id	sid	tid	token	lemma	pos
ejde-984	1	1	electronic	electronic	ADJ
ejde-984	1	2	journal	journal	NOUN
ejde-984	1	3	of	of	ADP
ejde-984	1	4	differential	differential	ADJ
ejde-984	1	5	equations	equation	NOUN
ejde-984	1	6	,	,	PUNCT
ejde-984	1	7	vol	vol	NOUN
ejde-984	1	8	.	.	PUNCT
ejde-984	1	9	2025	2025	NUM
ejde-984	1	10	(	(	PUNCT
ejde-984	1	11	2025	2025	NUM
ejde-984	1	12	)	)	PUNCT
ejde-984	1	13	,	,	PUNCT
ejde-984	1	14	no	no	INTJ
ejde-984	1	15	.	.	NOUN
ejde-984	1	16	75	75	NUM
ejde-984	1	17	,	,	PUNCT
ejde-984	1	18	pp	pp	PROPN
ejde-984	1	19	.	.	PUNCT
ejde-984	2	1	1–16	1–16	PROPN
ejde-984	2	2	.	.	PUNCT
ejde-984	3	1	issn	issn	PROPN
ejde-984	3	2	:	:	PUNCT
ejde-984	3	3	1072	1072	NUM
ejde-984	3	4	-	-	SYM
ejde-984	3	5	6691	6691	NUM
ejde-984	3	6	.	.	PUNCT
ejde-984	4	1	url	url	PROPN
ejde-984	4	2	:	:	PUNCT
ejde-984	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-984	4	4	,	,	PUNCT
ejde-984	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-984	4	6	doi	doi	PROPN
ejde-984	4	7	:	:	PUNCT
ejde-984	4	8	10.58997	10.58997	NUM
ejde-984	4	9	/	/	SYM
ejde-984	4	10	ejde.2025.75	ejde.2025.75	ADV
ejde-984	4	11	normalized	normalize	VERB
ejde-984	4	12	solutions	solution	NOUN
ejde-984	4	13	of	of	ADP
ejde-984	4	14	fractional	fractional	PROPN
ejde-984	4	15	kirchhoff	kirchhoff	NOUN
ejde-984	4	16	equations	equation	NOUN
ejde-984	4	17	in	in	ADP
ejde-984	4	18	the	the	DET
ejde-984	4	19	defocusing	defocuse	VERB
ejde-984	4	20	case	case	NOUN
ejde-984	4	21	zhenyu	zhenyu	PROPN
ejde-984	4	22	guo	guo	PROPN
ejde-984	4	23	,	,	PUNCT
ejde-984	4	24	tianqing	tianqe	VERB
ejde-984	4	25	zhang	zhang	PROPN
ejde-984	4	26	abstract	abstract	NOUN
ejde-984	4	27	.	.	PUNCT
ejde-984	5	1	in	in	ADP
ejde-984	5	2	this	this	DET
ejde-984	5	3	article	article	NOUN
ejde-984	5	4	,	,	PUNCT
ejde-984	5	5	we	we	PRON
ejde-984	5	6	focus	focus	VERB
ejde-984	5	7	on	on	ADP
ejde-984	5	8	the	the	DET
ejde-984	5	9	normalized	normalize	VERB
ejde-984	5	10	solutions	solution	NOUN
ejde-984	5	11	to	to	ADP
ejde-984	5	12	the	the	DET
ejde-984	5	13	fractional	fractional	PROPN
ejde-984	5	14	kirchhoff	kirchhoff	NOUN
ejde-984	5	15	equations	equation	NOUN
ejde-984	5	16	with	with	ADP
ejde-984	5	17	subcritical	subcritical	ADJ
ejde-984	5	18	nonlinearities	nonlinearitie	NOUN
ejde-984	5	19	in	in	ADP
ejde-984	5	20	the	the	DET
ejde-984	5	21	defocusing	defocuse	VERB
ejde-984	5	22	case	case	NOUN
ejde-984	5	23	.	.	PUNCT
ejde-984	6	1	by	by	ADP
ejde-984	6	2	applying	apply	VERB
ejde-984	6	3	distinct	distinct	ADJ
ejde-984	6	4	suppositions	supposition	NOUN
ejde-984	6	5	to	to	ADP
ejde-984	6	6	the	the	DET
ejde-984	6	7	coefficients	coefficient	NOUN
ejde-984	6	8	of	of	ADP
ejde-984	6	9	nonlinearities	nonlinearitie	NOUN
ejde-984	6	10	,	,	PUNCT
ejde-984	6	11	namely	namely	ADV
ejde-984	6	12	q	q	X
ejde-984	6	13	<	<	X
ejde-984	6	14	p	p	X
ejde-984	6	15	,	,	PUNCT
ejde-984	6	16	we	we	PRON
ejde-984	6	17	prove	prove	VERB
ejde-984	6	18	the	the	DET
ejde-984	6	19	existence	existence	NOUN
ejde-984	6	20	and	and	CCONJ
ejde-984	6	21	nonexistence	nonexistence	NOUN
ejde-984	6	22	of	of	ADP
ejde-984	6	23	normalized	normalize	VERB
ejde-984	6	24	solutions	solution	NOUN
ejde-984	6	25	.	.	PUNCT
ejde-984	7	1	also	also	ADV
ejde-984	7	2	we	we	PRON
ejde-984	7	3	obtain	obtain	VERB
ejde-984	7	4	new	new	ADJ
ejde-984	7	5	results	result	NOUN
ejde-984	7	6	on	on	ADP
ejde-984	7	7	the	the	DET
ejde-984	7	8	characterization	characterization	NOUN
ejde-984	7	9	of	of	ADP
ejde-984	7	10	ground	ground	NOUN
ejde-984	7	11	states	state	NOUN
ejde-984	7	12	of	of	ADP
ejde-984	7	13	the	the	DET
ejde-984	7	14	fractional	fractional	PROPN
ejde-984	7	15	kirchhoff	kirchhoff	PROPN
ejde-984	7	16	equations	equation	NOUN
ejde-984	7	17	.	.	PUNCT
ejde-984	8	1	1	1	X
ejde-984	8	2	.	.	X
ejde-984	8	3	introduction	introduction	NOUN
ejde-984	8	4	the	the	DET
ejde-984	8	5	aim	aim	NOUN
ejde-984	8	6	of	of	ADP
ejde-984	8	7	article	article	NOUN
ejde-984	8	8	is	be	AUX
ejde-984	8	9	to	to	PART
ejde-984	8	10	study	study	VERB
ejde-984	8	11	the	the	DET
ejde-984	8	12	fractional	fractional	PROPN
ejde-984	8	13	kirchhoff	kirchhoff	PROPN
ejde-984	8	14	equation	equation	NOUN
ejde-984	8	15	(	(	PUNCT
ejde-984	8	16	a+	a+	PRON
ejde-984	8	17	b	b	PROPN
ejde-984	8	18	∫	∫	PROPN
ejde-984	8	19	r3	r3	PROPN
ejde-984	8	20	|(−∆)s/2u|2dx	|(−∆)s/2u|2dx	NUM
ejde-984	8	21	)	)	PUNCT
ejde-984	8	22	(	(	PUNCT
ejde-984	8	23	−∆)su	−∆)su	NOUN
ejde-984	8	24	=	=	PUNCT
ejde-984	8	25	λu+	λu+	INTJ
ejde-984	8	26	µ|u|q−2u+	µ|u|q−2u+	PROPN
ejde-984	8	27	|u|p−2u	|u|p−2u	PROPN
ejde-984	8	28	in	in	ADP
ejde-984	8	29	r3	r3	PROPN
ejde-984	8	30	(	(	PUNCT
ejde-984	8	31	1.1	1.1	NUM
ejde-984	8	32	)	)	PUNCT
ejde-984	8	33	with	with	ADP
ejde-984	8	34	a	a	DET
ejde-984	8	35	prescribed	prescribe	VERB
ejde-984	8	36	mass	mass	NOUN
ejde-984	8	37	∫	∫	PROPN
ejde-984	8	38	r3	r3	PROPN
ejde-984	8	39	u2dx	u2dx	PROPN
ejde-984	8	40	=	=	PROPN
ejde-984	8	41	c2	c2	PROPN
ejde-984	8	42	.	.	PUNCT
ejde-984	9	1	(	(	PUNCT
ejde-984	9	2	1.2	1.2	NUM
ejde-984	9	3	)	)	PUNCT
ejde-984	9	4	in	in	ADP
ejde-984	9	5	this	this	DET
ejde-984	9	6	article	article	NOUN
ejde-984	9	7	,	,	PUNCT
ejde-984	9	8	except	except	SCONJ
ejde-984	9	9	for	for	ADP
ejde-984	9	10	additional	additional	ADJ
ejde-984	9	11	statements	statement	NOUN
ejde-984	9	12	,	,	PUNCT
ejde-984	9	13	we	we	PRON
ejde-984	9	14	assume	assume	VERB
ejde-984	9	15	that	that	SCONJ
ejde-984	9	16	a	a	DET
ejde-984	9	17	,	,	PUNCT
ejde-984	9	18	b	b	NOUN
ejde-984	9	19	,	,	PUNCT
ejde-984	9	20	c	c	NOUN
ejde-984	9	21	>	>	X
ejde-984	9	22	0	0	NUM
ejde-984	9	23	,	,	PUNCT
ejde-984	9	24	0	0	PUNCT
ejde-984	9	25	<	<	X
ejde-984	9	26	s	s	X
ejde-984	9	27	<	<	X
ejde-984	9	28	1	1	NUM
ejde-984	9	29	,	,	PUNCT
ejde-984	9	30	2	2	NUM
ejde-984	9	31	<	<	X
ejde-984	9	32	q	q	X
ejde-984	9	33	<	<	X
ejde-984	9	34	p	p	X
ejde-984	9	35	≤	≤	ADJ
ejde-984	9	36	2∗s	2∗s	NUM
ejde-984	9	37	=	=	SYM
ejde-984	9	38	6	6	NUM
ejde-984	9	39	3−2s	3−2s	NUM
ejde-984	9	40	(	(	PUNCT
ejde-984	9	41	2∗s	2∗s	NUM
ejde-984	9	42	is	be	AUX
ejde-984	9	43	the	the	DET
ejde-984	9	44	sobolev	sobolev	ADJ
ejde-984	9	45	critical	critical	ADJ
ejde-984	9	46	exponent	exponent	NOUN
ejde-984	9	47	)	)	PUNCT
ejde-984	9	48	,	,	PUNCT
ejde-984	9	49	n	n	NOUN
ejde-984	9	50	=	=	SYM
ejde-984	9	51	3	3	NUM
ejde-984	9	52	,	,	PUNCT
ejde-984	9	53	λ	λ	PROPN
ejde-984	9	54	∈	∈	NOUN
ejde-984	9	55	r	r	NOUN
ejde-984	9	56	is	be	AUX
ejde-984	9	57	a	a	DET
ejde-984	9	58	lagrange	lagrange	NOUN
ejde-984	9	59	multiplier	multipli	ADJ
ejde-984	9	60	and	and	CCONJ
ejde-984	9	61	µ	µ	X
ejde-984	9	62	<	<	X
ejde-984	9	63	0	0	NUM
ejde-984	9	64	is	be	AUX
ejde-984	9	65	a	a	DET
ejde-984	9	66	parameter	parameter	NOUN
ejde-984	9	67	.	.	PUNCT
ejde-984	10	1	for	for	ADP
ejde-984	10	2	equation	equation	NOUN
ejde-984	10	3	(	(	PUNCT
ejde-984	10	4	1.1	1.1	NUM
ejde-984	10	5	)	)	PUNCT
ejde-984	10	6	,	,	PUNCT
ejde-984	10	7	we	we	PRON
ejde-984	10	8	name	name	VERB
ejde-984	10	9	the	the	DET
ejde-984	10	10	focusing	focus	VERB
ejde-984	10	11	case	case	NOUN
ejde-984	10	12	when	when	SCONJ
ejde-984	10	13	µ	µ	X
ejde-984	10	14	>	>	X
ejde-984	10	15	0	0	NUM
ejde-984	10	16	,	,	PUNCT
ejde-984	10	17	and	and	CCONJ
ejde-984	10	18	defocusing	defocuse	VERB
ejde-984	10	19	case	case	NOUN
ejde-984	10	20	when	when	SCONJ
ejde-984	10	21	µ	µ	X
ejde-984	10	22	<	<	X
ejde-984	10	23	0	0	NUM
ejde-984	10	24	.	.	PUNCT
ejde-984	11	1	in	in	ADP
ejde-984	11	2	equation	equation	NOUN
ejde-984	11	3	(	(	PUNCT
ejde-984	11	4	1.1	1.1	NUM
ejde-984	11	5	)	)	PUNCT
ejde-984	11	6	,	,	PUNCT
ejde-984	11	7	(	(	PUNCT
ejde-984	11	8	−∆)s	−∆)s	X
ejde-984	11	9	with	with	ADP
ejde-984	11	10	s	s	X
ejde-984	11	11	∈	∈	PROPN
ejde-984	11	12	(	(	PUNCT
ejde-984	11	13	0	0	NUM
ejde-984	11	14	,	,	PUNCT
ejde-984	11	15	1	1	NUM
ejde-984	11	16	)	)	PUNCT
ejde-984	11	17	)	)	PUNCT
ejde-984	11	18	,	,	PUNCT
ejde-984	11	19	namely	namely	ADV
ejde-984	11	20	the	the	DET
ejde-984	11	21	fractional	fractional	PROPN
ejde-984	11	22	laplacian	laplacian	NOUN
ejde-984	11	23	,	,	PUNCT
ejde-984	11	24	is	be	AUX
ejde-984	11	25	generally	generally	ADV
ejde-984	11	26	specified	specify	VERB
ejde-984	11	27	as	as	ADP
ejde-984	11	28	(	(	PUNCT
ejde-984	11	29	−∆)sv(x	−∆)sv(x	NUM
ejde-984	11	30	)	)	PUNCT
ejde-984	12	1	=	=	SYM
ejde-984	12	2	csp.v	csp.v	PROPN
ejde-984	12	3	.	.	PUNCT
ejde-984	12	4	∫	∫	PROPN
ejde-984	12	5	r3	r3	PROPN
ejde-984	12	6	v(x)−	v(x)−	PROPN
ejde-984	12	7	v(y	v(y	PROPN
ejde-984	12	8	)	)	PUNCT
ejde-984	12	9	|x−	|x−	PART
ejde-984	13	1	y|3	y|3	PROPN
ejde-984	13	2	+	+	PROPN
ejde-984	13	3	2s	2s	X
ejde-984	13	4	dy	dy	NOUN
ejde-984	13	5	=	=	SYM
ejde-984	13	6	cs	cs	PROPN
ejde-984	13	7	lim	lim	PROPN
ejde-984	13	8	ϵ→0	ϵ→0	PROPN
ejde-984	13	9	+	+	CCONJ
ejde-984	13	10	∫	∫	PROPN
ejde-984	13	11	r3\bϵ(x	r3\bϵ(x	X
ejde-984	13	12	)	)	PUNCT
ejde-984	13	13	v(x)−	v(x)−	PROPN
ejde-984	13	14	v(y	v(y	PROPN
ejde-984	13	15	)	)	PUNCT
ejde-984	13	16	|x−	|x−	PART
ejde-984	14	1	y|3	y|3	PROPN
ejde-984	14	2	+	+	PROPN
ejde-984	14	3	2s	2s	X
ejde-984	14	4	dy	dy	NOUN
ejde-984	14	5	=	=	NOUN
ejde-984	14	6	−1	−1	NOUN
ejde-984	14	7	2	2	NUM
ejde-984	14	8	cs	cs	PROPN
ejde-984	14	9	∫	∫	PROPN
ejde-984	14	10	r3	r3	PROPN
ejde-984	14	11	v(x+	v(x+	PROPN
ejde-984	14	12	y	y	PROPN
ejde-984	14	13	)	)	PUNCT
ejde-984	14	14	+	+	CCONJ
ejde-984	14	15	v(x−	v(x−	NOUN
ejde-984	14	16	y)−	y)−	PROPN
ejde-984	14	17	2v(x	2v(x	NUM
ejde-984	14	18	)	)	PUNCT
ejde-984	14	19	|y|3	|y|3	PROPN
ejde-984	14	20	+	+	PROPN
ejde-984	14	21	2s	2s	X
ejde-984	14	22	dy	dy	NOUN
ejde-984	14	23	(	(	PUNCT
ejde-984	14	24	1.3	1.3	NUM
ejde-984	14	25	)	)	PUNCT
ejde-984	14	26	for	for	ADP
ejde-984	14	27	v	v	NOUN
ejde-984	14	28	∈	∈	PROPN
ejde-984	14	29	s(r3	s(r3	NOUN
ejde-984	14	30	)	)	PUNCT
ejde-984	14	31	,	,	PUNCT
ejde-984	14	32	where	where	SCONJ
ejde-984	14	33	s(r3	s(r3	NOUN
ejde-984	14	34	)	)	PUNCT
ejde-984	14	35	denotes	denote	VERB
ejde-984	14	36	the	the	DET
ejde-984	14	37	schwartz	schwartz	PROPN
ejde-984	14	38	space	space	NOUN
ejde-984	14	39	of	of	ADP
ejde-984	14	40	rapidly	rapidly	ADV
ejde-984	14	41	decaying	decay	VERB
ejde-984	14	42	c∞	c∞	PROPN
ejde-984	14	43	function	function	NOUN
ejde-984	14	44	,	,	PUNCT
ejde-984	14	45	bϵ(x	bϵ(x	NUM
ejde-984	14	46	)	)	PUNCT
ejde-984	14	47	represents	represent	VERB
ejde-984	14	48	an	an	DET
ejde-984	14	49	open	open	ADJ
ejde-984	14	50	ball	ball	NOUN
ejde-984	14	51	of	of	ADP
ejde-984	14	52	radius	radius	NOUN
ejde-984	14	53	ϵ	ϵ	PROPN
ejde-984	14	54	centered	center	VERB
ejde-984	14	55	at	at	ADP
ejde-984	14	56	x	x	X
ejde-984	14	57	,	,	PUNCT
ejde-984	14	58	p.v	p.v	PROPN
ejde-984	14	59	.	.	PROPN
ejde-984	14	60	is	be	AUX
ejde-984	14	61	the	the	DET
ejde-984	14	62	principle	principle	ADJ
ejde-984	14	63	value	value	NOUN
ejde-984	14	64	,	,	PUNCT
ejde-984	14	65	which	which	PRON
ejde-984	14	66	is	be	AUX
ejde-984	14	67	defined	define	VERB
ejde-984	14	68	by	by	ADP
ejde-984	14	69	the	the	DET
ejde-984	14	70	latter	latter	ADJ
ejde-984	14	71	expression	expression	NOUN
ejde-984	14	72	in	in	ADP
ejde-984	14	73	(	(	PUNCT
ejde-984	14	74	1.3	1.3	NUM
ejde-984	14	75	)	)	PUNCT
ejde-984	14	76	,	,	PUNCT
ejde-984	14	77	cs	cs	PROPN
ejde-984	14	78	=	=	SYM
ejde-984	14	79	(	(	PUNCT
ejde-984	14	80	∫	∫	PROPN
ejde-984	14	81	r3	r3	PROPN
ejde-984	14	82	1−	1−	NUM
ejde-984	14	83	cos(ξ1	cos(ξ1	NOUN
ejde-984	14	84	)	)	PUNCT
ejde-984	14	85	|ξ|3	|ξ|3	PROPN
ejde-984	14	86	+	+	PROPN
ejde-984	14	87	2s	2s	PROPN
ejde-984	14	88	dξ	dξ	PROPN
ejde-984	14	89	)	)	PUNCT
ejde-984	14	90	−1	−1	NOUN
ejde-984	14	91	.	.	PUNCT
ejde-984	15	1	for	for	ADP
ejde-984	15	2	u	u	PROPN
ejde-984	15	3	∈	∈	PROPN
ejde-984	15	4	s(r3	s(r3	NOUN
ejde-984	15	5	)	)	PUNCT
ejde-984	15	6	,	,	PUNCT
ejde-984	15	7	the	the	DET
ejde-984	15	8	fractional	fractional	ADJ
ejde-984	15	9	laplacian	laplacian	NOUN
ejde-984	15	10	(	(	PUNCT
ejde-984	15	11	−∆)s	−∆)s	PRON
ejde-984	15	12	can	can	AUX
ejde-984	15	13	be	be	AUX
ejde-984	15	14	regulated	regulate	VERB
ejde-984	15	15	by	by	ADP
ejde-984	15	16	the	the	DET
ejde-984	15	17	fourier	fourier	NOUN
ejde-984	15	18	transform	transform	NOUN
ejde-984	15	19	(	(	PUNCT
ejde-984	15	20	−∆)su	−∆)su	NOUN
ejde-984	15	21	=	=	SYM
ejde-984	15	22	f−1(|ξ|2sfu	f−1(|ξ|2sfu	PROPN
ejde-984	15	23	)	)	PUNCT
ejde-984	15	24	,	,	PUNCT
ejde-984	15	25	f	f	PROPN
ejde-984	15	26	denotes	denote	VERB
ejde-984	15	27	the	the	DET
ejde-984	15	28	usual	usual	ADJ
ejde-984	15	29	fourier	fourier	NOUN
ejde-984	15	30	transform	transform	NOUN
ejde-984	15	31	.	.	PUNCT
ejde-984	16	1	2020	2020	NUM
ejde-984	16	2	mathematics	mathematic	NOUN
ejde-984	16	3	subject	subject	ADJ
ejde-984	16	4	classification	classification	NOUN
ejde-984	16	5	.	.	PUNCT
ejde-984	17	1	35a01	35a01	NOUN
ejde-984	17	2	,	,	PUNCT
ejde-984	17	3	35r11	35r11	NUM
ejde-984	17	4	.	.	PUNCT
ejde-984	18	1	key	key	ADJ
ejde-984	18	2	words	word	NOUN
ejde-984	18	3	and	and	CCONJ
ejde-984	18	4	phrases	phrase	NOUN
ejde-984	18	5	.	.	PUNCT
ejde-984	19	1	fractional	fractional	PROPN
ejde-984	19	2	kirchhoff	kirchhoff	PROPN
ejde-984	19	3	equations	equation	NOUN
ejde-984	19	4	;	;	PUNCT
ejde-984	19	5	defocusing	defocuse	VERB
ejde-984	19	6	case	case	NOUN
ejde-984	19	7	;	;	PUNCT
ejde-984	19	8	subritical	subritical	ADJ
ejde-984	19	9	nonlinear	nonlinear	ADJ
ejde-984	19	10	terms	term	NOUN
ejde-984	19	11	;	;	PUNCT
ejde-984	19	12	normalized	normalize	VERB
ejde-984	19	13	solutions	solution	NOUN
ejde-984	19	14	.	.	PUNCT
ejde-984	20	1	©	©	PROPN
ejde-984	20	2	2025	2025	NUM
ejde-984	20	3	.	.	PUNCT
ejde-984	21	1	this	this	DET
ejde-984	21	2	work	work	NOUN
ejde-984	21	3	is	be	AUX
ejde-984	21	4	licensed	license	VERB
ejde-984	21	5	under	under	ADP
ejde-984	21	6	a	a	DET
ejde-984	21	7	cc	cc	NOUN
ejde-984	21	8	by	by	ADP
ejde-984	21	9	4.0	4.0	NUM
ejde-984	21	10	license	license	NOUN
ejde-984	21	11	.	.	PUNCT
ejde-984	22	1	submitted	submit	VERB
ejde-984	22	2	october	october	PROPN
ejde-984	22	3	2	2	NUM
ejde-984	22	4	,	,	PUNCT
ejde-984	22	5	2024	2024	NUM
ejde-984	22	6	.	.	PUNCT
ejde-984	23	1	published	publish	VERB
ejde-984	23	2	july	july	PROPN
ejde-984	23	3	16	16	NUM
ejde-984	23	4	,	,	PUNCT
ejde-984	23	5	2025	2025	NUM
ejde-984	23	6	.	.	PUNCT
ejde-984	24	1	1	1	NUM
ejde-984	24	2	2	2	NUM
ejde-984	24	3	z.	z.	X
ejde-984	24	4	guo	guo	PROPN
ejde-984	24	5	,	,	PUNCT
ejde-984	24	6	t.	t.	PROPN
ejde-984	24	7	zhang	zhang	PROPN
ejde-984	24	8	ejde-2025/75	ejde-2025/75	ADJ
ejde-984	24	9	when	when	SCONJ
ejde-984	24	10	setting	set	VERB
ejde-984	24	11	a	a	DET
ejde-984	24	12	=	=	SYM
ejde-984	24	13	1	1	NUM
ejde-984	24	14	,	,	PUNCT
ejde-984	24	15	s	s	PART
ejde-984	24	16	=	=	SYM
ejde-984	24	17	1	1	NUM
ejde-984	24	18	and	and	CCONJ
ejde-984	24	19	b	b	X
ejde-984	24	20	=	=	SYM
ejde-984	24	21	0	0	PUNCT
ejde-984	25	1	(	(	PUNCT
ejde-984	25	2	that	that	PRON
ejde-984	25	3	is	be	AUX
ejde-984	25	4	to	to	PART
ejde-984	25	5	say	say	VERB
ejde-984	25	6	,	,	PUNCT
ejde-984	25	7	(	(	PUNCT
ejde-984	25	8	1.1	1.1	NUM
ejde-984	25	9	)	)	PUNCT
ejde-984	25	10	turns	turn	VERB
ejde-984	25	11	into	into	ADP
ejde-984	25	12	the	the	DET
ejde-984	25	13	schrödinger	schrödinger	NOUN
ejde-984	25	14	equation	equation	NOUN
ejde-984	25	15	)	)	PUNCT
ejde-984	25	16	,	,	PUNCT
ejde-984	25	17	in	in	ADP
ejde-984	25	18	1997	1997	NUM
ejde-984	25	19	,	,	PUNCT
ejde-984	25	20	jeanjean	jeanjean	PROPN
ejde-984	26	1	[	[	X
ejde-984	26	2	13	13	NUM
ejde-984	26	3	]	]	PUNCT
