONJ
ejde-636	641	40	statistical	statistical	ADJ
ejde-636	641	41	sciences	science	NOUN
ejde-636	641	42	,	,	PUNCT
ejde-636	641	43	university	university	PROPN
ejde-636	641	44	of	of	ADP
ejde-636	641	45	texas	texas	PROPN
ejde-636	641	46	rio	rio	PROPN
ejde-636	641	47	grande	grande	PROPN
ejde-636	641	48	valley	valley	PROPN
ejde-636	641	49	,	,	PUNCT
ejde-636	641	50	edinburg	edinburg	PROPN
ejde-636	641	51	,	,	PUNCT
ejde-636	641	52	tx	tx	PROPN
ejde-636	641	53	78539	78539	NUM
ejde-636	641	54	,	,	PUNCT
ejde-636	641	55	usa	usa	PROPN
ejde-636	641	56	email	email	NOUN
ejde-636	641	57	address	address	NOUN
ejde-636	641	58	:	:	PUNCT
ejde-636	641	59	zhaosheng.feng@utrgv.edu	zhaosheng.feng@utrgv.edu	PROPN
ejde-636	641	60	1	1	NUM
ejde-636	641	61	.	.	PUNCT
ejde-636	642	1	introduction	introduction	NOUN
ejde-636	642	2	and	and	CCONJ
ejde-636	642	3	main	main	ADJ
ejde-636	642	4	results	result	NOUN
ejde-636	642	5	2	2	NUM
ejde-636	642	6	.	.	PUNCT
ejde-636	642	7	preliminary	preliminary	ADJ
ejde-636	642	8	results	result	NOUN
ejde-636	642	9	3	3	NUM
ejde-636	642	10	.	.	X
ejde-636	642	11	case	case	NOUN
ejde-636	642	12	2	2	NUM
ejde-636	642	13	<	<	NOUN
ejde-636	642	14	q<2	q<2	X
ejde-636	642	15	+	+	NOUN
ejde-636	642	16	4n	4n	NOUN
ejde-636	642	17	<	<	X
ejde-636	642	18	p	p	X
ejde-636	642	19	<	<	X
ejde-636	642	20	p4	p4	ADJ
ejde-636	642	21	*	*	ADJ
ejde-636	642	22	3.1	3.1	NUM
ejde-636	642	23	.	.	PUNCT
ejde-636	643	1	properties	property	NOUN
ejde-636	643	2	of	of	ADP
ejde-636	643	3	mapping	map	VERB
ejde-636	643	4	cmp	cmp	NOUN
ejde-636	643	5	,	,	PUNCT
ejde-636	643	6	q(c	q(c	PROPN
ejde-636	643	7	)	)	PUNCT
ejde-636	643	8	3.2	3.2	NUM
ejde-636	643	9	.	.	PUNCT
ejde-636	644	1	proof	proof	NOUN
ejde-636	644	2	of	of	ADP
ejde-636	644	3	theorem	theorem	ADJ
ejde-636	644	4	1.2	1.2	NUM
ejde-636	644	5	4	4	NUM
ejde-636	644	6	.	.	PUNCT
ejde-636	645	1	case	case	NOUN
ejde-636	645	2	pq	pq	PROPN
ejde-636	645	3	<	<	X
ejde-636	645	4	p<4	p<4	NOUN
ejde-636	645	5	*	*	PROPN
ejde-636	645	6	4.1	4.1	NUM
ejde-636	645	7	.	.	PUNCT
ejde-636	646	1	monotonicity	monotonicity	NOUN
ejde-636	646	2	of	of	ADP
ejde-636	646	3	ground	ground	NOUN
ejde-636	646	4	state	state	NOUN
ejde-636	646	5	energy	energy	PROPN
ejde-636	646	6	mp	mp	PROPN
ejde-636	646	7	,	,	PUNCT
ejde-636	646	8	q(c	q(c	PROPN
ejde-636	646	9	)	)	PUNCT
ejde-636	646	10	4.2	4.2	NUM
ejde-636	646	11	.	.	PUNCT
ejde-636	647	1	ground	ground	NOUN
ejde-636	647	2	states	state	NOUN
ejde-636	647	3	acknowledgments	acknowledgment	NOUN
ejde-636	647	4	references	reference	NOUN