ejde-984	26	4	studied	study	VERB
ejde-984	26	5	the	the	DET
ejde-984	26	6	existence	existence	NOUN
ejde-984	26	7	of	of	ADP
ejde-984	26	8	the	the	DET
ejde-984	26	9	normalized	normalize	VERB
ejde-984	26	10	solutions	solution	NOUN
ejde-984	26	11	to	to	ADP
ejde-984	26	12	the	the	DET
ejde-984	26	13	schrödinger	schrödinger	NOUN
ejde-984	26	14	equation	equation	NOUN
ejde-984	26	15	in	in	ADP
ejde-984	26	16	the	the	DET
ejde-984	26	17	case	case	NOUN
ejde-984	26	18	of	of	ADP
ejde-984	26	19	the	the	DET
ejde-984	26	20	corresponding	corresponding	ADJ
ejde-984	26	21	energy	energy	NOUN
ejde-984	26	22	functional	functional	ADJ
ejde-984	26	23	f	f	X
ejde-984	26	24	(	(	PUNCT
ejde-984	26	25	u	u	NOUN
ejde-984	26	26	)	)	PUNCT
ejde-984	26	27	=	=	SYM
ejde-984	26	28	1	1	NUM
ejde-984	26	29	2	2	NUM
ejde-984	26	30	∫	∫	PROPN
ejde-984	26	31	rn	rn	PROPN
ejde-984	26	32	|∇u(x)|2dx−	|∇u(x)|2dx−	PROPN
ejde-984	27	1	m∑	m∑	VERB
ejde-984	27	2	i=1	i=1	PROPN
ejde-984	28	1	ai	ai	VERB
ejde-984	28	2	σi	σi	X
ejde-984	28	3	+	+	CCONJ
ejde-984	28	4	2	2	NUM
ejde-984	28	5	∫	∫	PROPN
ejde-984	28	6	rn	rn	PROPN
ejde-984	28	7	|u(x)|σi+2dx	|u(x)|σi+2dx	PROPN
ejde-984	28	8	is	be	AUX
ejde-984	28	9	unbounded	unbounded	ADJ
ejde-984	28	10	from	from	ADP
ejde-984	28	11	below	below	ADP
ejde-984	28	12	on	on	ADP
ejde-984	28	13	the	the	DET
ejde-984	28	14	l2	l2	NOUN
ejde-984	28	15	-	-	PUNCT
ejde-984	28	16	constraint	constraint	NOUN
ejde-984	28	17	set	set	NOUN
ejde-984	28	18	s(c	s(c	NOUN
ejde-984	28	19	)	)	PUNCT
ejde-984	29	1	=	=	SYM
ejde-984	29	2	{	{	PUNCT
ejde-984	29	3	u	u	NOUN
ejde-984	29	4	∈	∈	PROPN
ejde-984	29	5	h1(rn	h1(rn	PROPN
ejde-984	29	6	)	)	PUNCT
ejde-984	29	7	,	,	PUNCT
ejde-984	29	8	∥u∥l2(rn	∥u∥l2(rn	NOUN
ejde-984	29	9	)	)	PUNCT
ejde-984	29	10	=	=	PUNCT
ejde-984	30	1	c	c	X
ejde-984	30	2	}	}	PUNCT
ejde-984	30	3	initially	initially	ADV
ejde-984	30	4	.	.	PUNCT
ejde-984	31	1	lately	lately	ADV
ejde-984	31	2	,	,	PUNCT
ejde-984	31	3	such	such	ADJ
ejde-984	31	4	type	type	NOUN
ejde-984	31	5	of	of	ADP
ejde-984	31	6	problems	problem	NOUN
ejde-984	31	7	have	have	AUX
ejde-984	31	8	attracted	attract	VERB
ejde-984	31	9	extensive	extensive	ADJ
ejde-984	31	10	attention	attention	NOUN
ejde-984	31	11	in	in	ADP
ejde-984	31	12	the	the	DET
ejde-984	31	13	field	field	NOUN
ejde-984	31	14	of	of	ADP
ejde-984	31	15	partial	partial	ADJ
ejde-984	31	16	differential	differential	NOUN
ejde-984	31	17	equations	equation	NOUN
ejde-984	31	18	.	.	PUNCT
ejde-984	32	1	for	for	ADP
ejde-984	32	2	instance	instance	NOUN
ejde-984	32	3	,	,	PUNCT
ejde-984	32	4	soave	soave	PROPN
ejde-984	33	1	[	[	X
ejde-984	33	2	24	24	NUM
ejde-984	33	3	]	]	PUNCT
ejde-984	33	4	gave	give	VERB
ejde-984	33	5	the	the	DET
ejde-984	33	6	existence	existence	NOUN
ejde-984	33	7	and	and	CCONJ
ejde-984	33	8	some	some	DET
ejde-984	33	9	properties	property	NOUN
ejde-984	33	10	of	of	ADP
ejde-984	33	11	ground	ground	NOUN
ejde-984	33	12	states	state	NOUN
ejde-984	33	13	for	for	ADP
ejde-984	33	14	the	the	DET
ejde-984	33	15	nonlinear	nonlinear	ADJ
ejde-984	33	16	schrödinger	schrödinger	NOUN
ejde-984	33	17	equation	equation	NOUN
ejde-984	33	18	with	with	ADP
ejde-984	33	19	combined	combined	ADJ
ejde-984	33	20	power	power	NOUN
ejde-984	33	21	nonlinearities	nonlinearitie	NOUN
ejde-984	33	22	−∆u	−∆u	X
ejde-984	33	23	=	=	PUNCT
ejde-984	33	24	λu+	λu+	INTJ
ejde-984	33	25	µ|u|q−2u+	µ|u|q−2u+	PROPN
ejde-984	33	26	|u|p−2u	|u|p−2u	PROPN
ejde-984	33	27	in	in	ADP
ejde-984	33	28	rn	rn	PROPN
ejde-984	33	29	,	,	PUNCT
ejde-984	33	30	n	n	PRON
ejde-984	33	31	≥	≥	NOUN
ejde-984	33	32	1	1	NUM
ejde-984	33	33	,	,	PUNCT
ejde-984	33	34	on	on	ADP
ejde-984	33	35	the	the	DET
ejde-984	33	36	normalized	normalize	VERB
ejde-984	33	37	manifold	manifold	ADJ
ejde-984	33	38	∫	∫	PROPN
ejde-984	33	39	rn	rn	PROPN
ejde-984	33	40	|u|2dx	|u|2dx	PROPN
ejde-984	33	41	=	=	PROPN
ejde-984	33	42	a2	a2	PROPN
ejde-984	33	43	.	.	PUNCT
ejde-984	34	1	as	as	ADP
ejde-984	34	2	for	for	ADP
ejde-984	34	3	other	other	ADJ
ejde-984	34	4	results	result	NOUN
ejde-984	34	5	of	of	ADP
ejde-984	34	6	the	the	DET
ejde-984	34	7	schrödinger	schrödinger	NOUN
ejde-984	34	8	equation	equation	NOUN
ejde-984	34	9	,	,	PUNCT
ejde-984	34	10	we	we	PRON
ejde-984	34	11	refer	refer	VERB
ejde-984	34	12	readers	reader	NOUN
ejde-984	34	13	to	to	ADP
ejde-984	34	14	[	[	X
ejde-984	34	15	4	4	NUM
ejde-984	34	16	,	,	PUNCT
ejde-984	34	17	11	11	NUM
ejde-984	34	18	,	,	PUNCT
ejde-984	34	19	25	25	NUM
ejde-984	34	20	,	,	PUNCT
ejde-984	34	21	27	27	NUM
ejde-984	34	22	]	]	PUNCT
ejde-984	34	23	and	and	CCONJ
ejde-984	34	24	the	the	DET
ejde-984	34	25	references	reference	NOUN
ejde-984	34	26	therein	therein	ADV
ejde-984	34	27	.	.	PUNCT
ejde-984	35	1	when	when	SCONJ
ejde-984	35	2	considering	consider	VERB
ejde-984	35	3	the	the	DET
ejde-984	35	4	case	case	NOUN
ejde-984	35	5	a	a	DET
ejde-984	35	6	=	=	SYM
ejde-984	35	7	1	1	NUM
ejde-984	35	8	,	,	PUNCT
ejde-984	35	9	s	s	VERB
ejde-984	35	10	̸=	̸=	PROPN
ejde-984	35	11	1	1	NUM
ejde-984	35	12	and	and	CCONJ
ejde-984	35	13	b	b	X
ejde-984	35	14	=	=	SYM
ejde-984	35	15	0	0	NUM
ejde-984	35	16	,	,	PUNCT
ejde-984	35	17	i.e.	i.e.	X
ejde-984	35	18	,	,	PUNCT
ejde-984	35	19	for	for	ADP
ejde-984	35	20	the	the	DET
ejde-984	35	21	fractional	fractional	PROPN
ejde-984	35	22	schrödinger	schrödinger	NOUN
ejde-984	35	23	equations	equation	NOUN
ejde-984	35	24	,	,	PUNCT
ejde-984	35	25	see	see	VERB
ejde-984	35	26	[	[	X
ejde-984	35	27	18	18	NUM
ejde-984	35	28	,	,	PUNCT
ejde-984	35	29	29	29	NUM
ejde-984	35	30	]	]	PUNCT
ejde-984	35	31	and	and	CCONJ
ejde-984	35	32	the	the	DET
ejde-984	35	33	references	reference	NOUN
ejde-984	35	34	therein	therein	ADV
ejde-984	35	35	for	for	ADP
ejde-984	35	36	results	result	NOUN
ejde-984	35	37	about	about	ADP
ejde-984	35	38	the	the	DET
ejde-984	35	39	normalized	normalize	VERB
ejde-984	35	40	solutions	solution	NOUN
ejde-984	35	41	to	to	ADP
ejde-984	35	42	the	the	DET
ejde-984	35	43	fractional	fractional	PROPN
ejde-984	35	44	schrödinger	schrödinger	NOUN
ejde-984	35	45	equations	equation	NOUN
ejde-984	35	46	.	.	PUNCT
ejde-984	36	1	when	when	SCONJ
ejde-984	36	2	b	b	X
ejde-984	36	3	>	>	X
ejde-984	36	4	0	0	PUNCT
ejde-984	36	5	and	and	CCONJ
ejde-984	36	6	s	s	X
ejde-984	36	7	=	=	SYM
ejde-984	36	8	1	1	NUM
ejde-984	36	9	,	,	PUNCT
ejde-984	36	10	equation	equation	NOUN
ejde-984	36	11	(	(	PUNCT
ejde-984	36	12	1.1	1.1	NUM
ejde-984	36	13	)	)	PUNCT
ejde-984	36	14	becomes	become	VERB
ejde-984	36	15	the	the	DET
ejde-984	36	16	classic	classic	ADJ
ejde-984	36	17	kirchhoff	kirchhoff	NOUN
ejde-984	36	18	model	model	NOUN
ejde-984	36	19	;	;	PUNCT
ejde-984	36	20	this	this	DET
ejde-984	36	21	type	type	NOUN
ejde-984	36	22	of	of	ADP
ejde-984	36	23	problems	problem	NOUN
ejde-984	36	24	has	have	AUX
ejde-984	36	25	also	also	ADV
ejde-984	36	26	been	be	AUX
ejde-984	36	27	researched	research	VERB
ejde-984	36	28	by	by	ADP
ejde-984	36	29	many	many	ADJ
ejde-984	36	30	authors	author	NOUN
ejde-984	36	31	[	[	X
ejde-984	36	32	1	1	NUM
ejde-984	36	33	,	,	PUNCT
ejde-984	36	34	14	14	NUM
ejde-984	36	35	,	,	PUNCT
ejde-984	36	36	21	21	NUM
ejde-984	36	37	]	]	PUNCT
ejde-984	36	38	.	.	PUNCT
ejde-984	37	1	in	in	ADP
ejde-984	37	2	fact	fact	NOUN
ejde-984	37	3	,	,	PUNCT
ejde-984	37	4	such	such	ADJ
ejde-984	37	5	model	model	NOUN
ejde-984	37	6	has	have	VERB
ejde-984	37	7	relation	relation	NOUN
ejde-984	37	8	to	to	ADP
ejde-984	37	9	the	the	DET
ejde-984	37	10	stationary	stationary	ADJ
ejde-984	37	11	solutions	solution	NOUN
ejde-984	37	12	of	of	ADP
ejde-984	37	13	equation	equation	NOUN
ejde-984	37	14	utt	utt	NOUN
ejde-984	38	1	−	−	PROPN
ejde-984	38	2	(	(	PUNCT
ejde-984	38	3	a+	a+	SYM
ejde-984	38	4	b	b	PROPN
ejde-984	38	5	∫	∫	PROPN
ejde-984	38	6	rn	rn	PROPN
ejde-984	38	7	|∇u|2dx	|∇u|2dx	PROPN
ejde-984	38	8	)	)	PUNCT
ejde-984	39	1	∆u	∆u	PROPN
ejde-984	39	2	=	=	SYM
ejde-984	39	3	f(x	f(x	PROPN
ejde-984	39	4	,	,	PUNCT
ejde-984	39	5	u	u	NOUN
ejde-984	39	6	)	)	PUNCT
ejde-984	39	7	.	.	PUNCT
ejde-984	40	1	(	(	PUNCT
ejde-984	40	2	1.4	1.4	NUM
ejde-984	40	3	)	)	PUNCT
ejde-984	40	4	this	this	DET
ejde-984	40	5	equation	equation	NOUN
ejde-984	40	6	comes	come	VERB
ejde-984	40	7	from	from	ADP
ejde-984	40	8	the	the	DET
ejde-984	40	9	traditional	traditional	ADJ
ejde-984	40	10	d’alembert	d’alembert	NOUN
ejde-984	40	11	wave	wave	NOUN
ejde-984	40	12	equation	equation	NOUN
ejde-984	40	13	which	which	PRON
ejde-984	40	14	was	be	AUX
ejde-984	40	15	given	give	VERB
ejde-984	40	16	by	by	ADP
ejde-984	40	17	kirchhoff	kirchhoff	NOUN
ejde-984	40	18	[	[	X
ejde-984	40	19	14	14	NUM
ejde-984	40	20	]	]	PUNCT
ejde-984	40	21	in	in	ADP
ejde-984	40	22	1876	1876	NUM
ejde-984	40	23	in	in	ADP
ejde-984	40	24	the	the	DET
ejde-984	40	25	process	process	NOUN
ejde-984	40	26	of	of	ADP
ejde-984	40	27	studying	study	VERB
ejde-984	40	28	the	the	DET
ejde-984	40	29	changes	change	NOUN
ejde-984	40	30	in	in	ADP
ejde-984	40	31	the	the	DET
ejde-984	40	32	length	length	NOUN
ejde-984	40	33	of	of	ADP
ejde-984	40	34	the	the	DET
ejde-984	40	35	string	string	NOUN
ejde-984	40	36	during	during	ADP
ejde-984	40	37	vibrations	vibration	NOUN
ejde-984	40	38	,	,	PUNCT
ejde-984	40	39	where	where	SCONJ
ejde-984	40	40	f(x	f(x	PROPN
ejde-984	40	41	,	,	PUNCT
ejde-984	40	42	u	u	NOUN
ejde-984	40	43	)	)	PUNCT
ejde-984	40	44	denotes	denote	VERB
ejde-984	40	45	a	a	DET
ejde-984	40	46	general	general	ADJ
ejde-984	40	47	nonlinear	nonlinear	ADJ
ejde-984	40	48	term	term	NOUN
ejde-984	40	49	.	.	PUNCT
ejde-984	41	1	additionally	additionally	ADV
ejde-984	41	2	,	,	PUNCT
ejde-984	41	3	it	it	PRON
ejde-984	41	4	deserves	deserve	VERB
ejde-984	41	5	attention	attention	NOUN
ejde-984	41	6	that	that	SCONJ
ejde-984	41	7	in	in	ADP
ejde-984	41	8	[	[	X
ejde-984	41	9	1	1	NUM
ejde-984	41	10	]	]	X
ejde-984	41	11	equation	equation	NOUN
ejde-984	41	12	(	(	PUNCT
ejde-984	41	13	1.4	1.4	NUM
ejde-984	41	14	)	)	PUNCT
ejde-984	41	15	models	model	NOUN
ejde-984	41	16	some	some	DET
ejde-984	41	17	physical	physical	ADJ
ejde-984	41	18	systems	system	NOUN
ejde-984	41	19	,	,	PUNCT
ejde-984	41	20	where	where	SCONJ
ejde-984	41	21	u	u	NOUN
ejde-984	41	22	explains	explain	VERB
ejde-984	41	23	a	a	DET
ejde-984	41	24	process	process	NOUN
ejde-984	41	25	which	which	PRON
ejde-984	41	26	is	be	AUX
ejde-984	41	27	related	relate	VERB
ejde-984	41	28	to	to	ADP
ejde-984	41	29	the	the	DET
ejde-984	41	30	average	average	NOUN
ejde-984	41	31	of	of	ADP
ejde-984	41	32	itself	itself	PRON
ejde-984	41	33	.	.	PUNCT
ejde-984	42	1	a	a	DET
ejde-984	42	2	lot	lot	NOUN
ejde-984	42	3	of	of	ADP
ejde-984	42	4	papers	paper	NOUN
ejde-984	42	5	about	about	ADP
ejde-984	42	6	the	the	DET
ejde-984	42	7	kirchhoff	kirchhoff	NOUN
ejde-984	42	8	type	type	NOUN
ejde-984	42	9	equations	equation	NOUN
ejde-984	42	10	emerged	emerge	VERB
ejde-984	42	11	with	with	ADP
ejde-984	42	12	the	the	DET
ejde-984	42	13	emergence	emergence	NOUN
ejde-984	42	14	of	of	ADP
ejde-984	42	15	this	this	DET
ejde-984	42	16	ground	ground	NOUN
ejde-984	42	17	breaking	break	VERB
ejde-984	42	18	article	article	NOUN
ejde-984	42	19	[	[	X
ejde-984	42	20	21	21	NUM
ejde-984	42	21	]	]	PUNCT
ejde-984	42	22	.	.	PUNCT
ejde-984	43	1	for	for	ADP
ejde-984	43	2	example	example	NOUN
ejde-984	43	3	,	,	PUNCT
ejde-984	43	4	considering	consider	VERB
ejde-984	43	5	a	a	DET
ejde-984	43	6	kirchhoff	kirchhoff	NOUN
ejde-984	43	7	model	model	NOUN
ejde-984	43	8	,	,	PUNCT
ejde-984	43	9	together	together	ADV
ejde-984	43	10	with	with	ADP
ejde-984	43	11	a	a	DET
ejde-984	43	12	critical	critical	ADJ
ejde-984	43	13	trudinger	trudinger	NOUN
ejde-984	43	14	-	-	PUNCT
ejde-984	43	15	moser	moser	PROPN
ejde-984	43	16	nonlinearity	nonlinearity	PROPN
ejde-984	43	17	f(x	f(x	PROPN
ejde-984	43	18	,	,	PUNCT
ejde-984	43	19	u	u	NOUN
ejde-984	43	20	)	)	PUNCT
ejde-984	43	21	,	,	PUNCT
ejde-984	43	22	a	a	DET
ejde-984	43	23	class	class	NOUN
ejde-984	43	24	of	of	ADP
ejde-984	43	25	fractional	fractional	PROPN
ejde-984	43	26	kirchhoff	kirchhoff	NOUN
ejde-984	43	27	-	-	PUNCT
ejde-984	43	28	type	type	NOUN
ejde-984	43	29	equation	equation	NOUN
ejde-984	43	30	with	with	ADP
ejde-984	43	31	trudinger	trudinger	NOUN
ejde-984	43	32	-	-	PUNCT
ejde-984	43	33	moser	moser	PROPN
ejde-984	43	34	nonlinearity	nonlinearity	NOUN
ejde-984	43	35	was	be	AUX
ejde-984	43	36	discussed	discuss	VERB
ejde-984	43	37	by	by	ADP
ejde-984	43	38	xiang	xiang	PROPN
ejde-984	43	39	,	,	PUNCT
ejde-984	43	40	rădulescu	rădulescu	PROPN
ejde-984	43	41	and	and	CCONJ
ejde-984	43	42	zhang	zhang	PROPN
ejde-984	44	1	[	[	X
ejde-984	44	2	20	20	NUM
ejde-984	44	3	]	]	PUNCT
ejde-984	44	4	.	.	PUNCT
ejde-984	45	1	using	use	VERB
ejde-984	45	2	appropriate	appropriate	ADJ
ejde-984	45	3	assumptions	assumption	NOUN
ejde-984	45	4	on	on	ADP
ejde-984	45	5	the	the	DET
ejde-984	45	6	potential	potential	ADJ
ejde-984	45	7	function	function	NOUN
ejde-984	45	8	v	v	NOUN
ejde-984	45	9	and	and	CCONJ
ejde-984	45	10	some	some	DET
ejde-984	45	11	energy	energy	NOUN
ejde-984	45	12	estimates	estimate	VERB
ejde-984	45	13	techniques	technique	NOUN
ejde-984	45	14	,	,	PUNCT
ejde-984	45	15	chen	chen	PROPN
ejde-984	45	16	and	and	CCONJ
ejde-984	45	17	huang	huang	PROPN
ejde-984	46	1	[	[	X
ejde-984	46	2	6	6	NUM
ejde-984	46	3	]	]	PUNCT
ejde-984	46	4	obtained	obtain	VERB
ejde-984	46	5	the	the	DET
ejde-984	46	6	existence	existence	NOUN
ejde-984	46	7	results	result	NOUN
ejde-984	46	8	of	of	ADP
ejde-984	46	9	normalized	normalize	VERB
ejde-984	46	10	solutions	solution	NOUN
ejde-984	46	11	for	for	ADP
ejde-984	46	12	a	a	DET
ejde-984	46	13	fractional	fractional	ADJ
ejde-984	46	14	kirchhoff	kirchhoff	NOUN
ejde-984	46	15	-	-	PUNCT
ejde-984	46	16	type	type	NOUN
ejde-984	46	17	equation	equation	NOUN
ejde-984	46	18	(	(	PUNCT
ejde-984	46	19	a+	a+	PRON
ejde-984	46	20	b	b	X
ejde-984	46	21	∫	∫	PROPN
ejde-984	46	22	rn	rn	PROPN
ejde-984	46	23	|(−∆)s/2u|2dx	|(−∆)s/2u|2dx	NUM
ejde-984	46	24	)	)	PUNCT
ejde-984	46	25	(	(	PUNCT
ejde-984	46	26	−∆)su+	−∆)su+	NOUN
ejde-984	46	27	v	v	X
ejde-984	46	28	(	(	PUNCT
ejde-984	46	29	x)u	x)u	PUNCT
ejde-984	46	30	=	=	SYM
ejde-984	47	1	c|u|p−2u+	c|u|p−2u+	NOUN
ejde-984	47	2	µu	µu	VERB
ejde-984	47	3	in	in	ADP
ejde-984	47	4	rn	rn	PROPN
ejde-984	47	5	with	with	ADP
ejde-984	47	6	doubly	doubly	ADV
ejde-984	47	7	critical	critical	ADJ
ejde-984	47	8	exponents	exponent	NOUN
ejde-984	47	9	(	(	PUNCT
ejde-984	47	10	when	when	SCONJ
ejde-984	47	11	considering	consider	VERB
ejde-984	47	12	the	the	DET
ejde-984	47	13	case	case	NOUN
ejde-984	47	14	n	n	NOUN
ejde-984	47	15	=	=	SYM
ejde-984	47	16	4s	4s	NUM
ejde-984	47	17	,	,	PUNCT
ejde-984	47	18	the	the	DET
ejde-984	47	19	critical	critical	ADJ
ejde-984	47	20	sobolev	sobolev	NOUN
ejde-984	47	21	exponent	exponent	NOUN
ejde-984	48	1	2∗s	2∗s	NUM
ejde-984	48	2	=	=	SYM
ejde-984	48	3	2n	2n	NUM
ejde-984	48	4	n−2s	n−2s	NOUN
ejde-984	48	5	and	and	CCONJ
ejde-984	48	6	the	the	DET
ejde-984	48	7	fractional	fractional	ADJ
ejde-984	48	8	gagliardo	gagliardo	NOUN
ejde-984	48	9	-	-	PUNCT
ejde-984	48	10	nirenberg	nirenberg	PROPN
ejde-984	48	11	-	-	PUNCT
ejde-984	48	12	sobolev	sobolev	NOUN
ejde-984	48	13	critical	critical	ADJ
ejde-984	48	14	exponent	exponent	NOUN
ejde-984	48	15	2∗gns	2∗gns	NUM
ejde-984	48	16	=	=	SYM
ejde-984	48	17	2n+8s	2n+8s	NUM
ejde-984	48	18	n	n	NOUN
ejde-984	48	19	are	be	AUX
ejde-984	48	20	equal	equal	ADJ
ejde-984	48	21	,	,	PUNCT
ejde-984	48	22	and	and	CCONJ
ejde-984	48	23	2n	2n	NUM
ejde-984	48	24	n−2s	n−2s	NOUN
ejde-984	49	1	=	=	SYM
ejde-984	49	2	2n+8s	2n+8s	NUM
ejde-984	49	3	n	n	NOUN
ejde-984	49	4	=	=	SYM
ejde-984	49	5	4	4	NUM
ejde-984	49	6	)	)	PUNCT
ejde-984	49	7	.	.	PUNCT
ejde-984	50	1	in	in	ADP
ejde-984	50	2	addition	addition	NOUN
ejde-984	50	3	,	,	PUNCT
ejde-984	50	4	in	in	ADP
ejde-984	50	5	2024	2024	NUM
ejde-984	50	6	,	,	PUNCT
ejde-984	50	7	the	the	DET
ejde-984	50	8	existence	existence	NOUN
ejde-984	50	9	of	of	ADP
ejde-984	50	10	the	the	DET
ejde-984	50	11	normalized	normalize	VERB
ejde-984	50	12	solutions	solution	NOUN
ejde-984	50	13	to	to	ADP
ejde-984	50	14	the	the	DET
ejde-984	50	15	fractional	fractional	PROPN
ejde-984	50	16	kirchhoff	kirchhoff	NOUN
ejde-984	50	17	equation	equation	NOUN
ejde-984	50	18	with	with	ADP
ejde-984	50	19	subcritical	subcritical	ADJ
ejde-984	50	20	nonlinearity	nonlinearity	NOUN
ejde-984	50	21	(	(	PUNCT
ejde-984	50	22	a+	a+	PRON
ejde-984	50	23	b	b	PROPN
ejde-984	50	24	∫	∫	PROPN
ejde-984	50	25	rn	rn	PROPN
ejde-984	50	26	|(−∆)s/2u|2dx	|(−∆)s/2u|2dx	NUM
ejde-984	50	27	)	)	PUNCT
ejde-984	50	28	(	(	PUNCT
ejde-984	50	29	−∆)su	−∆)su	NOUN
ejde-984	50	30	=	=	PUNCT
ejde-984	50	31	λu+	λu+	INTJ
ejde-984	50	32	µ|u|q−2u+	µ|u|q−2u+	PROPN
ejde-984	50	33	|u|p−2u	|u|p−2u	PROPN
ejde-984	50	34	in	in	ADP
ejde-984	50	35	rn	rn	PROPN
ejde-984	50	36	was	be	AUX
ejde-984	50	37	studied	study	VERB
ejde-984	50	38	in	in	ADP
ejde-984	50	39	[	[	X
ejde-984	50	40	8	8	NUM
ejde-984	50	41	]	]	PUNCT
ejde-984	50	42	in	in	ADP
ejde-984	50	43	r3	r3	PROPN
ejde-984	50	44	with	with	ADP
ejde-984	50	45	s	s	X
ejde-984	50	46	∈	∈	PROPN
ejde-984	50	47	(	(	PUNCT
ejde-984	50	48	3/4	3/4	NUM
ejde-984	50	49	,	,	PUNCT
ejde-984	50	50	1	1	NUM
ejde-984	50	51	)	)	PUNCT
ejde-984	50	52	and	and	CCONJ
ejde-984	50	53	µ	µ	X
ejde-984	50	54	<	<	X
ejde-984	50	55	0	0	NUM
ejde-984	50	56	.	.	PUNCT
ejde-984	51	1	if	if	SCONJ
ejde-984	51	2	the	the	DET
ejde-984	51	3	constraint	constraint	NOUN
ejde-984	51	4	condition	condition	NOUN
ejde-984	51	5	(	(	PUNCT
ejde-984	51	6	1.2	1.2	NUM
ejde-984	51	7	)	)	PUNCT
ejde-984	51	8	is	be	AUX
ejde-984	51	9	considered	consider	VERB
ejde-984	51	10	on	on	ADP
ejde-984	51	11	this	this	DET
ejde-984	51	12	basis	basis	NOUN
ejde-984	51	13	,	,	PUNCT
ejde-984	51	14	some	some	DET
ejde-984	51	15	universal	universal	ADJ
ejde-984	51	16	methods	method	NOUN
ejde-984	51	17	do	do	AUX
ejde-984	51	18	not	not	PART
ejde-984	51	19	take	take	VERB
ejde-984	51	20	effect	effect	NOUN
ejde-984	51	21	.	.	PUNCT
ejde-984	52	1	as	as	ADP
ejde-984	52	2	a	a	DET
ejde-984	52	3	result	result	NOUN
ejde-984	52	4	,	,	PUNCT
ejde-984	52	5	we	we	PRON
ejde-984	52	6	need	need	VERB
ejde-984	52	7	to	to	PART
ejde-984	52	8	establish	establish	VERB
ejde-984	52	9	extra	extra	ADJ
ejde-984	52	10	claims	claim	NOUN
ejde-984	52	11	to	to	PART
ejde-984	52	12	solve	solve	VERB
ejde-984	52	13	the	the	DET
ejde-984	52	14	technical	technical	ADJ
ejde-984	52	15	obstacles	obstacle	NOUN
ejde-984	52	16	.	.	PUNCT
ejde-984	53	1	as	as	SCONJ
ejde-984	53	2	is	be	AUX
ejde-984	53	3	known	know	VERB
ejde-984	53	4	(	(	PUNCT
ejde-984	53	5	such	such	ADJ
ejde-984	53	6	as	as	ADP
ejde-984	53	7	in	in	ADP
ejde-984	53	8	[	[	PUNCT
ejde-984	53	9	13	13	NUM
ejde-984	53	10	]	]	NUM
ejde-984	53	11	)	)	PUNCT
ejde-984	53	12	,	,	PUNCT
ejde-984	53	13	(	(	PUNCT
ejde-984	53	14	1.2	1.2	NUM
ejde-984	53	15	)	)	PUNCT
ejde-984	53	16	has	have	VERB
ejde-984	53	17	definite	definite	ADJ
ejde-984	53	18	physical	physical	ADJ
ejde-984	53	19	motivations	motivation	NOUN
ejde-984	53	20	.	.	PUNCT
ejde-984	54	1	consequently	consequently	ADV
ejde-984	54	2	,	,	PUNCT
ejde-984	54	3	it	it	PRON
ejde-984	54	4	has	have	AUX
ejde-984	54	5	sparked	spark	VERB
ejde-984	54	6	a	a	DET
ejde-984	54	7	wave	wave	NOUN
ejde-984	54	8	of	of	ADP
ejde-984	54	9	research	research	NOUN
ejde-984	54	10	on	on	ADP
ejde-984	54	11	the	the	DET
ejde-984	54	12	normalized	normalize	VERB
ejde-984	54	13	solutions	solution	NOUN
ejde-984	54	14	.	.	PUNCT
ejde-984	55	1	more	more	ADV
ejde-984	55	2	specifically	specifically	ADV
ejde-984	55	3	,	,	PUNCT
ejde-984	55	4	the	the	DET
ejde-984	55	5	practical	practical	ADJ
ejde-984	55	6	application	application	NOUN
ejde-984	55	7	background	background	NOUN
ejde-984	55	8	of	of	ADP
ejde-984	55	9	operator	operator	NOUN
ejde-984	55	10	(	(	PUNCT
ejde-984	55	11	−∆)s	−∆)s	PROPN
ejde-984	55	12	includes	include	VERB
ejde-984	55	13	the	the	DET
ejde-984	55	14	following	follow	VERB
ejde-984	55	15	aspects	aspect	NOUN
ejde-984	55	16	such	such	ADJ
ejde-984	55	17	as	as	ADP
ejde-984	55	18	fractional	fractional	ADJ
ejde-984	55	19	quantum	quantum	ADJ
ejde-984	55	20	mechanics	mechanic	NOUN
ejde-984	55	21	[	[	X
ejde-984	55	22	15	15	NUM
ejde-984	55	23	]	]	PUNCT
ejde-984	55	24	,	,	PUNCT
ejde-984	55	25	physics	physics	NOUN
ejde-984	55	26	and	and	CCONJ
ejde-984	55	27	chemistry	chemistry	NOUN
ejde-984	56	1	[	[	X
ejde-984	56	2	19	19	NUM
ejde-984	56	3	]	]	PUNCT
ejde-984	56	4	,	,	PUNCT
ejde-984	56	5	conformal	conformal	ADJ
ejde-984	56	6	geometry	geometry	NOUN
ejde-984	56	7	and	and	CCONJ
ejde-984	56	8	minimal	minimal	ADJ
ejde-984	56	9	surfaces	surface	NOUN
ejde-984	56	10	[	[	X
ejde-984	56	11	5	5	NUM
ejde-984	56	12	]	]	PUNCT
ejde-984	56	13	,	,	PUNCT
ejde-984	56	14	obstacle	obstacle	NOUN
ejde-984	56	15	problems	problem	NOUN
ejde-984	56	16	[	[	X
ejde-984	56	17	23	23	NUM
ejde-984	56	18	]	]	PUNCT
ejde-984	56	19	.	.	PUNCT
ejde-984	57	1	caffarelli	caffarelli	PROPN
ejde-984	57	2	ejde-2025/75	ejde-2025/75	ADJ
ejde-984	57	3	normalized	normalize	VERB
ejde-984	57	4	solutions	solution	NOUN
ejde-984	57	5	of	of	ADP
ejde-984	57	6	fractional	fractional	PROPN
ejde-984	57	7	kirchhoff	kirchhoff	NOUN
ejde-984	57	8	equations	equation	NOUN
ejde-984	57	9	3	3	NUM
ejde-984	57	10	and	and	CCONJ
ejde-984	57	11	silvestre	silvestre	PROPN
ejde-984	57	12	[	[	X
ejde-984	57	13	2	2	X
ejde-984	57	14	]	]	PUNCT
ejde-984	57	15	adopted	adopt	VERB
ejde-984	57	16	the	the	DET
ejde-984	57	17	extension	extension	NOUN
ejde-984	57	18	method	method	NOUN
ejde-984	57	19	which	which	PRON
ejde-984	57	20	converted	convert	VERB
ejde-984	57	21	this	this	DET
ejde-984	57	22	nonlocal	nonlocal	ADJ
ejde-984	57	23	problem	problem	NOUN
ejde-984	57	24	(	(	PUNCT
ejde-984	57	25	due	due	ADP
ejde-984	57	26	to	to	ADP
ejde-984	57	27	the	the	DET
ejde-984	57	28	nonlocal	nonlocal	ADJ
ejde-984	57	29	feature	feature	NOUN
ejde-984	57	30	of	of	ADP
ejde-984	57	31	the	the	DET
ejde-984	57	32	operator	operator	NOUN
ejde-984	57	33	(	(	PUNCT
ejde-984	57	34	−∆)s	−∆)s	PROPN
ejde-984	57	35	on	on	ADP
ejde-984	57	36	rn	rn	PROPN
ejde-984	57	37	(	(	PUNCT
ejde-984	57	38	n	n	CCONJ
ejde-984	57	39	≥	≥	NOUN
ejde-984	57	40	1	1	NUM
ejde-984	57	41	)	)	PUNCT
ejde-984	57	42	)	)	PUNCT
ejde-984	57	43	to	to	ADP
ejde-984	57	44	a	a	DET
ejde-984	57	45	local	local	ADJ
ejde-984	57	46	one	one	NUM
ejde-984	57	47	in	in	ADP
ejde-984	57	48	higher	high	ADJ
ejde-984	57	49	dimensions	dimension	NOUN
ejde-984	57	50	.	.	PUNCT
ejde-984	58	1	what	what	PRON
ejde-984	58	2	makes	make	VERB
ejde-984	58	3	this	this	PRON
ejde-984	58	4	amusing	amusing	ADJ
ejde-984	58	5	,	,	PUNCT
ejde-984	58	6	of	of	ADP
ejde-984	58	7	course	course	NOUN
ejde-984	58	8	,	,	PUNCT
ejde-984	58	9	is	be	AUX
ejde-984	58	10	that	that	SCONJ
ejde-984	58	11	this	this	DET
ejde-984	58	12	method	method	NOUN
ejde-984	58	13	can	can	AUX
ejde-984	58	14	go	go	VERB
ejde-984	58	15	for	for	ADP
ejde-984	58	16	nonlinear	nonlinear	ADJ
ejde-984	58	17	equations	equation	NOUN
ejde-984	58	18	with	with	ADP
ejde-984	58	19	a	a	DET
ejde-984	58	20	fractional	fractional	ADJ
ejde-984	58	21	laplacian	laplacian	NOUN
ejde-984	58	22	,	,	PUNCT
ejde-984	58	23	we	we	PRON
ejde-984	58	24	refer	refer	VERB
ejde-984	58	25	readers	reader	NOUN
ejde-984	58	26	to	to	ADP
ejde-984	58	27	[	[	X
ejde-984	58	28	7	7	NUM
ejde-984	58	29	,	,	PUNCT
ejde-984	58	30	9	9	NUM
ejde-984	58	31	]	]	PUNCT
ejde-984	58	32	and	and	CCONJ
ejde-984	58	33	the	the	DET
ejde-984	58	34	references	reference	NOUN
ejde-984	58	35	therein	therein	ADV
ejde-984	58	36	.	.	PUNCT
ejde-984	59	1	about	about	ADP
ejde-984	59	2	equation	equation	NOUN
ejde-984	59	3	(	(	PUNCT
ejde-984	59	4	1.1	1.1	NUM
ejde-984	59	5	)	)	PUNCT
ejde-984	59	6	,	,	PUNCT
ejde-984	59	7	there	there	PRON
ejde-984	59	8	are	be	VERB
ejde-984	59	9	usually	usually	ADV
ejde-984	59	10	two	two	NUM
ejde-984	59	11	classes	class	NOUN
ejde-984	59	12	of	of	ADP
ejde-984	59	13	treatments	treatment	NOUN
ejde-984	59	14	.	.	PUNCT
ejde-984	60	1	on	on	ADP
ejde-984	60	2	one	one	NUM
ejde-984	60	3	hand	hand	NOUN
ejde-984	60	4	,	,	PUNCT
ejde-984	60	5	we	we	PRON
ejde-984	60	6	can	can	AUX
ejde-984	60	7	think	think	VERB
ejde-984	60	8	of	of	ADP
ejde-984	60	9	it	it	PRON
ejde-984	60	10	as	as	ADP
ejde-984	60	11	a	a	DET
ejde-984	60	12	fixed	fix	VERB
ejde-984	60	13	frequency	frequency	NOUN
ejde-984	60	14	problem	problem	NOUN
ejde-984	60	15	,	,	PUNCT
ejde-984	60	16	in	in	ADP
ejde-984	60	17	other	other	ADJ
ejde-984	60	18	words	word	NOUN
ejde-984	60	19	,	,	PUNCT
ejde-984	60	20	we	we	PRON
ejde-984	60	21	look	look	VERB
ejde-984	60	22	for	for	ADP
ejde-984	60	23	solutions	solution	NOUN
ejde-984	60	24	u	u	NOUN
ejde-984	60	25	∈	∈	PROPN
ejde-984	60	26	hs(r3	hs(r3	PRON
ejde-984	60	27	)	)	PUNCT
ejde-984	60	28	by	by	ADP
ejde-984	60	29	hunting	hunt	VERB
ejde-984	60	30	for	for	ADP
ejde-984	60	31	critical	critical	ADJ
ejde-984	60	32	points	point	NOUN
ejde-984	60	33	of	of	ADP
ejde-984	60	34	the	the	DET
ejde-984	60	35	action	action	NOUN
ejde-984	60	36	functional	functional	ADJ
ejde-984	60	37	m	m	NOUN
ejde-984	60	38	:	:	PUNCT
ejde-984	60	39	hs(r3	hs(r3	NUM
ejde-984	60	40	)	)	PUNCT
ejde-984	61	1	→	→	SYM
ejde-984	61	2	r	r	X
ejde-984	61	3	:	:	PUNCT
ejde-984	61	4	m(u	m(u	NUM
ejde-984	61	5	)	)	PUNCT
ejde-984	61	6	:	:	PUNCT
ejde-984	62	1	=	=	PUNCT
ejde-984	62	2	a	a	DET
ejde-984	62	3	2	2	NUM
ejde-984	62	4	∫	∫	NOUN
ejde-984	62	5	r3	r3	PROPN
ejde-984	62	6	|(−∆)s/2u|2dx+	|(−∆)s/2u|2dx+	PROPN
ejde-984	62	7	b	b	X
ejde-984	62	8	4	4	NUM
ejde-984	62	9	(	(	PUNCT
ejde-984	62	10	∫	∫	PROPN
ejde-984	62	11	r3	r3	PROPN
ejde-984	62	12	|(−∆)s/2u|2dx	|(−∆)s/2u|2dx	VERB
ejde-984	62	13	)	)	PUNCT
ejde-984	62	14	2	2	NUM
ejde-984	62	15	−	−	PROPN
ejde-984	62	16	λ	λ	NOUN
ejde-984	62	17	2	2	NUM
ejde-984	62	18	∫	∫	NOUN
ejde-984	62	19	r3	r3	PROPN
ejde-984	62	20	|u|2dx−	|u|2dx−	PROPN
ejde-984	62	21	µ	µ	PROPN
ejde-984	62	22	q	q	ADJ
ejde-984	62	23	∫	∫	PROPN
ejde-984	62	24	r3	r3	PROPN
ejde-984	62	25	|u|qdx−	|u|qdx−	NOUN
ejde-984	62	26	1	1	NUM
ejde-984	62	27	p	p	NOUN
ejde-984	62	28	∫	∫	PROPN
ejde-984	62	29	r3	r3	PROPN
ejde-984	62	30	|u|pdx	|u|pdx	NUM
ejde-984	62	31	,	,	PUNCT
ejde-984	62	32	(	(	PUNCT
ejde-984	62	33	1.5	1.5	NUM
ejde-984	62	34	)	)	PUNCT
ejde-984	62	35	where	where	SCONJ
ejde-984	62	36	λ	λ	PROPN
ejde-984	62	37	∈	∈	PROPN
ejde-984	62	38	r	r	NOUN
ejde-984	62	39	is	be	AUX
ejde-984	62	40	a	a	DET
ejde-984	62	41	fixed	fix	VERB
ejde-984	62	42	frequency	frequency	NOUN
ejde-984	62	43	,	,	PUNCT
ejde-984	62	44	readers	reader	NOUN
ejde-984	62	45	can	can	AUX
ejde-984	62	46	see	see	VERB
ejde-984	62	47	[	[	X
ejde-984	62	48	12	12	NUM
ejde-984	62	49	,	,	PUNCT
ejde-984	62	50	17	17	NUM
ejde-984	62	51	]	]	PUNCT
ejde-984	62	52	for	for	ADP
ejde-984	62	53	more	more	ADJ
ejde-984	62	54	results	result	NOUN
ejde-984	62	55	.	.	PUNCT
ejde-984	63	1	on	on	ADP
ejde-984	63	2	the	the	DET
ejde-984	63	3	other	other	ADJ
ejde-984	63	4	hand	hand	NOUN
ejde-984	63	5	,	,	PUNCT
ejde-984	63	6	we	we	PRON
ejde-984	63	7	can	can	AUX
ejde-984	63	8	also	also	ADV
ejde-984	63	9	search	search	VERB
ejde-984	63	10	for	for	ADP
ejde-984	63	11	solutions	solution	NOUN
ejde-984	63	12	to	to	ADP
ejde-984	63	13	(	(	PUNCT
ejde-984	63	14	1.1	1.1	NUM
ejde-984	63	15	)	)	PUNCT
ejde-984	63	16	with	with	ADP
ejde-984	63	17	a	a	DET
ejde-984	63	18	prescribed	prescribed	ADJ
ejde-984	63	19	l2	l2	NOUN
ejde-984	63	20	norm	norm	NOUN
ejde-984	63	21	.	.	PUNCT
ejde-984	64	1	in	in	ADP
ejde-984	64	2	this	this	DET
ejde-984	64	3	case	case	NOUN
ejde-984	64	4	,	,	PUNCT
ejde-984	64	5	we	we	PRON
ejde-984	64	6	see	see	VERB
ejde-984	64	7	λ	λ	X
ejde-984	64	8	∈	∈	NOUN
ejde-984	64	9	r	r	NOUN
ejde-984	64	10	as	as	ADP
ejde-984	64	11	part	part	NOUN
ejde-984	64	12	of	of	ADP
ejde-984	64	13	unknown	unknown	ADJ
ejde-984	64	14	quantity	quantity	NOUN
ejde-984	64	15	.	.	PUNCT
ejde-984	65	1	as	as	SCONJ
ejde-984	65	2	everyone	everyone	PRON
ejde-984	65	3	knows	know	VERB
ejde-984	65	4	,	,	PUNCT
ejde-984	65	5	equation	equation	NOUN
ejde-984	65	6	(	(	PUNCT
ejde-984	65	7	1.1	1.1	NUM
ejde-984	65	8	)	)	PUNCT
ejde-984	65	9	has	have	VERB
ejde-984	65	10	roots	root	NOUN
ejde-984	65	11	in	in	ADP
ejde-984	65	12	the	the	DET
ejde-984	65	13	standing	standing	NOUN
ejde-984	65	14	wave	wave	NOUN
ejde-984	65	15	type	type	NOUN
ejde-984	65	16	solution	solution	NOUN
ejde-984	65	17	ψ(x	ψ(x	PROPN
ejde-984	65	18	,	,	PUNCT
ejde-984	65	19	t	t	PROPN
ejde-984	65	20	)	)	PUNCT
ejde-984	65	21	=	=	SYM
ejde-984	65	22	e−iλtu(x	e−iλtu(x	PROPN
ejde-984	65	23	)	)	PUNCT
ejde-984	65	24	,	,	PUNCT
ejde-984	66	1	λ	λ	X
ejde-984	66	2	∈	∈	NOUN
ejde-984	66	3	r	r	NOUN
ejde-984	66	4	to	to	ADP
ejde-984	66	5	the	the	DET
ejde-984	66	6	time	time	NOUN
ejde-984	66	7	-	-	PUNCT
ejde-984	66	8	dependent	dependent	ADJ
ejde-984	66	9	nonlinear	nonlinear	ADJ
ejde-984	66	10	fractional	fractional	ADJ
ejde-984	66	11	equation	equation	NOUN
ejde-984	66	12	defined	define	VERB
ejde-984	66	13	by	by	ADP
ejde-984	66	14	:	:	PUNCT
ejde-984	66	15	i	i	PRON
ejde-984	66	16	∂ψ	∂ψ	VERB
ejde-984	67	1	∂t	∂t	PROPN
ejde-984	67	2	=	=	PRON
ejde-984	67	3	(	(	PUNCT
ejde-984	67	4	a+	a+	PRON
ejde-984	67	5	b	b	PROPN
ejde-984	67	6	∫	∫	PROPN
ejde-984	67	7	r3	r3	PROPN
ejde-984	67	8	|(−∆)s/2ψ|2dx	|(−∆)s/2ψ|2dx	VERB
ejde-984	67	9	)	)	PUNCT
ejde-984	67	10	(	(	PUNCT
ejde-984	67	11	−∆)sψ	−∆)sψ	NOUN
ejde-984	67	12	−	−	PROPN
ejde-984	68	1	f(|ψ|)ψ	f(|ψ|)ψ	INTJ
ejde-984	68	2	,	,	PUNCT
ejde-984	68	3	in	in	ADP
ejde-984	68	4	r3	r3	PROPN
ejde-984	68	5	,	,	PUNCT
ejde-984	68	6	(	(	PUNCT
ejde-984	68	7	1.6	1.6	NUM
ejde-984	68	8	)	)	PUNCT
ejde-984	68	9	where	where	SCONJ
ejde-984	68	10	s	s	VERB
ejde-984	68	11	∈	∈	PROPN
ejde-984	68	12	(	(	PUNCT
ejde-984	68	13	0	0	NUM
ejde-984	68	14	,	,	PUNCT
ejde-984	68	15	1	1	NUM
ejde-984	68	16	)	)	PUNCT
ejde-984	68	17	,	,	PUNCT
ejde-984	68	18	i	i	PRON
ejde-984	68	19	represents	represent	VERB
ejde-984	68	20	the	the	DET
ejde-984	68	21	imaginary	imaginary	ADJ
ejde-984	68	22	unit	unit	NOUN
ejde-984	68	23	,	,	PUNCT
ejde-984	68	24	and	and	CCONJ
ejde-984	68	25	ψ	ψ	X
ejde-984	68	26	=	=	NOUN
ejde-984	68	27	ψ(x	ψ(x	PROPN
ejde-984	68	28	,	,	PUNCT
ejde-984	68	29	t	t	PROPN
ejde-984	68	30	)	)	PUNCT
ejde-984	68	31	:	:	PUNCT
ejde-984	68	32	r3×	r3×	VERB
ejde-984	68	33	[	[	X
ejde-984	68	34	0,+∞	0,+∞	NUM
ejde-984	68	35	)	)	PUNCT
ejde-984	68	36	→	→	SYM
ejde-984	68	37	c.	c.	PROPN
ejde-984	68	38	it	it	PRON
ejde-984	68	39	is	be	AUX
ejde-984	68	40	easy	easy	ADJ
ejde-984	68	41	to	to	PART
ejde-984	68	42	see	see	VERB
ejde-984	68	43	that	that	SCONJ
ejde-984	68	44	ψ	ψ	ADP
ejde-984	68	45	solves	solve	NOUN
ejde-984	68	46	(	(	PUNCT
ejde-984	68	47	1.6	1.6	NUM
ejde-984	68	48	)	)	PUNCT
ejde-984	68	49	if	if	SCONJ
ejde-984	68	50	and	and	CCONJ
ejde-984	68	51	only	only	ADV
ejde-984	68	52	if	if	SCONJ
ejde-984	68	53	the	the	DET
ejde-984	68	54	standing	standing	ADJ
ejde-984	68	55	wave	wave	NOUN
ejde-984	68	56	u(x	u(x	NOUN
ejde-984	68	57	)	)	PUNCT
ejde-984	68	58	satisfies	satisfie	NOUN
ejde-984	68	59	(	(	PUNCT
ejde-984	68	60	1.1	1.1	NUM
ejde-984	68	61	)	)	PUNCT
ejde-984	68	62	with	with	ADP
ejde-984	68	63	f(u	f(u	PROPN
ejde-984	68	64	)	)	PUNCT
ejde-984	68	65	=	=	PUNCT
ejde-984	69	1	µuq−2+up−2	µuq−2+up−2	PROPN
ejde-984	69	2	.	.	PUNCT
ejde-984	70	1	after	after	ADP
ejde-984	70	2	computations	computation	NOUN
ejde-984	70	3	,	,	PUNCT
ejde-984	70	4	we	we	PRON
ejde-984	70	5	can	can	AUX
ejde-984	70	6	see	see	VERB
ejde-984	70	7	that	that	SCONJ
ejde-984	70	8	solutions	solution	NOUN
ejde-984	70	9	ψ	ψ	ADP
ejde-984	70	10	∈	∈	PROPN
ejde-984	70	11	c([0	c([0	NOUN
ejde-984	70	12	,	,	PUNCT
ejde-984	70	13	t	t	PROPN
ejde-984	70	14	)	)	PUNCT
ejde-984	70	15	;	;	PUNCT
ejde-984	70	16	hs(r3	hs(r3	NUM
ejde-984	70	17	)	)	PUNCT
ejde-984	70	18	)	)	PUNCT
ejde-984	70	19	to	to	ADP
ejde-984	70	20	(	(	PUNCT
ejde-984	70	21	1.6	1.6	NUM
ejde-984	70	22	)	)	PUNCT
ejde-984	70	23	has	have	VERB
ejde-984	70	24	conservation	conservation	NOUN
ejde-984	70	25	of	of	ADP
ejde-984	70	26	mass	mass	NOUN
ejde-984	70	27	along	along	ADP
ejde-984	70	28	time	time	NOUN
ejde-984	70	29	,	,	PUNCT
ejde-984	70	30	therefore	therefore	ADV
ejde-984	70	31	this	this	DET
ejde-984	70	32	method	method	NOUN
ejde-984	70	33	is	be	AUX
ejde-984	70	34	extremely	extremely	ADV
ejde-984	70	35	significative	significative	ADJ
ejde-984	70	36	from	from	ADP
ejde-984	70	37	the	the	DET
ejde-984	70	38	physical	physical	ADJ
ejde-984	70	39	perspective	perspective	NOUN
ejde-984	70	40	.	.	PUNCT
ejde-984	71	1	now	now	ADV
ejde-984	71	2	mention	mention	VERB
ejde-984	71	3	some	some	DET
ejde-984	71	4	publications	publication	NOUN
ejde-984	71	5	that	that	PRON
ejde-984	71	6	consider	consider	VERB
ejde-984	71	7	normalized	normalize	VERB
ejde-984	71	8	solutions	solution	NOUN
ejde-984	71	9	to	to	ADP
ejde-984	71	10	(	(	PUNCT
ejde-984	71	11	1.1	1.1	NUM
ejde-984	71	12	)	)	PUNCT
ejde-984	71	13	.	.	PUNCT
ejde-984	72	1	li	li	PROPN
ejde-984	72	2	,	,	PUNCT
ejde-984	72	3	luo	luo	PROPN
ejde-984	72	4	and	and	CCONJ
ejde-984	72	5	yang	yang	PROPN
ejde-984	73	1	[	[	X
ejde-984	73	2	16	16	NUM
ejde-984	73	3	]	]	PUNCT
ejde-984	73	4	proved	prove	VERB
ejde-984	73	5	the	the	DET
ejde-984	73	6	existence	existence	NOUN
ejde-984	73	7	and	and	CCONJ
ejde-984	73	8	properties	property	NOUN
ejde-984	73	9	of	of	ADP
ejde-984	73	10	solutions	solution	NOUN
ejde-984	73	11	to	to	ADP
ejde-984	73	12	(	(	PUNCT
ejde-984	73	13	1.1	1.1	NUM
ejde-984	73	14	)	)	PUNCT
ejde-984	73	15	with	with	ADP
ejde-984	73	16	s	s	NOUN
ejde-984	73	17	=	=	SYM
ejde-984	73	18	1	1	NUM
ejde-984	73	19	under	under	ADP
ejde-984	74	1	normalized	normalize	VERB
ejde-984	74	2	constraint∫	constraint∫	PROPN
ejde-984	74	3	r3	r3	PROPN
ejde-984	74	4	|u|2dx	|u|2dx	VERB
ejde-984	74	5	=	=	SYM
ejde-984	74	6	c2	c2	PROPN
ejde-984	74	7	when	when	SCONJ
ejde-984	74	8	a	a	DET
ejde-984	74	9	,	,	PUNCT
ejde-984	74	10	b	b	X
ejde-984	74	11	>	>	X
ejde-984	74	12	0	0	NUM
ejde-984	74	13	and	and	CCONJ
ejde-984	74	14	µ	µ	X
ejde-984	74	15	>	>	X
ejde-984	74	16	0	0	NUM
ejde-984	74	17	,	,	PUNCT
ejde-984	74	18	namely	namely	ADV
ejde-984	74	19	the	the	DET
ejde-984	74	20	focusing	focus	VERB
ejde-984	74	21	case	case	NOUN
ejde-984	74	22	.	.	PUNCT
ejde-984	75	1	as	as	ADV
ejde-984	75	2	far	far	ADV
ejde-984	75	3	as	as	SCONJ
ejde-984	75	4	we	we	PRON
ejde-984	75	5	know	know	VERB
ejde-984	75	6	,	,	PUNCT
ejde-984	75	7	the	the	DET
ejde-984	75	8	defocusing	defocuse	VERB
ejde-984	75	9	case	case	NOUN
ejde-984	75	10	of	of	ADP
ejde-984	75	11	problem	problem	NOUN
ejde-984	75	12	(	(	PUNCT
ejde-984	75	13	1.1	1.1	NUM
ejde-984	75	14	)	)	PUNCT
ejde-984	75	15	with	with	ADP
ejde-984	75	16	the	the	DET
ejde-984	75	17	condition	condition	NOUN
ejde-984	75	18	(	(	PUNCT
ejde-984	75	19	1.2	1.2	NUM
ejde-984	75	20	)	)	PUNCT
ejde-984	75	21	was	be	AUX
ejde-984	75	22	mainly	mainly	ADV
ejde-984	75	23	studied	study	VERB
ejde-984	75	24	by	by	ADP
ejde-984	75	25	soave	soave	PROPN
ejde-984	75	26	[	[	X
ejde-984	75	27	24	24	NUM
ejde-984	75	28	]	]	PUNCT
ejde-984	75	29	with	with	ADP
ejde-984	75	30	b	b	PROPN
ejde-984	75	31	=	=	SYM
ejde-984	75	32	0	0	PROPN
ejde-984	75	33	.	.	PUNCT
ejde-984	76	1	the	the	DET
ejde-984	76	2	situation	situation	NOUN
ejde-984	76	3	b	b	PROPN
ejde-984	76	4	>	>	X
ejde-984	76	5	0	0	NUM
ejde-984	76	6	,	,	PUNCT
ejde-984	76	7	µ	µ	X
ejde-984	76	8	<	<	X
ejde-984	76	9	0	0	NUM
ejde-984	76	10	and	and	CCONJ
ejde-984	76	11	s	s	X
ejde-984	76	12	=	=	SYM
ejde-984	76	13	1	1	NUM
ejde-984	76	14	was	be	AUX
ejde-984	76	15	studied	study	VERB
ejde-984	76	16	in	in	ADP
ejde-984	76	17	[	[	X
ejde-984	76	18	3	3	NUM
ejde-984	76	19	]	]	PUNCT
ejde-984	76	20	.	.	PUNCT
ejde-984	77	1	the	the	DET
ejde-984	77	2	defocusing	defocuse	VERB
ejde-984	77	3	case	case	NOUN
ejde-984	77	4	of	of	ADP
ejde-984	77	5	fractional	fractional	PROPN
ejde-984	77	6	kirchhoff	kirchhoff	NOUN
ejde-984	77	7	equation	equation	NOUN
ejde-984	77	8	was	be	AUX
ejde-984	77	9	part	part	NOUN
ejde-984	77	10	of	of	ADP
ejde-984	77	11	ding	ding	NOUN
ejde-984	77	12	’s	’s	PART
ejde-984	77	13	result	result	NOUN
ejde-984	77	14	[	[	X
ejde-984	77	15	8	8	NUM
ejde-984	77	16	]	]	PUNCT
ejde-984	77	17	.	.	PUNCT
ejde-984	78	1	we	we	PRON
ejde-984	78	2	further	far	ADV
ejde-984	78	3	extended	extend	VERB
ejde-984	78	4	his	his	PRON
ejde-984	78	5	results	result	NOUN
ejde-984	78	6	(	(	PUNCT
ejde-984	78	7	see	see	VERB
ejde-984	78	8	theorem	theorem	ADJ
ejde-984	78	9	2.8	2.8	NUM
ejde-984	78	10	and	and	CCONJ
ejde-984	78	11	2.10	2.10	NUM
ejde-984	78	12	)	)	PUNCT
ejde-984	78	13	.	.	PUNCT
ejde-984	79	1	in	in	ADP
ejde-984	79	2	this	this	DET
ejde-984	79	3	paper	paper	NOUN
ejde-984	79	4	,	,	PUNCT
ejde-984	79	5	we	we	PRON
ejde-984	79	6	take	take	VERB
ejde-984	79	7	the	the	DET
ejde-984	79	8	case	case	NOUN
ejde-984	79	9	of	of	ADP
ejde-984	79	10	b	b	PROPN
ejde-984	79	11	>	>	X
ejde-984	79	12	0	0	NUM
ejde-984	79	13	,	,	PUNCT
ejde-984	79	14	µ	µ	X
ejde-984	79	15	<	<	X
ejde-984	79	16	0	0	NUM
ejde-984	79	17	and	and	CCONJ
ejde-984	79	18	s	s	PROPN
ejde-984	79	19	∈	∈	PROPN
ejde-984	79	20	(	(	PUNCT
ejde-984	79	21	0	0	NUM
ejde-984	79	22	,	,	PUNCT
ejde-984	79	23	1	1	NUM
ejde-984	79	24	)	)	PUNCT
ejde-984	79	25	into	into	ADP
ejde-984	79	26	consideration	consideration	NOUN
ejde-984	79	27	.	.	PUNCT
ejde-984	80	1	before	before	ADP
ejde-984	80	2	presenting	present	VERB
ejde-984	80	3	the	the	DET
ejde-984	80	4	main	main	ADJ
ejde-984	80	5	results	result	NOUN
ejde-984	80	6	of	of	ADP
ejde-984	80	7	our	our	PRON
ejde-984	80	8	paper	paper	NOUN
ejde-984	80	9	,	,	PUNCT
ejde-984	80	10	let	let	VERB
ejde-984	80	11	us	we	PRON
ejde-984	80	12	first	first	ADV
ejde-984	80	13	recall	recall	VERB
ejde-984	80	14	that	that	SCONJ
ejde-984	80	15	the	the	DET
ejde-984	80	16	fractional	fractional	ADJ
ejde-984	80	17	sobolev	sobolev	NOUN
ejde-984	80	18	space	space	NOUN
ejde-984	80	19	hs(r3	hs(r3	PUNCT
ejde-984	80	20	)	)	PUNCT
ejde-984	80	21	can	can	AUX
ejde-984	80	22	be	be	AUX
ejde-984	80	23	defined	define	VERB
ejde-984	80	24	as	as	SCONJ
ejde-984	80	25	follows	follow	VERB
ejde-984	80	26	:	:	PUNCT
ejde-984	80	27	hs(r3	hs(r3	NUM
ejde-984	80	28	)	)	PUNCT
ejde-984	81	1	=	=	PRON
ejde-984	81	2	{	{	PUNCT
ejde-984	81	3	u	u	NOUN
ejde-984	81	4	∈	∈	NOUN
ejde-984	81	5	l2(r3	l2(r3	NOUN
ejde-984	81	6	)	)	PUNCT
ejde-984	81	7	:	:	PUNCT
ejde-984	81	8	∫	∫	PROPN
ejde-984	81	9	r3	r3	PROPN
ejde-984	81	10	|(−∆	|(−∆	PROPN
ejde-984	81	11	)	)	PUNCT
ejde-984	81	12	s	s	PART
ejde-984	81	13	2u|2dx	2u|2dx	NUM
ejde-984	81	14	<	<	X
ejde-984	82	1	+	+	NOUN
ejde-984	82	2	∞	∞	NUM
ejde-984	82	3	}	}	PUNCT
ejde-984	82	4	with	with	ADP
ejde-984	82	5	the	the	DET
ejde-984	82	6	norm	norm	NOUN
ejde-984	82	7	∥u∥2	∥u∥2	NOUN
ejde-984	82	8	=	=	SYM
ejde-984	82	9	∫	∫	PROPN
ejde-984	82	10	r3	r3	PROPN
ejde-984	82	11	(	(	PUNCT
ejde-984	82	12	|(−∆)s/2u|2	|(−∆)s/2u|2	PROPN
ejde-984	82	13	+	+	NUM
ejde-984	82	14	|u|2)dx	|u|2)dx	NOUN
ejde-984	82	15	,	,	PUNCT
ejde-984	82	16	where	where	SCONJ
ejde-984	82	17	∫	∫	PROPN
ejde-984	82	18	r3	r3	PROPN
ejde-984	82	19	|(−∆	|(−∆	PROPN
ejde-984	82	20	)	)	PUNCT
ejde-984	82	21	s	s	PART
ejde-984	82	22	2u|2dx	2u|2dx	NUM
ejde-984	82	23	=	=	PUNCT
ejde-984	83	1	∫∫	∫∫	ADV
ejde-984	83	2	r6	r6	ADJ
ejde-984	83	3	|u(x)−	|u(x)−	PROPN
ejde-984	83	4	u(y)|2	u(y)|2	PROPN
ejde-984	83	5	|x−	|x−	PROPN
ejde-984	83	6	y|3	y|3	PROPN
ejde-984	83	7	+	+	PROPN
ejde-984	83	8	2s	2s	PROPN
ejde-984	83	9	dxdy	dxdy	NOUN
ejde-984	83	10	.	.	PUNCT
ejde-984	84	1	also	also	ADV
ejde-984	84	2	we	we	PRON
ejde-984	84	3	define	define	VERB
ejde-984	84	4	hs	hs	INTJ
ejde-984	84	5	r	r	PROPN
ejde-984	84	6	(	(	PUNCT
ejde-984	84	7	r3	r3	PROPN
ejde-984	84	8	)	)	PUNCT
ejde-984	84	9	=	=	PRON
ejde-984	84	10	{	{	PUNCT
ejde-984	84	11	u	u	NOUN
ejde-984	84	12	∈	∈	PROPN
ejde-984	84	13	hs(r3	hs(r3	NUM
ejde-984	84	14	)	)	PUNCT
ejde-984	84	15	:	:	PUNCT
ejde-984	84	16	u(x	u(x	PROPN
ejde-984	84	17	)	)	PUNCT
ejde-984	84	18	=	=	SYM
ejde-984	84	19	u(|x|	u(|x|	PROPN
ejde-984	84	20	)	)	PUNCT
ejde-984	84	21	,	,	PUNCT
ejde-984	84	22	x	x	PUNCT
ejde-984	84	23	∈	∈	PROPN
ejde-984	84	24	r3	r3	PROPN
ejde-984	84	25	}	}	PUNCT
ejde-984	84	26	.	.	PUNCT
ejde-984	85	1	in	in	ADP
ejde-984	85	2	this	this	DET
ejde-984	85	3	paper	paper	NOUN
ejde-984	85	4	,	,	PUNCT
ejde-984	85	5	we	we	PRON
ejde-984	85	6	denote	denote	VERB
ejde-984	85	7	by	by	ADP
ejde-984	85	8	|	|	ADV
ejde-984	85	9	·	·	PUNCT
ejde-984	85	10	|p	|p	VERB
ejde-984	85	11	the	the	DET
ejde-984	85	12	usual	usual	ADJ
ejde-984	85	13	norm	norm	NOUN
ejde-984	85	14	in	in	ADP
ejde-984	85	15	the	the	DET
ejde-984	85	16	lp(r3	lp(r3	ADJ
ejde-984	85	17	)	)	PUNCT
ejde-984	85	18	space	space	NOUN
ejde-984	85	19	,	,	PUNCT
ejde-984	85	20	sr	sr	PROPN
ejde-984	85	21	c	c	PROPN
ejde-984	85	22	=	=	SYM
ejde-984	85	23	sc	sc	PROPN
ejde-984	85	24	∩hs	∩hs	ADJ
ejde-984	85	25	r	r	NOUN
ejde-984	85	26	and	and	CCONJ
ejde-984	85	27	u∗	u∗	VERB
ejde-984	85	28	the	the	DET
ejde-984	85	29	symmetric	symmetric	ADJ
ejde-984	85	30	decreasing	decrease	VERB
ejde-984	85	31	rearrangement	rearrangement	NOUN
ejde-984	85	32	of	of	ADP
ejde-984	85	33	the	the	DET
ejde-984	85	34	modulus	modulus	NOUN
ejde-984	85	35	of	of	ADP
ejde-984	85	36	u	u	NOUN
ejde-984	85	37	∈	∈	PROPN
ejde-984	85	38	hs(r3	hs(r3	NUM
ejde-984	85	39	)	)	PUNCT
ejde-984	85	40	.	.	PUNCT
ejde-984	86	1	the	the	DET
ejde-984	86	2	functional	functional	ADJ
ejde-984	86	3	eµ	eµ	NOUN
ejde-984	86	4	:	:	PUNCT
ejde-984	86	5	sc	sc	PROPN
ejde-984	86	6	→	→	PUNCT
ejde-984	86	7	r	r	NOUN
ejde-984	86	8	is	be	AUX
ejde-984	86	9	regulated	regulate	VERB
ejde-984	86	10	as	as	ADP
ejde-984	86	11	eµ(u	eµ(u	NOUN
ejde-984	86	12	)	)	PUNCT
ejde-984	86	13	=	=	SYM
ejde-984	87	1	a	a	DET
ejde-984	87	2	2	2	NUM
ejde-984	87	3	∫	∫	NOUN
ejde-984	87	4	r3	r3	PROPN
ejde-984	87	5	|(−∆)s/2u|2dx+	|(−∆)s/2u|2dx+	PROPN
ejde-984	87	6	b	b	X
ejde-984	87	7	4	4	NUM
ejde-984	87	8	(	(	PUNCT
ejde-984	87	9	∫	∫	PROPN
ejde-984	87	10	r3	r3	PROPN
ejde-984	87	11	|(−∆)s/2u|2dx	|(−∆)s/2u|2dx	VERB
ejde-984	87	12	)	)	PUNCT
ejde-984	87	13	2	2	NUM
ejde-984	87	14	−	−	PROPN
ejde-984	87	15	µ	µ	PRON
ejde-984	87	16	q	q	NOUN
ejde-984	87	17	∫	∫	PROPN
ejde-984	87	18	r3	r3	PROPN
ejde-984	87	19	|u|qdx−	|u|qdx−	NOUN
ejde-984	87	20	1	1	NUM
ejde-984	87	21	p	p	NOUN
ejde-984	87	22	∫	∫	PROPN
ejde-984	87	23	r3	r3	PROPN
ejde-984	87	24	|u|pdx	|u|pdx	NUM
ejde-984	87	25	,	,	PUNCT
ejde-984	87	26	where	where	SCONJ
ejde-984	87	27	sc	sc	PROPN
ejde-984	87	28	is	be	AUX
ejde-984	87	29	the	the	DET
ejde-984	87	30	constraint	constraint	NOUN
ejde-984	87	31	space	space	NOUN
ejde-984	87	32	sc	sc	PROPN
ejde-984	87	33	=	=	PUNCT
ejde-984	87	34	{	{	PUNCT
ejde-984	87	35	u	u	NOUN
ejde-984	87	36	∈	∈	PROPN
ejde-984	87	37	hs(r3	hs(r3	NUM
ejde-984	87	38	)	)	PUNCT
ejde-984	87	39	:	:	PUNCT
ejde-984	88	1	∫	∫	PROPN
ejde-984	88	2	r3	r3	PROPN
ejde-984	88	3	|u|2dx	|u|2dx	VERB
ejde-984	88	4	=	=	SYM
ejde-984	88	5	c2	c2	PROPN
ejde-984	88	6	}	}	PUNCT
ejde-984	88	7	.	.	PUNCT
ejde-984	89	1	4	4	NUM
ejde-984	89	2	z.	z.	PROPN
ejde-984	89	3	guo	guo	PROPN
ejde-984	89	4	,	,	PUNCT
ejde-984	89	5	t.	t.	PROPN
ejde-984	89	6	zhang	zhang	PROPN
ejde-984	89	7	ejde-2025/75	ejde-2025/75	NOUN
ejde-984	89	8	it	it	PRON
ejde-984	89	9	is	be	AUX
ejde-984	89	10	easy	easy	ADJ
ejde-984	89	11	to	to	PART
ejde-984	89	12	check	check	VERB
ejde-984	89	13	that	that	DET
ejde-984	89	14	eµ	eµ	NOUN
ejde-984	89	15	∈	∈	PROPN
ejde-984	89	16	c1(r3,r	c1(r3,r	ADJ
ejde-984	89	17	)	)	PUNCT
ejde-984	89	18	.	.	PUNCT
ejde-984	90	1	thus	thus	ADV
ejde-984	90	2	,	,	PUNCT
ejde-984	90	3	the	the	DET
ejde-984	90	4	weak	weak	ADJ
ejde-984	90	5	solutions	solution	NOUN
ejde-984	90	6	of	of	ADP
ejde-984	90	7	(	(	PUNCT
ejde-984	90	8	1.1	1.1	NUM
ejde-984	90	9	)	)	PUNCT
ejde-984	90	10	under	under	ADP
ejde-984	90	11	the	the	DET
ejde-984	90	12	constraint	constraint	NOUN
ejde-984	90	13	(	(	PUNCT
ejde-984	90	14	1.2	1.2	NUM
ejde-984	90	15	)	)	PUNCT
ejde-984	90	16	can	can	AUX
ejde-984	90	17	be	be	AUX
ejde-984	90	18	obtained	obtain	VERB
ejde-984	90	19	as	as	ADP
ejde-984	90	20	critical	critical	ADJ
ejde-984	90	21	points	point	NOUN
ejde-984	90	22	of	of	ADP
ejde-984	90	23	the	the	DET
ejde-984	90	24	functional	functional	ADJ
ejde-984	90	25	eµ.	eµ.	NOUN
ejde-984	90	26	we	we	PRON
ejde-984	90	27	can	can	AUX
ejde-984	90	28	easily	easily	ADV
ejde-984	90	29	prove	prove	VERB
ejde-984	90	30	that	that	SCONJ
ejde-984	90	31	,	,	PUNCT
ejde-984	90	32	if	if	SCONJ
ejde-984	90	33	u	u	PROPN
ejde-984	90	34	∈	∈	PROPN
ejde-984	90	35	hs(r3	hs(r3	PRON
ejde-984	90	36	)	)	PUNCT
ejde-984	90	37	is	be	AUX
ejde-984	90	38	a	a	DET
ejde-984	90	39	weak	weak	ADJ
ejde-984	90	40	solution	solution	NOUN
ejde-984	90	41	of	of	ADP
ejde-984	90	42	(	(	PUNCT
ejde-984	90	43	1.1	1.1	NUM
ejde-984	90	44	)	)	PUNCT
ejde-984	90	45	,	,	PUNCT
ejde-984	90	46	then	then	ADV
ejde-984	90	47	we	we	PRON
ejde-984	90	48	have	have	VERB
ejde-984	90	49	the	the	DET
ejde-984	90	50	pohožaev	pohožaev	NOUN
ejde-984	90	51	identity	identity	NOUN
ejde-984	90	52	pµ(u	pµ(u	NOUN
ejde-984	90	53	)	)	PUNCT
ejde-984	90	54	:	:	PUNCT
ejde-984	91	1	=	=	SYM
ejde-984	91	2	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	91	3	+	+	NUM
ejde-984	91	4	b|(−∆)s/2u|42	b|(−∆)s/2u|42	PROPN
ejde-984	91	5	−	−	NOUN
ejde-984	91	6	µδs	µδ	NOUN
ejde-984	91	7	,	,	PUNCT
ejde-984	91	8	q|u|qq	q|u|qq	NOUN
ejde-984	91	9	−	−	NOUN
ejde-984	91	10	δs	δs	NOUN
ejde-984	91	11	,	,	PUNCT
ejde-984	91	12	p|u|pp	p|u|pp	PROPN
ejde-984	91	13	=	=	SYM
ejde-984	91	14	0	0	PROPN
ejde-984	91	15	,	,	PUNCT
ejde-984	91	16	where	where	SCONJ
ejde-984	91	17	δs	δs	NOUN
ejde-984	91	18	,	,	PUNCT
ejde-984	91	19	p	p	NOUN
ejde-984	91	20	=	=	NOUN
ejde-984	91	21	3(p−2	3(p−2	NOUN
ejde-984	91	22	)	)	PUNCT
ejde-984	91	23	2sp	2sp	NOUN
ejde-984	91	24	,	,	PUNCT
ejde-984	91	25	δs	δs	NOUN
ejde-984	91	26	,	,	PUNCT
ejde-984	91	27	q	q	NOUN
ejde-984	91	28	=	=	SYM
ejde-984	91	29	3(q−2	3(q−2	NUM
ejde-984	91	30	)	)	PUNCT
ejde-984	91	31	2sq	2sq	NOUN
ejde-984	91	32	.	.	PUNCT
ejde-984	92	1	therefore	therefore	ADV
ejde-984	92	2	,	,	PUNCT
ejde-984	92	3	the	the	DET
ejde-984	92	4	critical	critical	ADJ
ejde-984	92	5	points	point	NOUN
ejde-984	92	6	of	of	ADP
ejde-984	92	7	eµ	eµ	NOUN
ejde-984	92	8	is	be	AUX
ejde-984	92	9	certainly	certainly	ADV
ejde-984	92	10	contained	contain	VERB
ejde-984	92	11	in	in	ADP
ejde-984	92	12	the	the	DET
ejde-984	92	13	pohožaev	pohožaev	NOUN
ejde-984	92	14	set	set	VERB
ejde-984	92	15	pc,µ	pc,µ	NOUN
ejde-984	93	1	=	=	SYM
ejde-984	93	2	{	{	PUNCT
ejde-984	93	3	u	u	PROPN
ejde-984	93	4	∈	∈	PROPN
ejde-984	93	5	sc	sc	PROPN
ejde-984	93	6	:	:	PUNCT
ejde-984	93	7	pµ(u	pµ(u	X
ejde-984	93	8	)	)	PUNCT
ejde-984	93	9	=	=	SYM
ejde-984	93	10	0	0	X
ejde-984	93	11	}	}	PUNCT
ejde-984	93	12	(	(	PUNCT
ejde-984	93	13	see	see	VERB
ejde-984	93	14	lemma	lemma	PROPN
ejde-984	93	15	2.3	2.3	NUM
ejde-984	93	16	for	for	ADP
ejde-984	93	17	a	a	DET
ejde-984	93	18	proof	proof	NOUN
ejde-984	93	19	)	)	PUNCT
ejde-984	93	20	.	.	PUNCT
ejde-984	94	1	by	by	ADP
ejde-984	94	2	simple	simple	ADJ
ejde-984	94	3	calculations	calculation	NOUN
ejde-984	94	4	,	,	PUNCT
ejde-984	94	5	we	we	PRON
ejde-984	94	6	can	can	AUX
ejde-984	94	7	show	show	VERB
ejde-984	94	8	that	that	DET
ejde-984	94	9	δs	δs	NOUN
ejde-984	94	10	,	,	PUNCT
ejde-984	94	11	p	p	NOUN
ejde-984	94	12	∈	∈	PROPN
ejde-984	94	13	(	(	PUNCT
ejde-984	94	14	0	0	NUM
ejde-984	94	15	,	,	PUNCT
ejde-984	94	16	1	1	NUM
ejde-984	94	17	)	)	PUNCT
ejde-984	94	18	(	(	PUNCT
ejde-984	94	19	when	when	SCONJ
ejde-984	94	20	2	2	NUM
ejde-984	94	21	<	<	X
ejde-984	94	22	p	p	X
ejde-984	94	23	<	<	X
ejde-984	94	24	2∗s	2∗s	NUM
ejde-984	94	25	)	)	PUNCT
ejde-984	94	26	and	and	CCONJ
ejde-984	94	27	qδs	qδs	PROPN
ejde-984	94	28	,	,	PUNCT
ejde-984	94	29	q	q	PROPN
ejde-984	94	30	≤	≤	NUM
ejde-984	94	31	4	4	NUM
ejde-984	94	32	<	<	X
ejde-984	94	33	pδs	pδs	PROPN
ejde-984	94	34	,	,	PUNCT
ejde-984	94	35	p	p	X
ejde-984	94	36	,	,	PUNCT
ejde-984	94	37	if	if	SCONJ
ejde-984	94	38	2	2	NUM
ejde-984	94	39	<	<	X
ejde-984	94	40	q	q	X
ejde-984	94	41	≤	≤	NUM
ejde-984	94	42	2	2	NUM
ejde-984	94	43	+	+	NUM
ejde-984	94	44	8s	8s	NUM
ejde-984	94	45	3	3	NUM
ejde-984	94	46	<	<	X
ejde-984	94	47	p	p	X
ejde-984	94	48	<	<	X
ejde-984	94	49	6	6	NUM
ejde-984	94	50	3−	3−	NUM
ejde-984	94	51	2s	2s	NOUN
ejde-984	94	52	,	,	PUNCT
ejde-984	94	53	where	where	SCONJ
ejde-984	94	54	2	2	NUM
ejde-984	94	55	+	+	NOUN
ejde-984	94	56	8s	8	NOUN
ejde-984	94	57	3	3	NUM
ejde-984	94	58	is	be	AUX
ejde-984	94	59	the	the	DET
ejde-984	94	60	mass	mass	ADJ
ejde-984	94	61	critical	critical	ADJ
ejde-984	94	62	exponent	exponent	NOUN
ejde-984	94	63	for	for	ADP
ejde-984	94	64	the	the	DET
ejde-984	94	65	kirchhoff	kirchhoff	NOUN
ejde-984	94	66	constrained	constrain	VERB
ejde-984	94	67	minimization	minimization	NOUN
ejde-984	94	68	problem	problem	NOUN
ejde-984	94	69	,	,	PUNCT
ejde-984	94	70	namely	namely	ADV
ejde-984	94	71	,	,	PUNCT
ejde-984	94	72	2	2	NUM
ejde-984	94	73	+	+	NUM
ejde-984	94	74	8s	8	NOUN
ejde-984	94	75	n	n	VERB
ejde-984	94	76	is	be	AUX
ejde-984	94	77	the	the	DET
ejde-984	94	78	threshold	threshold	NOUN
ejde-984	94	79	exponent	exponent	NOUN
ejde-984	94	80	for	for	ADP
ejde-984	94	81	many	many	ADJ
ejde-984	94	82	dynamic	dynamic	ADJ
ejde-984	94	83	problems	problem	NOUN
ejde-984	94	84	,	,	PUNCT
ejde-984	94	85	see	see	VERB
ejde-984	94	86	[	[	X
ejde-984	94	87	28	28	NUM
ejde-984	94	88	]	]	PUNCT
ejde-984	94	89	for	for	ADP
ejde-984	94	90	more	more	ADJ
ejde-984	94	91	information	information	NOUN
ejde-984	94	92	.	.	PUNCT
ejde-984	95	1	in	in	ADP
ejde-984	95	2	this	this	DET
ejde-984	95	3	article	article	NOUN
ejde-984	95	4	,	,	PUNCT
ejde-984	95	5	we	we	PRON
ejde-984	95	6	will	will	AUX
ejde-984	95	7	be	be	AUX
ejde-984	95	8	concerned	concern	VERB
ejde-984	95	9	with	with	ADP
ejde-984	95	10	ground	ground	NOUN
ejde-984	95	11	state	state	NOUN
ejde-984	95	12	solutions	solution	NOUN
ejde-984	95	13	,	,	PUNCT
ejde-984	95	14	wihci	wihci	NOUN
ejde-984	95	15	are	be	AUX
ejde-984	95	16	defined	define	VERB
ejde-984	95	17	as	as	ADP
ejde-984	95	18	follows	follow	VERB
ejde-984	95	19	.	.	PUNCT
ejde-984	96	1	definition	definition	NOUN
ejde-984	96	2	1.1	1.1	NUM
ejde-984	96	3	.	.	PUNCT
ejde-984	97	1	we	we	PRON
ejde-984	97	2	say	say	VERB
ejde-984	97	3	that	that	SCONJ
ejde-984	97	4	ũ	ũ	PROPN
ejde-984	97	5	is	be	AUX
ejde-984	97	6	a	a	DET
ejde-984	97	7	ground	ground	NOUN
ejde-984	97	8	state	state	NOUN
ejde-984	97	9	of	of	ADP
ejde-984	97	10	(	(	PUNCT
ejde-984	97	11	1.1	1.1	NUM
ejde-984	97	12	)	)	PUNCT
ejde-984	97	13	on	on	ADP
ejde-984	97	14	sc	sc	PROPN
ejde-984	97	15	if	if	SCONJ
ejde-984	97	16	it	it	PRON
ejde-984	97	17	is	be	AUX
ejde-984	97	18	a	a	DET
ejde-984	97	19	solution	solution	NOUN
ejde-984	97	20	to	to	ADP
ejde-984	97	21	(	(	PUNCT
ejde-984	97	22	1.1	1.1	NUM
ejde-984	97	23	)	)	PUNCT
ejde-984	97	24	having	have	VERB
ejde-984	97	25	minimal	minimal	ADJ
ejde-984	97	26	energy	energy	NOUN
ejde-984	97	27	among	among	ADP
ejde-984	97	28	all	all	DET
ejde-984	97	29	the	the	DET
ejde-984	97	30	solutions	solution	NOUN
ejde-984	97	31	which	which	PRON
ejde-984	97	32	belong	belong	VERB
ejde-984	97	33	to	to	ADP
ejde-984	97	34	sc	sc	PROPN
ejde-984	97	35	:	:	PUNCT
ejde-984	97	36	deµ|sc	deµ|sc	PROPN
ejde-984	97	37	(	(	PUNCT
ejde-984	97	38	ũ	ũ	PROPN
ejde-984	97	39	)	)	PUNCT
ejde-984	97	40	=	=	SYM
ejde-984	97	41	0	0	NUM
ejde-984	97	42	and	and	CCONJ
ejde-984	97	43	eµ(ũ	eµ(ũ	NOUN
ejde-984	97	44	)	)	PUNCT
ejde-984	97	45	=	=	SYM
ejde-984	97	46	inf{eµ(u	inf{eµ(u	NOUN
ejde-984	97	47	)	)	PUNCT
ejde-984	97	48	:	:	PUNCT
ejde-984	97	49	deµ|sc	deµ|sc	PROPN
ejde-984	97	50	(	(	PUNCT
ejde-984	97	51	u	u	NOUN
ejde-984	97	52	)	)	PUNCT
ejde-984	97	53	=	=	SYM
ejde-984	97	54	0	0	NUM
ejde-984	97	55	,	,	PUNCT
ejde-984	97	56	and	and	CCONJ
ejde-984	97	57	u	u	PROPN
ejde-984	97	58	∈	∈	PROPN
ejde-984	97	59	sc	sc	PROPN
ejde-984	97	60	}	}	PUNCT
ejde-984	97	61	.	.	PUNCT
ejde-984	98	1	and	and	CCONJ
ejde-984	98	2	the	the	DET
ejde-984	98	3	set	set	NOUN
ejde-984	98	4	of	of	ADP
ejde-984	98	5	ground	ground	NOUN
ejde-984	98	6	states	state	NOUN
ejde-984	98	7	will	will	AUX
ejde-984	98	8	be	be	AUX
ejde-984	98	9	denoted	denote	VERB
ejde-984	98	10	by	by	ADP
ejde-984	98	11	zc,µ.	zc,µ.	PROPN
ejde-984	98	12	to	to	PART
ejde-984	98	13	go	go	VERB
ejde-984	98	14	over	over	ADP
ejde-984	98	15	the	the	DET
ejde-984	98	16	obstacles	obstacle	NOUN
ejde-984	98	17	about	about	ADP
ejde-984	98	18	the	the	DET
ejde-984	98	19	convergence	convergence	NOUN
ejde-984	98	20	of	of	ADP
ejde-984	98	21	the	the	DET
ejde-984	98	22	palais	palais	PROPN
ejde-984	98	23	smale	smale	PROPN
ejde-984	98	24	(	(	PUNCT
ejde-984	98	25	hereinafter	hereinafter	NOUN
ejde-984	98	26	referred	refer	VERB
ejde-984	98	27	to	to	ADP
ejde-984	98	28	as	as	ADP
ejde-984	98	29	ps	ps	NOUN
ejde-984	98	30	)	)	PUNCT
ejde-984	98	31	sequence	sequence	NOUN
ejde-984	98	32	of	of	ADP
ejde-984	98	33	eµ	eµ	NOUN
ejde-984	98	34	,	,	PUNCT
ejde-984	98	35	we	we	PRON
ejde-984	98	36	build	build	VERB
ejde-984	98	37	the	the	DET
ejde-984	98	38	sequence	sequence	NOUN
ejde-984	98	39	{	{	PUNCT
ejde-984	98	40	un}n∈n	un}n∈n	NOUN
ejde-984	98	41	satisfying	satisfying	PROPN
ejde-984	98	42	pµ(un	pµ(un	PROPN
ejde-984	98	43	)	)	PUNCT
ejde-984	98	44	→	→	SYM
ejde-984	98	45	0	0	NUM
ejde-984	98	46	,	,	PUNCT
ejde-984	98	47	when	when	SCONJ
ejde-984	98	48	n→	n→	PUNCT
ejde-984	98	49	∞.	∞.	PROPN
ejde-984	98	50	thanks	thank	NOUN
ejde-984	98	51	to	to	ADP
ejde-984	98	52	the	the	DET
ejde-984	98	53	normalized	normalized	ADJ
ejde-984	98	54	condition	condition	NOUN
ejde-984	98	55	(	(	PUNCT
ejde-984	98	56	1.2	1.2	NUM
ejde-984	98	57	)	)	PUNCT
ejde-984	98	58	,	,	PUNCT
ejde-984	98	59	we	we	PRON
ejde-984	98	60	define	define	VERB
ejde-984	98	61	the	the	DET
ejde-984	98	62	dilations	dilation	NOUN
ejde-984	98	63	(	(	PUNCT
ejde-984	98	64	ω	ω	NOUN
ejde-984	98	65	∗	∗	NOUN
ejde-984	98	66	u)(x	u)(x	PROPN
ejde-984	98	67	)	)	PUNCT
ejde-984	98	68	=	=	SYM
ejde-984	98	69	e3ω/2u(eωx	e3ω/2u(eωx	PROPN
ejde-984	98	70	)	)	PUNCT
ejde-984	98	71	,	,	PUNCT
ejde-984	98	72	a.e	a.e	PROPN
ejde-984	98	73	.	.	PROPN
ejde-984	98	74	in	in	ADP
ejde-984	98	75	r3	r3	PROPN
ejde-984	98	76	which	which	PRON
ejde-984	98	77	retain	retain	VERB
ejde-984	98	78	the	the	DET
ejde-984	98	79	l2	l2	NOUN
ejde-984	98	80	norm	norm	NOUN
ejde-984	98	81	,	,	PUNCT
ejde-984	98	82	more	more	ADJ
ejde-984	98	83	precisely,∫	precisely,∫	NOUN
ejde-984	98	84	r3	r3	PROPN
ejde-984	98	85	(	(	PUNCT
ejde-984	98	86	ω	ω	NOUN
ejde-984	98	87	∗	∗	X
ejde-984	98	88	u)2dx	u)2dx	NOUN
ejde-984	99	1	=	=	SYM
ejde-984	99	2	∫	∫	PROPN
ejde-984	99	3	r3	r3	PROPN
ejde-984	99	4	u2dx	u2dx	PROPN
ejde-984	99	5	,	,	PUNCT
ejde-984	99	6	and	and	CCONJ
ejde-984	99	7	it	it	PRON
ejde-984	99	8	is	be	AUX
ejde-984	99	9	a	a	DET
ejde-984	99	10	continuous	continuous	ADJ
ejde-984	99	11	map	map	NOUN
ejde-984	99	12	from	from	ADP
ejde-984	99	13	r×hs(r3	r×hs(r3	NOUN
ejde-984	99	14	)	)	PUNCT
ejde-984	99	15	into	into	ADP
ejde-984	99	16	hs(r3	hs(r3	NUM
ejde-984	99	17	)	)	PUNCT
ejde-984	99	18	.	.	PUNCT
ejde-984	100	1	furthermore	furthermore	ADV
ejde-984	100	2	,	,	PUNCT
ejde-984	100	3	we	we	PRON
ejde-984	100	4	introduce	introduce	VERB
ejde-984	100	5	the	the	DET
ejde-984	100	6	following	follow	VERB
ejde-984	100	7	fiber	fiber	NOUN
ejde-984	100	8	map	map	NOUN
ejde-984	100	9	jµ	jµ	PROPN
ejde-984	100	10	u	u	NOUN
ejde-984	100	11	(	(	PUNCT
ejde-984	100	12	ω	ω	NOUN
ejde-984	100	13	)	)	PUNCT
ejde-984	100	14	:	:	PUNCT
ejde-984	101	1	=	=	SYM
ejde-984	101	2	eµ(ω	eµ(ω	NUM
ejde-984	101	3	∗	∗	NOUN
ejde-984	101	4	u	u	NOUN
ejde-984	101	5	)	)	PUNCT
ejde-984	101	6	=	=	SYM
ejde-984	101	7	ae2sω	ae2sω	PROPN
ejde-984	101	8	2	2	NUM
ejde-984	101	9	|(−∆)s/2u|22	|(−∆)s/2u|22	NUM
ejde-984	101	10	+	+	CCONJ
ejde-984	101	11	be4sω	be4sω	NUM
ejde-984	101	12	4	4	NUM
ejde-984	101	13	|(−∆)s/2u|42	|(−∆)s/2u|42	NUM
ejde-984	101	14	−	−	PROPN
ejde-984	101	15	µ	µ	DET
ejde-984	101	16	eqδs	eqδs	NOUN
ejde-984	101	17	,	,	PUNCT
ejde-984	101	18	qsω	qsω	PROPN
ejde-984	101	19	q	q	NOUN
ejde-984	101	20	|u|qq	|u|qq	NOUN
ejde-984	101	21	−	−	NOUN
ejde-984	101	22	epδs	epδs	ADJ
ejde-984	101	23	,	,	PUNCT
ejde-984	101	24	psω	psω	ADJ
ejde-984	101	25	p	p	PROPN
ejde-984	101	26	|u|pp	|u|pp	NOUN
ejde-984	101	27	,	,	PUNCT
ejde-984	101	28	where	where	SCONJ
ejde-984	101	29	δs	δs	NOUN
ejde-984	101	30	,	,	PUNCT
ejde-984	101	31	q	q	NOUN
ejde-984	101	32	=	=	SYM
ejde-984	101	33	3(q−2	3(q−2	NUM
ejde-984	101	34	)	)	PUNCT
ejde-984	101	35	2sq	2sq	NOUN
ejde-984	101	36	and	and	CCONJ
ejde-984	101	37	δs	δs	NOUN
ejde-984	101	38	,	,	PUNCT
ejde-984	101	39	p	p	NOUN
ejde-984	101	40	=	=	NOUN
ejde-984	101	41	3(p−2	3(p−2	NOUN
ejde-984	101	42	)	)	PUNCT
ejde-984	101	43	2sp	2sp	NOUN
ejde-984	101	44	.	.	PUNCT
ejde-984	102	1	by	by	ADP
ejde-984	102	2	using	use	VERB
ejde-984	102	3	the	the	DET
ejde-984	102	4	functional	functional	ADJ
ejde-984	102	5	jµ	jµ	PROPN
ejde-984	102	6	u	u	NOUN
ejde-984	102	7	,	,	PUNCT
ejde-984	102	8	we	we	PRON
ejde-984	102	9	cast	cast	VERB
ejde-984	102	10	a	a	DET
ejde-984	102	11	function	function	NOUN
ejde-984	102	12	into	into	ADP
ejde-984	102	13	the	the	DET
ejde-984	102	14	pohožaev	pohožaev	PROPN
ejde-984	102	15	set	set	PROPN
ejde-984	102	16	.	.	PUNCT
ejde-984	103	1	soave	soave	PROPN
ejde-984	104	1	[	[	X
ejde-984	104	2	24	24	NUM
ejde-984	104	3	]	]	PUNCT
ejde-984	104	4	and	and	CCONJ
ejde-984	104	5	li	li	PROPN
ejde-984	104	6	,	,	PUNCT
ejde-984	104	7	luo	luo	PROPN
ejde-984	104	8	and	and	CCONJ
ejde-984	104	9	yang	yang	PROPN
ejde-984	105	1	[	[	X
ejde-984	105	2	16	16	NUM
ejde-984	105	3	]	]	PUNCT
ejde-984	105	4	have	have	AUX
ejde-984	105	5	also	also	ADV
ejde-984	105	6	applied	apply	VERB
ejde-984	105	7	such	such	ADJ
ejde-984	105	8	idea	idea	NOUN
ejde-984	105	9	.	.	PUNCT
ejde-984	106	1	the	the	DET
ejde-984	106	2	main	main	ADJ
ejde-984	106	3	results	result	NOUN
ejde-984	106	4	of	of	ADP
ejde-984	106	5	this	this	DET
ejde-984	106	6	paper	paper	NOUN
ejde-984	106	7	are	be	AUX
ejde-984	106	8	organized	organize	VERB
ejde-984	106	9	as	as	SCONJ
ejde-984	106	10	follows	follow	VERB
ejde-984	106	11	:	:	PUNCT
ejde-984	106	12	•	•	INTJ
ejde-984	106	13	if	if	SCONJ
ejde-984	106	14	µ	µ	X
ejde-984	106	15	<	<	X
ejde-984	106	16	0	0	NUM
ejde-984	106	17	,	,	PUNCT
ejde-984	106	18	2	2	NUM
ejde-984	106	19	<	<	X
ejde-984	106	20	q	q	X
ejde-984	106	21	<	<	X
ejde-984	106	22	p	p	X
ejde-984	106	23	=	=	SYM
ejde-984	106	24	2	2	NUM
ejde-984	106	25	+	+	NUM
ejde-984	106	26	8s	8s	NUM
ejde-984	106	27	3	3	NUM
ejde-984	106	28	=	=	SYM
ejde-984	106	29	p̄	p̄	NOUN
ejde-984	106	30	,	,	PUNCT
ejde-984	106	31	we	we	PRON
ejde-984	106	32	prove	prove	VERB
ejde-984	106	33	that	that	SCONJ
ejde-984	106	34	(	(	PUNCT
ejde-984	106	35	1.1	1.1	NUM
ejde-984	106	36	)	)	PUNCT
ejde-984	106	37	under	under	ADP
ejde-984	106	38	the	the	DET
ejde-984	106	39	condition	condition	NOUN
ejde-984	106	40	(	(	PUNCT
ejde-984	106	41	1.2	1.2	NUM
ejde-984	106	42	)	)	PUNCT
ejde-984	106	43	does	do	AUX
ejde-984	106	44	not	not	PART
ejde-984	106	45	have	have	VERB
ejde-984	106	46	solution	solution	NOUN
ejde-984	106	47	.	.	PUNCT
ejde-984	107	1	•	•	NOUN
ejde-984	107	2	if	if	SCONJ
ejde-984	107	3	2	2	NUM
ejde-984	107	4	<	<	X
ejde-984	107	5	q	q	X
ejde-984	107	6	≤	≤	NUM
ejde-984	107	7	2	2	NUM
ejde-984	107	8	+	+	NUM
ejde-984	107	9	8s	8s	NUM
ejde-984	107	10	3	3	NUM
ejde-984	107	11	<	<	X
ejde-984	107	12	p	p	X
ejde-984	107	13	<	<	X
ejde-984	107	14	6	6	NUM
ejde-984	107	15	3−2s	3−2s	NUM
ejde-984	107	16	are	be	AUX
ejde-984	107	17	given	give	VERB
ejde-984	107	18	constants	constant	NOUN
ejde-984	107	19	and	and	CCONJ
ejde-984	107	20	µ	µ	PRON
ejde-984	107	21	<	<	X
ejde-984	107	22	0	0	NUM
ejde-984	107	23	satisifies	satisifie	NOUN
ejde-984	107	24	an	an	DET
ejde-984	107	25	additional	additional	ADJ
ejde-984	107	26	assumption	assumption	NOUN
ejde-984	107	27	,	,	PUNCT
ejde-984	107	28	we	we	PRON
ejde-984	107	29	prove	prove	VERB
ejde-984	107	30	that	that	SCONJ
ejde-984	107	31	there	there	PRON
ejde-984	107	32	exists	exist	VERB
ejde-984	107	33	λ	λ	X
ejde-984	107	34	<	<	X
ejde-984	107	35	0	0	NUM
ejde-984	107	36	such	such	ADJ
ejde-984	107	37	that	that	SCONJ
ejde-984	107	38	(	(	PUNCT
ejde-984	107	39	1.1	1.1	NUM
ejde-984	107	40	)	)	PUNCT
ejde-984	107	41	under	under	ADP
ejde-984	107	42	the	the	DET
ejde-984	107	43	condition	condition	NOUN
ejde-984	107	44	(	(	PUNCT
ejde-984	107	45	1.2	1.2	NUM
ejde-984	107	46	)	)	PUNCT
ejde-984	107	47	has	have	VERB
ejde-984	107	48	a	a	DET
ejde-984	107	49	solution	solution	NOUN
ejde-984	107	50	.	.	PUNCT
ejde-984	108	1	the	the	DET
ejde-984	108	2	solution	solution	NOUN
ejde-984	108	3	is	be	AUX
ejde-984	108	4	radially	radially	ADV
ejde-984	108	5	symmetric	symmetric	ADJ
ejde-984	108	6	,	,	PUNCT
ejde-984	108	7	and	and	CCONJ
ejde-984	108	8	is	be	AUX
ejde-984	108	9	a	a	DET
ejde-984	108	10	ground	ground	NOUN
ejde-984	108	11	state	state	NOUN
ejde-984	108	12	on	on	ADP
ejde-984	108	13	sc	sc	PROPN
ejde-984	108	14	.	.	PROPN
ejde-984	108	15	•	•	NOUN
ejde-984	108	16	if	if	SCONJ
ejde-984	108	17	2	2	NUM
ejde-984	108	18	<	<	X
ejde-984	108	19	q	q	X
ejde-984	108	20	≤	≤	NUM
ejde-984	108	21	2	2	NUM
ejde-984	108	22	+	+	NUM
ejde-984	108	23	8s	8s	NUM
ejde-984	108	24	3	3	NUM
ejde-984	108	25	<	<	X
ejde-984	108	26	p	p	X
ejde-984	108	27	<	<	X
ejde-984	108	28	6	6	NUM
ejde-984	108	29	3−2s	3−2s	NUM
ejde-984	108	30	are	be	AUX
ejde-984	108	31	given	give	VERB
ejde-984	108	32	constants	constant	NOUN
ejde-984	108	33	and	and	CCONJ
ejde-984	108	34	µ	µ	PRON
ejde-984	108	35	<	<	X
ejde-984	108	36	0	0	NUM
ejde-984	108	37	satisfies	satisfie	NOUN
ejde-984	108	38	an	an	DET
ejde-984	108	39	additional	additional	ADJ
ejde-984	108	40	assumption	assumption	NOUN
ejde-984	108	41	,	,	PUNCT
ejde-984	108	42	we	we	PRON
ejde-984	108	43	also	also	ADV
ejde-984	108	44	give	give	VERB
ejde-984	108	45	a	a	DET
ejde-984	108	46	characterization	characterization	NOUN
ejde-984	108	47	to	to	ADP
ejde-984	108	48	the	the	DET
ejde-984	108	49	set	set	NOUN
ejde-984	108	50	of	of	ADP
ejde-984	108	51	ground	ground	NOUN
ejde-984	108	52	states	state	NOUN
ejde-984	108	53	.	.	PUNCT
ejde-984	109	1	for	for	ADP
ejde-984	109	2	overcoming	overcome	VERB
ejde-984	109	3	some	some	DET
ejde-984	109	4	technical	technical	ADJ
ejde-984	109	5	difficulties	difficulty	NOUN
ejde-984	109	6	,	,	PUNCT
ejde-984	109	7	the	the	DET
ejde-984	109	8	main	main	ADJ
ejde-984	109	9	proof	proof	NOUN
ejde-984	109	10	of	of	ADP
ejde-984	109	11	our	our	PRON
ejde-984	109	12	results	result	NOUN
ejde-984	109	13	involves	involve	VERB
ejde-984	109	14	the	the	DET
ejde-984	109	15	techniques	technique	NOUN
ejde-984	109	16	used	use	VERB
ejde-984	109	17	by	by	ADP
ejde-984	109	18	soave	soave	PROPN
ejde-984	109	19	[	[	X
ejde-984	109	20	24	24	NUM
ejde-984	109	21	]	]	PUNCT
ejde-984	109	22	,	,	PUNCT
ejde-984	109	23	li	li	PROPN
ejde-984	109	24	,	,	PUNCT
ejde-984	109	25	luo	luo	PROPN
ejde-984	109	26	and	and	CCONJ
ejde-984	109	27	yang	yang	PROPN
ejde-984	110	1	[	[	X
ejde-984	110	2	16	16	NUM
ejde-984	110	3	]	]	PUNCT
ejde-984	110	4	.	.	PUNCT
ejde-984	111	1	this	this	DET
ejde-984	111	2	paper	paper	NOUN
ejde-984	111	3	is	be	AUX
ejde-984	111	4	organized	organize	VERB
ejde-984	111	5	as	as	SCONJ
ejde-984	111	6	follows	follow	VERB
ejde-984	111	7	.	.	PUNCT
ejde-984	112	1	in	in	ADP
ejde-984	112	2	section	section	NOUN
ejde-984	112	3	2	2	NUM
ejde-984	112	4	we	we	PRON
ejde-984	112	5	give	give	VERB
ejde-984	112	6	the	the	DET
ejde-984	112	7	notation	notation	NOUN
ejde-984	112	8	and	and	CCONJ
ejde-984	112	9	assumptions	assumption	NOUN
ejde-984	112	10	and	and	CCONJ
ejde-984	112	11	we	we	PRON
ejde-984	112	12	enunciate	enunciate	VERB
ejde-984	112	13	our	our	PRON
ejde-984	112	14	main	main	ADJ
ejde-984	112	15	results	result	NOUN
ejde-984	112	16	.	.	PUNCT
ejde-984	113	1	in	in	ADP
ejde-984	113	2	section	section	NOUN
ejde-984	113	3	3	3	NUM
ejde-984	113	4	we	we	PRON
ejde-984	113	5	prove	prove	VERB
ejde-984	113	6	some	some	DET
ejde-984	113	7	results	result	NOUN
ejde-984	113	8	concerning	concern	VERB
ejde-984	113	9	the	the	DET
ejde-984	113	10	subcritical	subcritical	ADJ
ejde-984	113	11	case	case	NOUN
ejde-984	113	12	(	(	PUNCT
ejde-984	113	13	namely	namely	ADV
ejde-984	113	14	,	,	PUNCT
ejde-984	113	15	the	the	DET
ejde-984	113	16	main	main	ADJ
ejde-984	113	17	results	result	NOUN
ejde-984	113	18	in	in	ADP
ejde-984	113	19	this	this	DET
ejde-984	113	20	paper	paper	NOUN
ejde-984	113	21	)	)	PUNCT
ejde-984	113	22	.	.	PUNCT
ejde-984	114	1	ejde-2025/75	ejde-2025/75	ADJ
ejde-984	114	2	normalized	normalize	VERB
ejde-984	114	3	solutions	solution	NOUN
ejde-984	114	4	of	of	ADP
ejde-984	114	5	fractional	fractional	PROPN
ejde-984	114	6	kirchhoff	kirchhoff	NOUN
ejde-984	114	7	equations	equation	NOUN
ejde-984	114	8	5	5	NUM
ejde-984	114	9	2	2	NUM
ejde-984	114	10	.	.	PUNCT
ejde-984	114	11	preliminaries	preliminary	NOUN
ejde-984	114	12	and	and	CCONJ
ejde-984	114	13	main	main	ADJ
ejde-984	114	14	results	result	NOUN
ejde-984	114	15	lemma	lemma	VERB
ejde-984	114	16	2.1	2.1	NUM
ejde-984	114	17	(	(	PUNCT
ejde-984	114	18	[	[	X
ejde-984	114	19	7	7	NUM
ejde-984	114	20	]	]	NUM
ejde-984	114	21	)	)	PUNCT
ejde-984	114	22	.	.	PUNCT
ejde-984	115	1	let	let	VERB
ejde-984	115	2	s	s	PRON
ejde-984	115	3	∈	∈	PROPN
ejde-984	115	4	(	(	PUNCT
ejde-984	115	5	0	0	NUM
ejde-984	115	6	,	,	PUNCT
ejde-984	115	7	1	1	NUM
ejde-984	115	8	)	)	PUNCT
ejde-984	115	9	and	and	CCONJ
ejde-984	115	10	p	p	NOUN
ejde-984	115	11	∈	∈	PROPN
ejde-984	116	1	[	[	X
ejde-984	116	2	1,+∞	1,+∞	NUM
ejde-984	116	3	)	)	PUNCT
ejde-984	116	4	be	be	VERB
ejde-984	116	5	such	such	ADJ
ejde-984	116	6	that	that	PRON
ejde-984	116	7	sp	sp	ADP
ejde-984	116	8	<	<	X
ejde-984	116	9	n	n	NOUN
ejde-984	116	10	.	.	PUNCT
ejde-984	117	1	then	then	ADV
ejde-984	117	2	,	,	PUNCT
ejde-984	117	3	there	there	PRON
ejde-984	117	4	exists	exist	VERB
ejde-984	117	5	a	a	DET
ejde-984	117	6	positive	positive	ADJ
ejde-984	117	7	constant	constant	ADJ
ejde-984	117	8	ss	ss	NOUN
ejde-984	117	9	=	=	PUNCT
ejde-984	117	10	ss(n	ss(n	PROPN
ejde-984	117	11	,	,	PUNCT
ejde-984	117	12	p	p	X
ejde-984	117	13	,	,	PUNCT
ejde-984	117	14	s	s	PART
ejde-984	117	15	)	)	PUNCT
ejde-984	117	16	such	such	ADJ
ejde-984	117	17	that	that	SCONJ
ejde-984	117	18	,	,	PUNCT
ejde-984	117	19	for	for	ADP
ejde-984	117	20	any	any	DET
ejde-984	117	21	measurable	measurable	ADJ
ejde-984	117	22	and	and	CCONJ
ejde-984	117	23	compactly	compactly	ADV
ejde-984	117	24	supported	support	VERB
ejde-984	117	25	function	function	NOUN
ejde-984	117	26	u	u	NOUN
ejde-984	117	27	:	:	PUNCT
ejde-984	117	28	rn	rn	PROPN
ejde-984	117	29	→	→	SYM
ejde-984	117	30	r	r	NOUN
ejde-984	117	31	,	,	PUNCT
ejde-984	117	32	we	we	PRON
ejde-984	117	33	have	have	AUX
ejde-984	117	34	ss|u|22∗s	ss|u|22∗s	ADJ
ejde-984	117	35	≤	≤	PUNCT
ejde-984	118	1	∫∫	∫∫	ADV
ejde-984	118	2	r2n	r2n	ADJ
ejde-984	118	3	|u(x)−	|u(x)−	PROPN
ejde-984	118	4	u(y)|2	u(y)|2	PROPN
ejde-984	118	5	|x−	|x−	PROPN
ejde-984	118	6	y|n+2s	y|n+2s	PROPN
ejde-984	118	7	dxdy	dxdy	PROPN
ejde-984	118	8	,	,	PUNCT
ejde-984	118	9	(	(	PUNCT
ejde-984	118	10	2.1	2.1	NUM
ejde-984	118	11	)	)	PUNCT
ejde-984	118	12	where	where	SCONJ
ejde-984	118	13	2∗s	2∗s	NUM
ejde-984	118	14	=	=	SYM
ejde-984	118	15	2n	2n	NUM
ejde-984	118	16	n−2s	n−2s	ADJ
ejde-984	118	17	is	be	AUX
ejde-984	118	18	the	the	DET
ejde-984	118	19	so	so	ADV
ejde-984	118	20	-	-	PUNCT
ejde-984	118	21	called	call	VERB
ejde-984	118	22	fractional	fractional	ADJ
ejde-984	118	23	critical	critical	ADJ
ejde-984	118	24	exponent	exponent	NOUN
ejde-984	118	25	.	.	PUNCT
ejde-984	119	1	moreover	moreover	ADV
ejde-984	119	2	,	,	PUNCT
ejde-984	119	3	(	(	PUNCT
ejde-984	119	4	2.1	2.1	NUM
ejde-984	119	5	)	)	PUNCT
ejde-984	119	6	becomes	become	VERB
ejde-984	119	7	equality	equality	NOUN
ejde-984	119	8	if	if	SCONJ
ejde-984	119	9	and	and	CCONJ
ejde-984	119	10	only	only	ADV
ejde-984	119	11	if	if	SCONJ
ejde-984	119	12	ũ	ũ	PROPN
ejde-984	119	13	=	=	X
ejde-984	119	14	k(µ̄2	k(µ̄2	PROPN
ejde-984	119	15	+	+	CCONJ
ejde-984	119	16	|x	|x	PROPN
ejde-984	119	17	−	−	PROPN
ejde-984	119	18	x0|2)−	x0|2)−	PUNCT
ejde-984	120	1	n−2s	n−2s	INTJ
ejde-984	120	2	2	2	NUM
ejde-984	120	3	with	with	ADP
ejde-984	120	4	k	k	PROPN
ejde-984	120	5	∈	∈	PROPN
ejde-984	120	6	r\{0	r\{0	PROPN
ejde-984	120	7	}	}	PUNCT
ejde-984	120	8	,	,	PUNCT
ejde-984	120	9	µ̄	µ̄	PROPN
ejde-984	120	10	>	>	X
ejde-984	120	11	0	0	PROPN
ejde-984	120	12	,	,	PUNCT
ejde-984	120	13	x0	x0	PROPN
ejde-984	120	14	∈	∈	PROPN
ejde-984	120	15	rn	rn	PROPN
ejde-984	120	16	fixed	fix	VERB
ejde-984	120	17	constants	constant	NOUN
ejde-984	120	18	,	,	PUNCT
ejde-984	120	19	ss	ss	PROPN
ejde-984	120	20	is	be	AUX
ejde-984	120	21	the	the	DET
ejde-984	120	22	best	good	ADJ
ejde-984	120	23	sobolev	sobolev	NOUN
ejde-984	120	24	embedding	embed	VERB
ejde-984	120	25	constant	constant	ADJ
ejde-984	120	26	.	.	PUNCT
ejde-984	121	1	it	it	PRON
ejde-984	121	2	is	be	AUX
ejde-984	121	3	known	know	VERB
ejde-984	121	4	that	that	SCONJ
ejde-984	121	5	if	if	SCONJ
ejde-984	121	6	p	p	PROPN
ejde-984	121	7	∈	∈	PROPN
ejde-984	121	8	(	(	PUNCT
ejde-984	121	9	2	2	NUM
ejde-984	121	10	,	,	PUNCT
ejde-984	121	11	2∗s	2∗s	NUM
ejde-984	121	12	)	)	PUNCT
ejde-984	121	13	,	,	PUNCT
ejde-984	121	14	then	then	ADV
ejde-984	121	15	there	there	PRON
ejde-984	121	16	exists	exist	VERB
ejde-984	121	17	an	an	DET
ejde-984	121	18	optimal	optimal	ADJ
ejde-984	121	19	constant	constant	ADJ
ejde-984	121	20	c(s	c(	NOUN
ejde-984	121	21	,	,	PUNCT
ejde-984	121	22	p	p	NOUN
ejde-984	121	23	)	)	PUNCT
ejde-984	121	24	such	such	ADJ
ejde-984	121	25	that	that	SCONJ
ejde-984	121	26	|u|p	|u|p	PROPN
ejde-984	121	27	≤	≤	ADJ
ejde-984	121	28	c(s	c(	NOUN
ejde-984	121	29	,	,	PUNCT
ejde-984	121	30	p)|(−∆)s/2u|δs	p)|(−∆)s/2u|δs	PROPN
ejde-984	121	31	,	,	PUNCT
ejde-984	121	32	p2	p2	PROPN
ejde-984	121	33	|u|1−δs	|u|1−δs	NOUN
ejde-984	121	34	,	,	PUNCT
ejde-984	121	35	p	p	NOUN
ejde-984	121	36	2	2	NUM
ejde-984	121	37	,	,	PUNCT
ejde-984	121	38	(	(	PUNCT
ejde-984	121	39	2.2	2.2	NUM
ejde-984	121	40	)	)	PUNCT
ejde-984	121	41	holds	hold	VERB
ejde-984	121	42	for	for	ADP
ejde-984	121	43	all	all	DET
ejde-984	121	44	u	u	NOUN
ejde-984	121	45	∈	∈	PROPN
ejde-984	121	46	hs(rn	hs(rn	PROPN
ejde-984	121	47	)	)	PUNCT
ejde-984	121	48	.	.	PUNCT
ejde-984	122	1	(	(	PUNCT
ejde-984	122	2	2.2	2.2	NUM
ejde-984	122	3	)	)	PUNCT
ejde-984	122	4	is	be	AUX
ejde-984	122	5	called	call	VERB
ejde-984	122	6	of	of	ADP
ejde-984	122	7	the	the	DET
ejde-984	122	8	fractional	fractional	ADJ
ejde-984	122	9	gagliardo	gagliardo	NOUN
ejde-984	122	10	-	-	PUNCT
ejde-984	122	11	nirenberg	nirenberg	NOUN
ejde-984	122	12	inequality	inequality	NOUN
ejde-984	122	13	.	.	PUNCT
ejde-984	123	1	lemma	lemma	PROPN
ejde-984	123	2	2.2	2.2	NUM
ejde-984	123	3	(	(	PUNCT
ejde-984	123	4	[	[	X
ejde-984	123	5	18	18	NUM
ejde-984	123	6	]	]	NUM
ejde-984	123	7	)	)	PUNCT
ejde-984	123	8	.	.	PUNCT
ejde-984	124	1	let	let	VERB
ejde-984	124	2	u	u	PRON
ejde-984	124	3	∈	∈	PROPN
ejde-984	124	4	hs(rn	hs(rn	PROPN
ejde-984	124	5	)	)	PUNCT
ejde-984	124	6	,	,	PUNCT
ejde-984	124	7	n	n	PRON
ejde-984	124	8	≥	≥	NUM
ejde-984	124	9	2	2	NUM
ejde-984	124	10	satisfy	satisfy	VERB
ejde-984	124	11	the	the	DET
ejde-984	124	12	equation	equation	NOUN
ejde-984	124	13	(	(	PUNCT
ejde-984	124	14	−∆)su	−∆)su	NOUN
ejde-984	124	15	=	=	PUNCT
ejde-984	124	16	g(u	g(u	PROPN
ejde-984	124	17	)	)	PUNCT
ejde-984	124	18	,	,	PUNCT
ejde-984	124	19	then	then	ADV
ejde-984	124	20	n	n	PRON
ejde-984	124	21	−	−	NOUN
ejde-984	125	1	2s	2s	NUM
ejde-984	125	2	2	2	NUM
ejde-984	125	3	∫	∫	PROPN
ejde-984	125	4	rn	rn	PROPN
ejde-984	125	5	|(−∆)s/2u|2dx	|(−∆)s/2u|2dx	PROPN
ejde-984	125	6	=	=	SYM
ejde-984	125	7	n	n	CCONJ
ejde-984	125	8	∫	∫	PROPN
ejde-984	125	9	rn	rn	PROPN
ejde-984	125	10	g(u)dx	g(u)dx	PROPN
ejde-984	125	11	,	,	PUNCT
ejde-984	125	12	where	where	SCONJ
ejde-984	125	13	g(u	g(u	PROPN
ejde-984	125	14	)	)	PUNCT
ejde-984	125	15	=	=	SYM
ejde-984	125	16	∫	∫	PROPN
ejde-984	125	17	u	u	NOUN
ejde-984	125	18	0	0	PROPN
ejde-984	125	19	g(t)dt	g(t)dt	PROPN
ejde-984	125	20	.	.	PUNCT
ejde-984	126	1	lemma	lemma	PROPN
ejde-984	126	2	2.3	2.3	NUM
ejde-984	126	3	.	.	PUNCT
ejde-984	127	1	let	let	VERB
ejde-984	127	2	p	p	PRON
ejde-984	127	3	,	,	PUNCT
ejde-984	127	4	q	q	NOUN
ejde-984	127	5	∈	∈	PROPN
ejde-984	127	6	(	(	PUNCT
ejde-984	127	7	2	2	NUM
ejde-984	127	8	,	,	PUNCT
ejde-984	127	9	2n	2n	NUM
ejde-984	127	10	n−2s	n−2s	NOUN
ejde-984	127	11	]	]	PUNCT
ejde-984	127	12	and	and	CCONJ
ejde-984	127	13	λ	λ	PROPN
ejde-984	127	14	,	,	PUNCT
ejde-984	127	15	µ	µ	PROPN
ejde-984	127	16	∈	∈	PROPN
ejde-984	127	17	r.	r.	NOUN
ejde-984	127	18	if	if	SCONJ
ejde-984	127	19	u	u	PROPN
ejde-984	127	20	∈	∈	PROPN
ejde-984	127	21	hs(rn	hs(rn	PROPN
ejde-984	127	22	)	)	PUNCT
ejde-984	127	23	is	be	AUX
ejde-984	127	24	a	a	DET
ejde-984	127	25	weak	weak	ADJ
ejde-984	127	26	solution	solution	NOUN
ejde-984	127	27	of	of	ADP
ejde-984	127	28	the	the	DET
ejde-984	127	29	equation	equation	NOUN
ejde-984	127	30	in	in	ADP
ejde-984	127	31	the	the	PRON
ejde-984	127	32	n	n	CCONJ
ejde-984	127	33	-dimensional	-dimensional	ADJ
ejde-984	127	34	space	space	NOUN
ejde-984	127	35	corresponding	correspond	VERB
ejde-984	127	36	to	to	ADP
ejde-984	127	37	equation	equation	NOUN
ejde-984	127	38	(	(	PUNCT
ejde-984	127	39	1.1	1.1	NUM
ejde-984	127	40	)	)	PUNCT
ejde-984	127	41	,	,	PUNCT
ejde-984	127	42	then	then	ADV
ejde-984	127	43	it	it	PRON
ejde-984	127	44	satisfies	satisfy	VERB
ejde-984	127	45	the	the	DET
ejde-984	127	46	pohožaev	pohožaev	NOUN
ejde-984	127	47	identity	identity	NOUN
ejde-984	127	48	pµ(u	pµ(u	NOUN
ejde-984	127	49	)	)	PUNCT
ejde-984	127	50	:	:	PUNCT
ejde-984	128	1	=	=	SYM
ejde-984	128	2	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	128	3	+	+	NUM
ejde-984	128	4	b|(−∆)s/2u|42	b|(−∆)s/2u|42	PROPN
ejde-984	128	5	−	−	NOUN
ejde-984	128	6	µδs	µδ	NOUN
ejde-984	128	7	,	,	PUNCT
ejde-984	128	8	q|u|qq	q|u|qq	NOUN
ejde-984	128	9	−	−	NOUN
ejde-984	128	10	δs	δs	NOUN
ejde-984	128	11	,	,	PUNCT
ejde-984	128	12	p|u|pp	p|u|pp	PROPN
ejde-984	128	13	=	=	SYM
ejde-984	128	14	0	0	NUM
ejde-984	128	15	,	,	PUNCT
ejde-984	128	16	(	(	PUNCT
ejde-984	128	17	2.3	2.3	NUM
ejde-984	128	18	)	)	PUNCT
ejde-984	128	19	where	where	SCONJ
ejde-984	128	20	δs	δs	NOUN
ejde-984	128	21	,	,	PUNCT
ejde-984	128	22	q	q	NOUN
ejde-984	128	23	=	=	SYM
ejde-984	128	24	n(q−2	n(q−2	NOUN
ejde-984	128	25	)	)	PUNCT
ejde-984	128	26	2sq	2sq	NOUN
ejde-984	128	27	and	and	CCONJ
ejde-984	128	28	δs	δs	NOUN
ejde-984	128	29	,	,	PUNCT
ejde-984	128	30	p	p	NOUN
ejde-984	128	31	=	=	SYM
ejde-984	128	32	n(p−2	n(p−2	PROPN
ejde-984	128	33	)	)	PUNCT
ejde-984	128	34	2sp	2sp	NOUN
ejde-984	128	35	.	.	PUNCT
ejde-984	129	1	proof	proof	NOUN
ejde-984	129	2	.	.	PUNCT
ejde-984	130	1	set	set	VERB
ejde-984	130	2	a	a	DET
ejde-984	130	3	=	=	X
ejde-984	130	4	a+	a+	PRON
ejde-984	130	5	b	b	NUM
ejde-984	130	6	∫	∫	PROPN
ejde-984	130	7	rn	rn	PROPN
ejde-984	130	8	|(−∆)s/2u|2dx	|(−∆)s/2u|2dx	PROPN
ejde-984	130	9	.	.	PUNCT
ejde-984	131	1	according	accord	VERB
ejde-984	131	2	to	to	ADP
ejde-984	131	3	the	the	DET
ejde-984	131	4	lemma	lemma	PROPN
ejde-984	131	5	2.2	2.2	NUM
ejde-984	131	6	,	,	PUNCT
ejde-984	131	7	we	we	PRON
ejde-984	131	8	have	have	VERB
ejde-984	131	9	(	(	PUNCT
ejde-984	131	10	−∆)su	−∆)su	NOUN
ejde-984	131	11	=	=	NOUN
ejde-984	131	12	1	1	NUM
ejde-984	131	13	a	a	DET
ejde-984	131	14	(	(	PUNCT
ejde-984	131	15	λu+	λu+	INTJ
ejde-984	131	16	µ|u|q−2u+	µ|u|q−2u+	PROPN
ejde-984	131	17	|u|p−2u	|u|p−2u	PROPN
ejde-984	131	18	)	)	PUNCT
ejde-984	131	19	.	.	PUNCT
ejde-984	132	1	(	(	PUNCT
ejde-984	132	2	2.4	2.4	NUM
ejde-984	132	3	)	)	PUNCT
ejde-984	132	4	let	let	VERB
ejde-984	132	5	f(u	f(u	PROPN
ejde-984	132	6	)	)	PUNCT
ejde-984	132	7	=	=	PUNCT
ejde-984	133	1	1	1	NUM
ejde-984	133	2	a	a	DET
ejde-984	133	3	(	(	PUNCT
ejde-984	133	4	λu+	λu+	INTJ
ejde-984	133	5	µ|u|q−2u+	µ|u|q−2u+	PROPN
ejde-984	133	6	|u|p−2u	|u|p−2u	PROPN
ejde-984	133	7	)	)	PUNCT
ejde-984	133	8	,	,	PUNCT
ejde-984	133	9	then	then	ADV
ejde-984	133	10	f	f	PROPN
ejde-984	133	11	(	(	PUNCT
ejde-984	133	12	u	u	NOUN
ejde-984	133	13	)	)	PUNCT
ejde-984	133	14	=	=	SYM
ejde-984	134	1	1	1	NUM
ejde-984	134	2	a	a	PRON
ejde-984	134	3	(	(	PUNCT
ejde-984	134	4	λ2	λ2	NOUN
ejde-984	134	5	|u|	|u|	PROPN
ejde-984	134	6	2	2	NUM
ejde-984	134	7	+	+	SYM
ejde-984	134	8	µ	µ	X
ejde-984	134	9	q	q	X
ejde-984	134	10	|u|	|u|	PROPN
ejde-984	134	11	q	q	NOUN
ejde-984	135	1	+	+	NUM
ejde-984	135	2	1	1	NUM
ejde-984	135	3	p	p	NOUN
ejde-984	135	4	|u|	|u|	PROPN
ejde-984	135	5	p	p	NOUN
ejde-984	135	6	)	)	PUNCT
ejde-984	135	7	.	.	PUNCT
ejde-984	136	1	according	accord	VERB
ejde-984	136	2	to	to	ADP
ejde-984	136	3	the	the	DET
ejde-984	136	4	assumption	assumption	NOUN
ejde-984	136	5	that	that	SCONJ
ejde-984	136	6	u	u	NOUN
ejde-984	136	7	is	be	AUX
ejde-984	136	8	a	a	DET
ejde-984	136	9	weak	weak	ADJ
ejde-984	136	10	solution	solution	NOUN
ejde-984	136	11	to	to	ADP
ejde-984	136	12	the	the	DET
ejde-984	136	13	equation	equation	NOUN
ejde-984	136	14	in	in	ADP
ejde-984	136	15	the	the	PRON
ejde-984	136	16	n	n	CCONJ
ejde-984	136	17	-dimensional	-dimensional	ADJ
ejde-984	136	18	space	space	NOUN
ejde-984	136	19	corresponding	correspond	VERB
ejde-984	136	20	to	to	ADP
ejde-984	136	21	equation	equation	NOUN
ejde-984	136	22	(	(	PUNCT
ejde-984	136	23	1.1	1.1	NUM
ejde-984	136	24	)	)	PUNCT
ejde-984	136	25	(	(	PUNCT
ejde-984	136	26	in	in	ADP
ejde-984	136	27	other	other	ADJ
ejde-984	136	28	words	word	NOUN
ejde-984	136	29	,	,	PUNCT
ejde-984	136	30	multiplying	multiply	VERB
ejde-984	136	31	the	the	DET
ejde-984	136	32	above	above	ADJ
ejde-984	136	33	equation	equation	NOUN
ejde-984	136	34	by	by	ADP
ejde-984	136	35	u	u	NOUN
ejde-984	136	36	and	and	CCONJ
ejde-984	136	37	integrating	integrating	NOUN
ejde-984	136	38	)	)	PUNCT
ejde-984	136	39	,	,	PUNCT
ejde-984	136	40	one	one	PRON
ejde-984	136	41	has	have	VERB
ejde-984	136	42	that	that	DET
ejde-984	136	43	∫	∫	PROPN
ejde-984	136	44	rn	rn	PROPN
ejde-984	136	45	a(−∆)su	a(−∆)su	PROPN
ejde-984	136	46	·	·	PUNCT
ejde-984	136	47	udx	udx	NOUN
ejde-984	136	48	=	=	SYM
ejde-984	136	49	(	(	PUNCT
ejde-984	136	50	λ|u|22	λ|u|22	X
ejde-984	136	51	+	+	CCONJ
ejde-984	136	52	µ|u|qq	µ|u|qq	PROPN
ejde-984	136	53	+	+	NUM
ejde-984	136	54	|u|pp	|u|pp	NOUN
ejde-984	136	55	)	)	PUNCT
ejde-984	136	56	.	.	PUNCT
ejde-984	137	1	(	(	PUNCT
ejde-984	137	2	2.5	2.5	NUM
ejde-984	137	3	)	)	PUNCT
ejde-984	137	4	the	the	DET
ejde-984	137	5	above	above	ADJ
ejde-984	137	6	equality	equality	NOUN
ejde-984	137	7	implies	imply	VERB
ejde-984	137	8	that	that	SCONJ
ejde-984	137	9	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	138	1	=	=	SYM
ejde-984	138	2	(	(	PUNCT
ejde-984	138	3	λ|u|22	λ|u|22	X
ejde-984	138	4	+	+	CCONJ
ejde-984	138	5	µ|u|qq	µ|u|qq	PROPN
ejde-984	138	6	+	+	NUM
ejde-984	138	7	|u|pp	|u|pp	NOUN
ejde-984	138	8	)	)	PUNCT
ejde-984	138	9	.	.	PUNCT
ejde-984	139	1	(	(	PUNCT
ejde-984	139	2	2.6	2.6	NUM
ejde-984	139	3	)	)	PUNCT
ejde-984	139	4	multiplying	multiply	VERB
ejde-984	139	5	by	by	ADP
ejde-984	139	6	n	n	ADV
ejde-984	140	1	n−2s	n−2s	ADV
ejde-984	140	2	,	,	PUNCT
ejde-984	140	3	it	it	PRON
ejde-984	140	4	holds	hold	VERB
ejde-984	140	5	that	that	SCONJ
ejde-984	140	6	n	n	VERB
ejde-984	140	7	n	n	ADV
ejde-984	140	8	−	−	NOUN
ejde-984	141	1	2s	2s	NUM
ejde-984	141	2	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	141	3	=	=	SYM
ejde-984	141	4	n	n	CCONJ
ejde-984	141	5	n	n	ADV
ejde-984	141	6	−	−	PROPN
ejde-984	141	7	2s	2s	PROPN
ejde-984	141	8	λ|u|22	λ|u|22	X
ejde-984	141	9	+	+	CCONJ
ejde-984	141	10	n	n	CCONJ
ejde-984	141	11	n	n	ADV
ejde-984	141	12	−	−	NOUN
ejde-984	141	13	2s	2s	NUM
ejde-984	141	14	µ|u|qq	µ|u|qq	NOUN
ejde-984	141	15	+	+	CCONJ
ejde-984	141	16	n	n	CCONJ
ejde-984	141	17	n	n	ADV
ejde-984	141	18	−	−	NOUN
ejde-984	142	1	2s	2s	NUM
ejde-984	142	2	|u|pp	|u|pp	PROPN
ejde-984	142	3	.	.	PUNCT
ejde-984	143	1	(	(	PUNCT
ejde-984	143	2	2.7	2.7	NUM
ejde-984	143	3	)	)	PUNCT
ejde-984	143	4	by	by	ADP
ejde-984	143	5	lemma	lemma	PROPN
ejde-984	143	6	2.2	2.2	NUM
ejde-984	143	7	,	,	PUNCT
ejde-984	143	8	we	we	PRON
ejde-984	143	9	have	have	VERB
ejde-984	143	10	that	that	PRON
ejde-984	143	11	|(−∆)s/2u|22	|(−∆)s/2u|22	ADP
ejde-984	143	12	=	=	SYM
ejde-984	143	13	2n	2n	NUM
ejde-984	143	14	a(n	a(n	ADP
ejde-984	143	15	−	−	PROPN
ejde-984	144	1	2s	2s	NUM
ejde-984	144	2	)	)	PUNCT
ejde-984	144	3	(	(	PUNCT
ejde-984	144	4	λ	λ	PROPN
ejde-984	144	5	2	2	NUM
ejde-984	144	6	|u|22	|u|22	PROPN
ejde-984	144	7	+	+	CCONJ
ejde-984	144	8	µ	µ	DET
ejde-984	144	9	q	q	NOUN
ejde-984	144	10	|u|qq	|u|qq	NOUN
ejde-984	144	11	+	+	CCONJ
ejde-984	144	12	1	1	NUM
ejde-984	144	13	p	p	NOUN
ejde-984	144	14	|u|pp	|u|pp	PROPN
ejde-984	144	15	)	)	PUNCT
ejde-984	144	16	.	.	PUNCT
ejde-984	145	1	(	(	PUNCT
ejde-984	145	2	2.8	2.8	NUM
ejde-984	145	3	)	)	PUNCT
ejde-984	145	4	combining	combine	VERB
ejde-984	145	5	the	the	DET
ejde-984	145	6	above	above	ADJ
ejde-984	145	7	two	two	NUM
ejde-984	145	8	equations	equation	NOUN
ejde-984	145	9	,	,	PUNCT
ejde-984	145	10	we	we	PRON
ejde-984	145	11	obtain	obtain	VERB
ejde-984	145	12	that	that	SCONJ
ejde-984	145	13	(	(	PUNCT
ejde-984	145	14	1−	1−	NUM
ejde-984	145	15	n	n	CCONJ
ejde-984	145	16	n	n	ADV
ejde-984	145	17	−	−	PROPN
ejde-984	145	18	2s	2s	NUM
ejde-984	145	19	)	)	PUNCT
ejde-984	145	20	a|(−∆)s/2u|22	a|(−∆)s/2u|22	X
ejde-984	146	1	=	=	SYM
ejde-984	146	2	µ	µ	X
ejde-984	146	3	(	(	PUNCT
ejde-984	146	4	2n	2n	NUM
ejde-984	146	5	q(n	q(n	PROPN
ejde-984	146	6	−	−	PROPN
ejde-984	146	7	2s	2s	NUM
ejde-984	146	8	)	)	PUNCT
ejde-984	146	9	−	−	PROPN
ejde-984	146	10	n	n	CCONJ
ejde-984	146	11	n	n	ADV
ejde-984	146	12	−	−	PROPN
ejde-984	146	13	2s	2s	NUM
ejde-984	146	14	)	)	PUNCT
ejde-984	146	15	|u|qq	|u|qq	NOUN
ejde-984	146	16	+	+	CCONJ
ejde-984	146	17	(	(	PUNCT
ejde-984	146	18	2n	2n	NUM
ejde-984	146	19	p(n	p(n	PROPN
ejde-984	146	20	−	−	PROPN
ejde-984	146	21	2s	2s	NUM
ejde-984	146	22	)	)	PUNCT
ejde-984	146	23	−	−	PROPN
ejde-984	146	24	n	n	CCONJ
ejde-984	146	25	n	n	ADV
ejde-984	146	26	−	−	PROPN
ejde-984	146	27	2s	2s	PROPN
ejde-984	146	28	)	)	PUNCT
ejde-984	146	29	|u|pp	|u|pp	PROPN
ejde-984	146	30	.	.	PROPN
ejde-984	147	1	from	from	ADP
ejde-984	147	2	the	the	DET
ejde-984	147	3	above	above	ADJ
ejde-984	147	4	equality	equality	NOUN
ejde-984	147	5	,	,	PUNCT
ejde-984	147	6	one	one	PRON
ejde-984	147	7	deduces	deduce	VERB
ejde-984	147	8	that	that	PRON
ejde-984	147	9	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	147	10	=	=	SYM
ejde-984	147	11	µδs	µδ	NOUN
ejde-984	147	12	,	,	PUNCT
ejde-984	147	13	q|u|qq	q|u|qq	PROPN
ejde-984	147	14	+	+	CCONJ
ejde-984	147	15	δs	δs	NOUN
ejde-984	147	16	,	,	PUNCT
ejde-984	147	17	p|u|pp	p|u|pp	PROPN
ejde-984	147	18	.	.	PUNCT
ejde-984	148	1	6	6	NUM
ejde-984	148	2	z.	z.	PROPN
ejde-984	148	3	guo	guo	PROPN
ejde-984	148	4	,	,	PUNCT
ejde-984	148	5	t.	t.	PROPN
ejde-984	148	6	zhang	zhang	PROPN
ejde-984	148	7	ejde-2025/75	ejde-2025/75	NOUN
ejde-984	148	8	thus	thus	ADV
ejde-984	148	9	,	,	PUNCT
ejde-984	148	10	the	the	DET
ejde-984	148	11	fractional	fractional	PROPN
ejde-984	148	12	pohožaev	pohožaev	PROPN
ejde-984	148	13	identity	identity	NOUN
ejde-984	148	14	holds	hold	NOUN
ejde-984	148	15	,	,	PUNCT
ejde-984	148	16	namely	namely	ADV
ejde-984	148	17	,	,	PUNCT
ejde-984	148	18	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	148	19	+	+	CCONJ
ejde-984	148	20	b|(−∆)s/2u|42	b|(−∆)s/2u|42	NOUN
ejde-984	148	21	=	=	SYM
ejde-984	148	22	µδs	µδ	NOUN
ejde-984	148	23	,	,	PUNCT
ejde-984	148	24	q|u|qq	q|u|qq	PROPN
ejde-984	148	25	+	+	CCONJ
ejde-984	148	26	δs	δs	NOUN
ejde-984	148	27	,	,	PUNCT
ejde-984	148	28	p|u|pp	p|u|pp	PROPN
ejde-984	148	29	.	.	PUNCT
ejde-984	149	1	□	□	PUNCT
ejde-984	149	2	for	for	ADP
ejde-984	149	3	convenience	convenience	NOUN
ejde-984	149	4	,	,	PUNCT
ejde-984	149	5	we	we	PRON
ejde-984	149	6	decompose	decompose	VERB
ejde-984	149	7	the	the	DET
ejde-984	149	8	set	set	NOUN
ejde-984	149	9	pc,µ	pc,µ	NOUN
ejde-984	149	10	into	into	ADP
ejde-984	149	11	three	three	NUM
ejde-984	149	12	disjoint	disjoint	NOUN
ejde-984	149	13	sets	set	NOUN
ejde-984	149	14	as	as	SCONJ
ejde-984	149	15	follows	follow	VERB
ejde-984	149	16	:	:	PUNCT
ejde-984	149	17	pc,µ	pc,µ	PUNCT
ejde-984	149	18	=	=	SYM
ejde-984	149	19	p+	p+	VERB
ejde-984	149	20	c,µ	c,µ	NOUN
ejde-984	149	21	∪	∪	ADJ
ejde-984	149	22	p0	p0	NOUN
ejde-984	149	23	c,µ	c,µ	NOUN
ejde-984	149	24	∪	∪	ADP
ejde-984	149	25	p−	p−	NOUN
ejde-984	149	26	c,µ	c,µ	NOUN
ejde-984	149	27	,	,	PUNCT
ejde-984	149	28	where	where	SCONJ
ejde-984	149	29	p+	p+	VERB
ejde-984	149	30	c,µ	c,µ	NOUN
ejde-984	149	31	=	=	SYM
ejde-984	149	32	{	{	PUNCT
ejde-984	149	33	u	u	NOUN
ejde-984	149	34	∈	∈	PROPN
ejde-984	149	35	pc,µ	pc,µ	NOUN
ejde-984	149	36	,	,	PUNCT
ejde-984	149	37	2a|(−∆)s/2u|22	2a|(−∆)s/2u|22	NUM
ejde-984	149	38	+	+	SYM
ejde-984	149	39	4b|(−∆)s/2u|42	4b|(−∆)s/2u|42	NUM
ejde-984	149	40	−	−	PROPN
ejde-984	149	41	µqδ2s	µqδ2s	NOUN
ejde-984	149	42	,	,	PUNCT
ejde-984	149	43	q|u|qq	q|u|qq	PROPN
ejde-984	149	44	−	−	PROPN
ejde-984	149	45	pδ2s	pδ2s	NOUN
ejde-984	149	46	,	,	PUNCT
ejde-984	149	47	p|u|pp	p|u|pp	NOUN
ejde-984	149	48	>	>	X
ejde-984	149	49	0	0	NUM
ejde-984	149	50	}	}	PUNCT
ejde-984	149	51	=	=	SYM
ejde-984	149	52	{	{	PUNCT
ejde-984	149	53	u	u	NOUN
ejde-984	149	54	∈	∈	PROPN
ejde-984	149	55	pc,µ	pc,µ	NOUN
ejde-984	149	56	,	,	PUNCT
ejde-984	149	57	(	(	PUNCT
ejde-984	149	58	j	j	PROPN
ejde-984	149	59	µ	µ	X
ejde-984	149	60	u	u	NOUN
ejde-984	149	61	)	)	PUNCT
ejde-984	149	62	′′	′′	PROPN
ejde-984	149	63	(	(	PUNCT
ejde-984	149	64	0	0	NUM
ejde-984	149	65	)	)	PUNCT
ejde-984	149	66	>	>	X
ejde-984	149	67	0	0	NUM
ejde-984	149	68	}	}	PUNCT
ejde-984	149	69	,	,	PUNCT
ejde-984	149	70	p−	p−	NOUN
ejde-984	149	71	c,µ	c,µ	NOUN
ejde-984	149	72	=	=	SYM
ejde-984	149	73	{	{	PUNCT
ejde-984	149	74	u	u	NOUN
ejde-984	149	75	∈	∈	PROPN
ejde-984	149	76	pc,µ	pc,µ	NOUN
ejde-984	149	77	,	,	PUNCT
ejde-984	149	78	2a|(−∆)s/2u|22	2a|(−∆)s/2u|22	NUM
ejde-984	149	79	+	+	SYM
ejde-984	149	80	4b|(−∆)s/2u|42	4b|(−∆)s/2u|42	NUM
ejde-984	149	81	−	−	PROPN
ejde-984	149	82	µqδ2s	µqδ2s	NOUN
ejde-984	149	83	,	,	PUNCT
ejde-984	149	84	q|u|qq	q|u|qq	PROPN
ejde-984	149	85	−	−	PROPN
ejde-984	149	86	pδ2s	pδ2s	NOUN
ejde-984	149	87	,	,	PUNCT
ejde-984	149	88	p|u|pp	p|u|pp	NOUN
ejde-984	149	89	<	<	X
ejde-984	149	90	0	0	NUM
ejde-984	149	91	}	}	PUNCT
ejde-984	149	92	=	=	SYM
ejde-984	149	93	{	{	PUNCT
ejde-984	149	94	u	u	NOUN
ejde-984	149	95	∈	∈	PROPN
ejde-984	149	96	pc,µ	pc,µ	NOUN
ejde-984	149	97	,	,	PUNCT
ejde-984	149	98	(	(	PUNCT
ejde-984	149	99	j	j	PROPN
ejde-984	149	100	µ	µ	X
ejde-984	149	101	u	u	NOUN
ejde-984	149	102	)	)	PUNCT
ejde-984	149	103	′′	′′	PROPN
ejde-984	149	104	(	(	PUNCT
ejde-984	149	105	0	0	NUM
ejde-984	149	106	)	)	PUNCT
ejde-984	149	107	<	<	X
ejde-984	149	108	0	0	NUM
ejde-984	149	109	}	}	PUNCT
ejde-984	149	110	,	,	PUNCT
ejde-984	149	111	p0	p0	NOUN
ejde-984	149	112	c,µ	c,µ	NOUN
ejde-984	149	113	=	=	SYM
ejde-984	149	114	{	{	PUNCT
ejde-984	149	115	u	u	NOUN
ejde-984	149	116	∈	∈	PROPN
ejde-984	149	117	pc,µ	pc,µ	NOUN
ejde-984	149	118	,	,	PUNCT
ejde-984	149	119	2a|(−∆)s/2u|22	2a|(−∆)s/2u|22	NUM
ejde-984	149	120	+	+	SYM
ejde-984	149	121	4b|(−∆)s/2u|42	4b|(−∆)s/2u|42	NUM
ejde-984	149	122	−	−	PROPN
ejde-984	149	123	µqδ2s	µqδ2s	NOUN
ejde-984	149	124	,	,	PUNCT
ejde-984	149	125	q|u|qq	q|u|qq	PROPN
ejde-984	149	126	−	−	PROPN
ejde-984	149	127	pδ2s	pδ2s	NOUN
ejde-984	149	128	,	,	PUNCT
ejde-984	149	129	p|u|pp	p|u|pp	NOUN
ejde-984	149	130	=	=	PUNCT
ejde-984	149	131	0	0	X
ejde-984	149	132	}	}	PUNCT
ejde-984	149	133	=	=	SYM
ejde-984	149	134	{	{	PUNCT
ejde-984	149	135	u	u	NOUN
ejde-984	149	136	∈	∈	PROPN
ejde-984	149	137	pc,µ	pc,µ	NOUN
ejde-984	149	138	,	,	PUNCT
ejde-984	149	139	(	(	PUNCT
ejde-984	149	140	j	j	PROPN
ejde-984	149	141	µ	µ	X
ejde-984	149	142	u	u	NOUN
ejde-984	149	143	)	)	PUNCT
ejde-984	149	144	′′	′′	PROPN
ejde-984	149	145	(	(	PUNCT
ejde-984	149	146	0	0	NUM
ejde-984	149	147	)	)	PUNCT
ejde-984	149	148	=	=	SYM
ejde-984	149	149	0	0	X
ejde-984	149	150	}	}	PUNCT
ejde-984	149	151	,	,	PUNCT
ejde-984	149	152	where	where	SCONJ
ejde-984	149	153	(	(	PUNCT
ejde-984	149	154	jµ	jµ	PROPN
ejde-984	149	155	u	u	NOUN
ejde-984	149	156	)	)	PUNCT
ejde-984	149	157	′′	′′	PROPN
ejde-984	149	158	(	(	PUNCT
ejde-984	149	159	0	0	NUM
ejde-984	149	160	)	)	PUNCT
ejde-984	149	161	=	=	NOUN
ejde-984	149	162	(	(	PUNCT
ejde-984	149	163	2a|(−∆)s/2u|22	2a|(−∆)s/2u|22	NUM
ejde-984	149	164	+	+	CCONJ
ejde-984	149	165	4b|(−∆)s/2u|42	4b|(−∆)s/2u|42	NUM
ejde-984	149	166	−	−	PROPN
ejde-984	149	167	µqδ2s	µqδ2s	NOUN
ejde-984	149	168	,	,	PUNCT
ejde-984	149	169	q|u|qq	q|u|qq	PROPN
ejde-984	149	170	−	−	PROPN
ejde-984	149	171	pδ2s	pδ2s	NOUN
ejde-984	149	172	,	,	PUNCT
ejde-984	149	173	p|u|pp	p|u|pp	PROPN
ejde-984	149	174	)	)	PUNCT
ejde-984	149	175	s2	s2	PROPN
ejde-984	149	176	.	.	PUNCT
ejde-984	150	1	next	next	ADV
ejde-984	150	2	we	we	PRON
ejde-984	150	3	state	state	VERB
ejde-984	150	4	a	a	DET
ejde-984	150	5	lemma	lemma	PROPN
ejde-984	150	6	that	that	PRON
ejde-984	150	7	is	be	AUX
ejde-984	150	8	a	a	DET
ejde-984	150	9	type	type	NOUN
ejde-984	150	10	of	of	ADP
ejde-984	150	11	minimax	minimax	NOUN
ejde-984	150	12	principle	principle	NOUN
ejde-984	150	13	.	.	PUNCT
ejde-984	151	1	but	but	CCONJ
ejde-984	151	2	first	first	ADV
ejde-984	151	3	,	,	PUNCT
ejde-984	151	4	we	we	PRON
ejde-984	151	5	state	state	VERB
ejde-984	151	6	a	a	DET
ejde-984	151	7	related	related	ADJ
ejde-984	151	8	definition	definition	NOUN
ejde-984	151	9	.	.	PUNCT
ejde-984	152	1	definition	definition	NOUN
ejde-984	152	2	2.4	2.4	NUM
ejde-984	152	3	.	.	PUNCT
ejde-984	153	1	let	let	VERB
ejde-984	153	2	x	x	PRON
ejde-984	153	3	be	be	AUX
ejde-984	153	4	a	a	DET
ejde-984	153	5	topological	topological	ADJ
ejde-984	153	6	space	space	NOUN
ejde-984	153	7	and	and	CCONJ
ejde-984	153	8	b	b	NOUN
ejde-984	153	9	be	be	AUX
ejde-984	153	10	a	a	DET
ejde-984	153	11	closed	closed	ADJ
ejde-984	153	12	subset	subset	NOUN
ejde-984	153	13	of	of	ADP
ejde-984	153	14	x.	x.	NOUN
ejde-984	153	15	we	we	PRON
ejde-984	153	16	say	say	VERB
ejde-984	153	17	that	that	SCONJ
ejde-984	153	18	a	a	DET
ejde-984	153	19	class	class	NOUN
ejde-984	153	20	f	f	NOUN
ejde-984	153	21	of	of	ADP
ejde-984	153	22	compact	compact	ADJ
ejde-984	153	23	subsets	subset	NOUN
ejde-984	153	24	of	of	ADP
ejde-984	153	25	x	x	X
ejde-984	153	26	is	be	AUX
ejde-984	153	27	a	a	DET
ejde-984	153	28	homotopy	homotopy	NOUN
ejde-984	153	29	-	-	PUNCT
ejde-984	153	30	stable	stable	ADJ
ejde-984	153	31	family	family	NOUN
ejde-984	153	32	with	with	ADP
ejde-984	153	33	extended	extended	ADJ
ejde-984	153	34	boundary	boundary	ADJ
ejde-984	153	35	b	b	NOUN
ejde-984	153	36	if	if	SCONJ
ejde-984	153	37	for	for	ADP
ejde-984	153	38	any	any	DET
ejde-984	153	39	set	set	NOUN
ejde-984	153	40	a	a	PRON
ejde-984	153	41	in	in	ADP
ejde-984	153	42	f	f	PROPN
ejde-984	153	43	and	and	CCONJ
ejde-984	153	44	any	any	DET
ejde-984	153	45	η	η	PROPN
ejde-984	153	46	∈	∈	PROPN
ejde-984	153	47	c([0	c([0	NOUN
ejde-984	153	48	,	,	PUNCT
ejde-984	153	49	1]×x;x	1]×x;x	NUM
ejde-984	153	50	)	)	PUNCT
ejde-984	153	51	satisfying	satisfy	VERB
ejde-984	153	52	η(t	η(t	NOUN
ejde-984	153	53	,	,	PUNCT
ejde-984	153	54	x	x	NOUN
ejde-984	153	55	)	)	PUNCT
ejde-984	153	56	=	=	PUNCT
ejde-984	154	1	x	x	PROPN
ejde-984	154	2	for	for	ADP
ejde-984	154	3	all	all	DET
ejde-984	154	4	(	(	PUNCT
ejde-984	154	5	t	t	PROPN
ejde-984	154	6	,	,	PUNCT
ejde-984	154	7	x	x	NOUN
ejde-984	154	8	)	)	PUNCT
ejde-984	154	9	∈	∈	PROPN
ejde-984	154	10	(	(	PUNCT
ejde-984	154	11	{	{	PUNCT
ejde-984	154	12	0	0	NUM
ejde-984	154	13	}	}	PUNCT
ejde-984	154	14	×x	×x	X
ejde-984	154	15	)	)	PUNCT
ejde-984	154	16	∪	∪	X
ejde-984	154	17	(	(	PUNCT
ejde-984	154	18	[	[	X
ejde-984	154	19	0	0	NUM
ejde-984	154	20	,	,	PUNCT
ejde-984	154	21	1]×b	1]×b	NUM
ejde-984	154	22	)	)	PUNCT
ejde-984	154	23	,	,	PUNCT
ejde-984	154	24	we	we	PRON
ejde-984	154	25	have	have	VERB
ejde-984	154	26	that	that	SCONJ
ejde-984	154	27	η({1	η({1	PRON
ejde-984	154	28	}	}	PUNCT
ejde-984	154	29	×a	×a	NOUN
ejde-984	154	30	)	)	PUNCT
ejde-984	154	31	is	be	AUX
ejde-984	154	32	in	in	ADP
ejde-984	154	33	f	f	PROPN
ejde-984	154	34	.	.	PUNCT
ejde-984	155	1	lemma	lemma	PROPN
ejde-984	155	2	2.5	2.5	NUM
ejde-984	155	3	(	(	PUNCT
ejde-984	155	4	[	[	X
ejde-984	155	5	10	10	NUM
ejde-984	155	6	,	,	PUNCT
ejde-984	155	7	theorem	theorem	VERB
ejde-984	155	8	5.2	5.2	NUM
ejde-984	155	9	]	]	PUNCT
ejde-984	155	10	)	)	PUNCT
ejde-984	155	11	.	.	PUNCT
ejde-984	156	1	let	let	VERB
ejde-984	156	2	φ	φ	PROPN
ejde-984	156	3	be	be	AUX
ejde-984	156	4	a	a	DET
ejde-984	156	5	c1	c1	NOUN
ejde-984	156	6	functional	functional	ADJ
ejde-984	156	7	on	on	ADP
ejde-984	156	8	a	a	DET
ejde-984	156	9	complete	complete	ADJ
ejde-984	156	10	connected	connect	VERB
ejde-984	156	11	c1	c1	NOUN
ejde-984	156	12	-	-	PUNCT
ejde-984	156	13	finsler	finsler	NOUN
ejde-984	156	14	manifold	manifold	NOUN
ejde-984	156	15	x	x	X
ejde-984	156	16	and	and	CCONJ
ejde-984	156	17	consider	consider	VERB
ejde-984	156	18	a	a	DET
ejde-984	156	19	homotopy	homotopy	NOUN
ejde-984	156	20	-	-	PUNCT
ejde-984	156	21	stable	stable	ADJ
ejde-984	156	22	family	family	NOUN
ejde-984	156	23	f	f	PROPN
ejde-984	156	24	with	with	ADP
ejde-984	156	25	an	an	DET
ejde-984	156	26	extended	extended	ADJ
ejde-984	156	27	closed	close	VERB
ejde-984	156	28	boundary	boundary	ADJ
ejde-984	156	29	b.	b.	PROPN
ejde-984	157	1	set	set	VERB
ejde-984	157	2	m	m	PROPN
ejde-984	157	3	=	=	SYM
ejde-984	157	4	m(φ	m(φ	PROPN
ejde-984	157	5	,	,	PUNCT
ejde-984	157	6	f	f	X
ejde-984	157	7	)	)	PUNCT
ejde-984	157	8	=	=	PUNCT
ejde-984	158	1	infa∈f	infa∈f	VERB
ejde-984	158	2	maxx∈a	maxx∈a	NOUN
ejde-984	158	3	φ(x	φ(x	NOUN
ejde-984	158	4	)	)	PUNCT
ejde-984	158	5	and	and	CCONJ
ejde-984	158	6	let	let	VERB
ejde-984	158	7	f	f	PRON
ejde-984	158	8	be	be	AUX
ejde-984	158	9	a	a	DET
ejde-984	158	10	closed	closed	ADJ
ejde-984	158	11	subset	subset	NOUN
ejde-984	158	12	of	of	ADP
ejde-984	158	13	x	x	SYM
ejde-984	158	14	satisfying	satisfy	VERB
ejde-984	158	15	(	(	PUNCT
ejde-984	158	16	1	1	NUM
ejde-984	158	17	)	)	PUNCT
ejde-984	158	18	a	a	DET
ejde-984	158	19	∩	∩	NOUN
ejde-984	158	20	f\b	f\b	X
ejde-984	158	21	̸=	̸=	PROPN
ejde-984	158	22	∅	∅	NOUN
ejde-984	158	23	for	for	ADP
ejde-984	158	24	each	each	DET
ejde-984	158	25	a	a	DET
ejde-984	158	26	∈	∈	PROPN
ejde-984	158	27	f	f	X
ejde-984	158	28	.	.	PUNCT
ejde-984	159	1	(	(	PUNCT
ejde-984	159	2	2	2	X
ejde-984	159	3	)	)	PUNCT
ejde-984	159	4	supφ(b	supφ(b	PROPN
ejde-984	159	5	)	)	PUNCT
ejde-984	159	6	≤	≤	NUM
ejde-984	159	7	m	m	VERB
ejde-984	159	8	≤	≤	PROPN
ejde-984	159	9	inf	inf	PROPN
ejde-984	159	10	φ(f	φ(f	PROPN
ejde-984	159	11	)	)	PUNCT
ejde-984	159	12	.	.	PUNCT
ejde-984	160	1	then	then	ADV
ejde-984	160	2	,	,	PUNCT
ejde-984	160	3	for	for	ADP
ejde-984	160	4	any	any	DET
ejde-984	160	5	sequence	sequence	NOUN
ejde-984	160	6	of	of	ADP
ejde-984	160	7	sets	set	NOUN
ejde-984	160	8	{	{	PUNCT
ejde-984	160	9	an}n	an}n	NOUN
ejde-984	160	10	in	in	ADP
ejde-984	160	11	f	f	PROPN
ejde-984	160	12	such	such	ADJ
ejde-984	160	13	that	that	SCONJ
ejde-984	160	14	limn	limn	PROPN
ejde-984	160	15	supan	supan	PROPN
ejde-984	160	16	φ	φ	PROPN
ejde-984	160	17	=	=	SYM
ejde-984	160	18	m	m	PROPN
ejde-984	160	19	,	,	PUNCT
ejde-984	160	20	there	there	PRON
ejde-984	160	21	exists	exist	VERB
ejde-984	160	22	a	a	DET
ejde-984	160	23	sequence	sequence	NOUN
ejde-984	160	24	{	{	PUNCT
ejde-984	160	25	xn}n	xn}n	VERB
ejde-984	160	26	in	in	ADP
ejde-984	160	27	x\b	x\b	PROPN
ejde-984	160	28	such	such	ADJ
ejde-984	160	29	that	that	SCONJ
ejde-984	160	30	lim	lim	PROPN
ejde-984	160	31	n→∞	n→∞	NUM
ejde-984	160	32	φ(xn	φ(xn	PROPN
ejde-984	160	33	)	)	PUNCT
ejde-984	160	34	=	=	SYM
ejde-984	160	35	m	m	PROPN
ejde-984	160	36	,	,	PUNCT
ejde-984	160	37	lim	lim	PROPN
ejde-984	160	38	n→∞	n→∞	X
ejde-984	161	1	∥dφ(xn)∥	∥dφ(xn)∥	PROPN
ejde-984	161	2	=	=	SYM
ejde-984	161	3	0	0	PROPN
ejde-984	161	4	,	,	PUNCT
ejde-984	161	5	lim	lim	PROPN
ejde-984	161	6	n→∞	n→∞	NUM
ejde-984	161	7	dist(xn	dist(xn	PROPN
ejde-984	161	8	,	,	PUNCT
ejde-984	161	9	f	f	PROPN
ejde-984	161	10	)	)	PUNCT
ejde-984	162	1	=	=	SYM
ejde-984	162	2	0	0	PROPN
ejde-984	162	3	,	,	PUNCT
ejde-984	162	4	lim	lim	PROPN
ejde-984	162	5	n→∞	n→∞	NUM
ejde-984	162	6	dist(xn	dist(xn	PROPN
ejde-984	162	7	,	,	PUNCT
ejde-984	162	8	an	an	NOUN
ejde-984	162	9	)	)	PUNCT
ejde-984	162	10	=	=	SYM
ejde-984	162	11	0	0	X
ejde-984	162	12	.	.	PUNCT
ejde-984	163	1	lemma	lemma	PROPN
ejde-984	163	2	2.6	2.6	NUM
ejde-984	163	3	(	(	PUNCT
ejde-984	163	4	[	[	X
ejde-984	163	5	18	18	NUM
ejde-984	163	6	]	]	NUM
ejde-984	163	7	)	)	PUNCT
ejde-984	163	8	.	.	PUNCT
ejde-984	164	1	let	let	VERB
ejde-984	164	2	n	n	PRON
ejde-984	164	3	≥	≥	NOUN
ejde-984	164	4	2	2	NUM
ejde-984	164	5	,	,	PUNCT
ejde-984	164	6	then	then	ADV
ejde-984	164	7	hs	hs	INTJ
ejde-984	164	8	r	r	PROPN
ejde-984	164	9	(	(	PUNCT
ejde-984	164	10	rn	rn	PROPN
ejde-984	164	11	)	)	PUNCT
ejde-984	164	12	is	be	AUX
ejde-984	164	13	compactly	compactly	ADV
ejde-984	164	14	embedding	embed	VERB
ejde-984	164	15	into	into	ADP
ejde-984	164	16	lp(rn	lp(rn	PROPN
ejde-984	164	17	)	)	PUNCT
ejde-984	164	18	for	for	ADP
ejde-984	164	19	p	p	PROPN
ejde-984	164	20	∈	∈	PROPN
ejde-984	164	21	(	(	PUNCT
ejde-984	164	22	2	2	NUM
ejde-984	164	23	,	,	PUNCT
ejde-984	164	24	2∗s	2∗s	NUM
ejde-984	164	25	)	)	PUNCT
ejde-984	164	26	.	.	PUNCT
ejde-984	165	1	lemma	lemma	PROPN
ejde-984	165	2	2.7	2.7	NUM
ejde-984	165	3	(	(	PUNCT
ejde-984	165	4	[	[	X
ejde-984	165	5	18	18	NUM
ejde-984	165	6	]	]	NUM
ejde-984	165	7	)	)	PUNCT
ejde-984	165	8	.	.	PUNCT
ejde-984	166	1	let	let	VERB
ejde-984	166	2	s	s	PRON
ejde-984	166	3	∈	∈	PROPN
ejde-984	166	4	(	(	PUNCT
ejde-984	166	5	0	0	NUM
ejde-984	166	6	,	,	PUNCT
ejde-984	166	7	1	1	NUM
ejde-984	166	8	)	)	PUNCT
ejde-984	166	9	.	.	PUNCT
ejde-984	167	1	for	for	ADP
ejde-984	167	2	any	any	DET
ejde-984	167	3	u	u	PROPN
ejde-984	167	4	∈	∈	PROPN
ejde-984	167	5	hs(rn	hs(rn	PROPN
ejde-984	167	6	)	)	PUNCT
ejde-984	167	7	,	,	PUNCT
ejde-984	167	8	the	the	DET
ejde-984	167	9	following	follow	VERB
ejde-984	167	10	inequality	inequality	NOUN
ejde-984	167	11	holds∫∫	holds∫∫	ADJ
ejde-984	167	12	r2n	r2n	NOUN
ejde-984	167	13	(	(	PUNCT
ejde-984	167	14	u∗(x)−	u∗(x)−	NOUN
ejde-984	167	15	u∗(y))2	u∗(y))2	PROPN
ejde-984	167	16	|x−	|x−	PROPN
ejde-984	167	17	y|n+2s	y|n+2s	PROPN
ejde-984	167	18	dxdy	dxdy	NOUN
ejde-984	167	19	≤	≤	VERB
ejde-984	168	1	∫∫	∫∫	ADV
ejde-984	168	2	r2n	r2n	NOUN
ejde-984	168	3	(	(	PUNCT
ejde-984	168	4	u(x)−	u(x)−	PROPN
ejde-984	168	5	u(y))2	u(y))2	PROPN
ejde-984	168	6	|x−	|x−	PROPN
ejde-984	168	7	y|n+2s	y|n+2s	PROPN
ejde-984	168	8	dxdy	dxdy	PROPN
ejde-984	168	9	.	.	PUNCT
ejde-984	169	1	next	next	ADV
ejde-984	169	2	,	,	PUNCT
ejde-984	169	3	we	we	PRON
ejde-984	169	4	state	state	VERB
ejde-984	169	5	the	the	DET
ejde-984	169	6	main	main	ADJ
ejde-984	169	7	results	result	NOUN
ejde-984	169	8	of	of	ADP
ejde-984	169	9	this	this	DET
ejde-984	169	10	paper	paper	NOUN
ejde-984	169	11	.	.	PUNCT
ejde-984	170	1	theorem	theorem	VERB
ejde-984	170	2	2.8	2.8	NUM
ejde-984	170	3	(	(	PUNCT
ejde-984	170	4	subcritical	subcritical	ADJ
ejde-984	170	5	case	case	NOUN
ejde-984	170	6	)	)	PUNCT
ejde-984	170	7	.	.	PUNCT
ejde-984	171	1	let	let	VERB
ejde-984	171	2	n	n	NOUN
ejde-984	171	3	=	=	SYM
ejde-984	171	4	3	3	NUM
ejde-984	171	5	and	and	CCONJ
ejde-984	171	6	2	2	NUM
ejde-984	171	7	<	<	X
ejde-984	171	8	q	q	X
ejde-984	171	9	<	<	X
ejde-984	171	10	p	p	X
ejde-984	171	11	=	=	SYM
ejde-984	171	12	2	2	NUM
ejde-984	171	13	+	+	NUM
ejde-984	171	14	8s	8s	NUM
ejde-984	171	15	3	3	NUM
ejde-984	171	16	=	=	SYM
ejde-984	171	17	p̄.	p̄.	NOUN
ejde-984	171	18	if	if	SCONJ
ejde-984	171	19	b	b	PROPN
ejde-984	171	20	≥	≥	X
ejde-984	171	21	4	4	NUM
ejde-984	171	22	p̄c(s	p̄c(s	ADJ
ejde-984	171	23	,	,	PUNCT
ejde-984	171	24	p̄	p̄	NUM
ejde-984	171	25	)	)	PUNCT
ejde-984	171	26	p̄cp̄−4	p̄cp̄−4	PROPN
ejde-984	171	27	,	,	PUNCT
ejde-984	171	28	then	then	ADV
ejde-984	171	29	there	there	PRON
ejde-984	171	30	is	be	VERB
ejde-984	171	31	no	no	DET
ejde-984	171	32	solution	solution	NOUN
ejde-984	171	33	to	to	ADP
ejde-984	171	34	problem	problem	NOUN
ejde-984	171	35	(	(	PUNCT
ejde-984	171	36	1.1)-(1.2	1.1)-(1.2	NUM
ejde-984	171	37	)	)	PUNCT
ejde-984	171	38	for	for	ADP
ejde-984	171	39	any	any	DET
ejde-984	171	40	µ	µ	X
ejde-984	171	41	<	<	X
ejde-984	171	42	0	0	NUM
ejde-984	171	43	.	.	PUNCT
ejde-984	172	1	now	now	ADV
ejde-984	172	2	,	,	PUNCT
ejde-984	172	3	we	we	PRON
ejde-984	172	4	define	define	VERB
ejde-984	172	5	the	the	DET
ejde-984	172	6	constant	constant	ADJ
ejde-984	172	7	c0	c0	NOUN
ejde-984	172	8	:	:	PUNCT
ejde-984	172	9	=	=	SYM
ejde-984	172	10	(	(	PUNCT
ejde-984	172	11	a	a	DET
ejde-984	172	12	δs	δs	NOUN
ejde-984	172	13	,	,	PUNCT
ejde-984	172	14	pc(s	pc(s	NUM
ejde-984	172	15	,	,	PUNCT
ejde-984	172	16	p)pcp(1−δs	p)pcp(1−δs	ADJ
ejde-984	172	17	,	,	PUNCT
ejde-984	172	18	p	p	NOUN
ejde-984	172	19	)	)	PUNCT
ejde-984	172	20	)	)	PUNCT
ejde-984	172	21	1	1	NUM
ejde-984	172	22	pδs	pδs	NOUN
ejde-984	172	23	,	,	PUNCT
ejde-984	172	24	p−2	p−2	PROPN
ejde-984	172	25	.	.	PUNCT
ejde-984	173	1	theorem	theorem	VERB
ejde-984	173	2	2.9	2.9	NUM
ejde-984	173	3	(	(	PUNCT
ejde-984	173	4	subcritical	subcritical	ADJ
ejde-984	173	5	case	case	NOUN
ejde-984	173	6	)	)	PUNCT
ejde-984	173	7	.	.	PUNCT
ejde-984	174	1	let	let	VERB
ejde-984	174	2	n	n	NOUN
ejde-984	174	3	=	=	SYM
ejde-984	174	4	3	3	NUM
ejde-984	174	5	and	and	CCONJ
ejde-984	174	6	2	2	NUM
ejde-984	174	7	<	<	X
ejde-984	174	8	q	q	X
ejde-984	174	9	≤	≤	NUM
ejde-984	174	10	2	2	NUM
ejde-984	174	11	+	+	NUM
ejde-984	174	12	8s	8s	NUM
ejde-984	174	13	3	3	NUM
ejde-984	174	14	<	<	X
ejde-984	174	15	p	p	X
ejde-984	174	16	<	<	X
ejde-984	174	17	2∗s	2∗s	NUM
ejde-984	174	18	=	=	SYM
ejde-984	174	19	6	6	NUM
ejde-984	174	20	3−2s	3−2s	NUM
ejde-984	174	21	be	be	AUX
ejde-984	174	22	given	give	VERB
ejde-984	174	23	constants	constant	NOUN
ejde-984	174	24	.	.	PUNCT
ejde-984	175	1	if	if	SCONJ
ejde-984	175	2	µ	µ	PRON
ejde-984	175	3	<	<	X
ejde-984	175	4	0	0	NUM
ejde-984	175	5	satisfies	satisfie	NOUN
ejde-984	175	6	(	(	PUNCT
ejde-984	175	7	1−	1−	NUM
ejde-984	175	8	1	1	NUM
ejde-984	175	9	δs	δs	NOUN
ejde-984	175	10	,	,	PUNCT
ejde-984	175	11	p	p	NOUN
ejde-984	175	12	)	)	PUNCT
ejde-984	175	13	(	(	PUNCT
ejde-984	175	14	a+	a+	X
ejde-984	175	15	bc2	bc2	NOUN
ejde-984	175	16	0	0	NUM
ejde-984	175	17	)	)	PUNCT
ejde-984	175	18	c	c	NOUN
ejde-984	175	19	2−qδs	2−qδs	NOUN
ejde-984	175	20	,	,	PUNCT
ejde-984	175	21	q	q	NOUN
ejde-984	175	22	0	0	NUM
ejde-984	175	23	+	+	CCONJ
ejde-984	175	24	µ	µ	X
ejde-984	175	25	(	(	PUNCT
ejde-984	175	26	δs	δs	NOUN
ejde-984	175	27	,	,	PUNCT
ejde-984	175	28	q	q	NOUN
ejde-984	175	29	δs	δs	NOUN
ejde-984	175	30	,	,	PUNCT
ejde-984	175	31	p	p	NOUN
ejde-984	175	32	−	−	PROPN
ejde-984	175	33	1	1	NUM
ejde-984	175	34	)	)	PUNCT
ejde-984	175	35	c(s	c(	NOUN
ejde-984	175	36	,	,	PUNCT
ejde-984	175	37	q)qcq(1−δs	q)qcq(1−δs	PROPN
ejde-984	175	38	,	,	PUNCT
ejde-984	175	39	q	q	NOUN
ejde-984	175	40	)	)	PUNCT
ejde-984	175	41	:	:	PUNCT
ejde-984	175	42	=	=	X
ejde-984	175	43	ϵ0	ϵ0	VERB
ejde-984	175	44	<	<	X
ejde-984	175	45	0	0	NUM
ejde-984	175	46	,	,	PUNCT
ejde-984	175	47	(	(	PUNCT
ejde-984	175	48	2.9	2.9	NUM
ejde-984	175	49	)	)	PUNCT
ejde-984	175	50	then	then	ADV
ejde-984	175	51	eµ|sc	eµ|sc	X
ejde-984	175	52	has	have	VERB
ejde-984	175	53	a	a	DET
ejde-984	175	54	critical	critical	ADJ
ejde-984	175	55	point	point	NOUN
ejde-984	175	56	ũ	ũ	PROPN
ejde-984	175	57	at	at	ADP
ejde-984	175	58	a	a	DET
ejde-984	175	59	positive	positive	ADJ
ejde-984	175	60	level	level	NOUN
ejde-984	175	61	m(c	m(c	PROPN
ejde-984	175	62	,	,	PUNCT
ejde-984	175	63	µ	µ	NOUN
ejde-984	175	64	)	)	PUNCT
ejde-984	175	65	=	=	SYM
ejde-984	175	66	infu∈pc,µ	infu∈pc,µ	PROPN
ejde-984	175	67	eµ(u	eµ(u	NOUN
ejde-984	175	68	)	)	PUNCT
ejde-984	175	69	>	>	X
ejde-984	175	70	0	0	PUNCT
ejde-984	176	1	satisfying	satisfy	VERB
ejde-984	176	2	:	:	PUNCT
ejde-984	176	3	ũ	ũ	PROPN
ejde-984	176	4	is	be	AUX
ejde-984	176	5	radially	radially	ADV
ejde-984	176	6	symmetric	symmetric	ADJ
ejde-984	176	7	,	,	PUNCT
ejde-984	176	8	it	it	PRON
ejde-984	176	9	solves	solve	VERB
ejde-984	176	10	(	(	PUNCT
ejde-984	176	11	1.1	1.1	NUM
ejde-984	176	12	)	)	PUNCT
ejde-984	176	13	for	for	ADP
ejde-984	176	14	some	some	DET
ejde-984	176	15	λ̃	λ̃	PROPN
ejde-984	176	16	<	<	X
ejde-984	176	17	0	0	PUNCT
ejde-984	177	1	and	and	CCONJ
ejde-984	177	2	it	it	PRON
ejde-984	177	3	is	be	AUX
ejde-984	177	4	a	a	DET
ejde-984	177	5	ground	ground	NOUN
ejde-984	177	6	state	state	NOUN
ejde-984	177	7	of	of	ADP
ejde-984	177	8	(	(	PUNCT
ejde-984	177	9	1.1	1.1	NUM
ejde-984	177	10	)	)	PUNCT
ejde-984	177	11	on	on	ADP
ejde-984	177	12	sc	sc	PROPN
ejde-984	177	13	,	,	PUNCT
ejde-984	177	14	where	where	SCONJ
ejde-984	177	15	ejde-2025/75	ejde-2025/75	ADJ
ejde-984	177	16	normalized	normalize	VERB
ejde-984	177	17	solutions	solution	NOUN
ejde-984	177	18	of	of	ADP
ejde-984	177	19	fractional	fractional	PROPN
ejde-984	177	20	kirchhoff	kirchhoff	NOUN
ejde-984	177	21	equations	equation	NOUN
ejde-984	177	22	7	7	NUM
ejde-984	177	23	c(s	c(	NOUN
ejde-984	177	24	,	,	PUNCT
ejde-984	177	25	q	q	X
ejde-984	177	26	)	)	PUNCT
ejde-984	177	27	is	be	AUX
ejde-984	177	28	an	an	DET
ejde-984	177	29	optimal	optimal	ADJ
ejde-984	177	30	constant	constant	ADJ
ejde-984	177	31	such	such	ADJ
ejde-984	177	32	that	that	SCONJ
ejde-984	177	33	the	the	DET
ejde-984	177	34	fractional	fractional	ADJ
ejde-984	177	35	gagliardo	gagliardo	NOUN
ejde-984	177	36	-	-	PUNCT
ejde-984	177	37	nirenberg	nirenberg	PROPN
ejde-984	177	38	inequality	inequality	NOUN
ejde-984	177	39	holds	hold	VERB
ejde-984	177	40	(	(	PUNCT
ejde-984	177	41	see	see	VERB
ejde-984	177	42	inequality	inequality	NOUN
ejde-984	177	43	(	(	PUNCT
ejde-984	177	44	2.2	2.2	NUM
ejde-984	177	45	)	)	PUNCT
ejde-984	177	46	)	)	PUNCT
ejde-984	177	47	.	.	PUNCT
ejde-984	178	1	the	the	DET
ejde-984	178	2	following	follow	VERB
ejde-984	178	3	theorem	theorem	NOUN
ejde-984	178	4	characterizes	characterize	VERB
ejde-984	178	5	the	the	DET
ejde-984	178	6	ground	ground	NOUN
ejde-984	178	7	states	state	NOUN
ejde-984	178	8	.	.	PUNCT
ejde-984	179	1	theorem	theorem	VERB
ejde-984	179	2	2.10	2.10	NUM
ejde-984	179	3	(	(	PUNCT
ejde-984	179	4	subcritical	subcritical	ADJ
ejde-984	179	5	case	case	NOUN
ejde-984	179	6	)	)	PUNCT
ejde-984	179	7	.	.	PUNCT
ejde-984	180	1	under	under	ADP
ejde-984	180	2	the	the	DET
ejde-984	180	3	assumptions	assumption	NOUN
ejde-984	180	4	of	of	ADP
ejde-984	180	5	theorem	theorem	NOUN
ejde-984	180	6	2.9	2.9	NUM
ejde-984	180	7	,	,	PUNCT
ejde-984	180	8	if	if	SCONJ
ejde-984	180	9	u	u	PROPN
ejde-984	180	10	∈	∈	PROPN
ejde-984	180	11	zc,µ	zc,µ	PROPN
ejde-984	180	12	(	(	PUNCT
ejde-984	180	13	see	see	VERB
ejde-984	180	14	definition	definition	NOUN
ejde-984	180	15	1.1	1.1	NUM
ejde-984	180	16	for	for	ADP
ejde-984	180	17	the	the	DET
ejde-984	180	18	definition	definition	NOUN
ejde-984	180	19	of	of	ADP
ejde-984	180	20	zc,µ	zc,µ	NUM
ejde-984	180	21	)	)	PUNCT
ejde-984	180	22	,	,	PUNCT
ejde-984	180	23	then	then	ADV
ejde-984	180	24	eiθ|u|	eiθ|u|	PROPN
ejde-984	180	25	∈	∈	PROPN
ejde-984	180	26	zc,µ	zc,µ	PROPN
ejde-984	180	27	for	for	ADP
ejde-984	180	28	each	each	DET
ejde-984	180	29	θ	θ	PROPN
ejde-984	180	30	∈	∈	PROPN
ejde-984	180	31	r.	r.	PROPN
ejde-984	180	32	moreover	moreover	ADV
ejde-984	180	33	,	,	PUNCT
ejde-984	180	34	if	if	SCONJ
ejde-984	180	35	u	u	NOUN
ejde-984	180	36	is	be	AUX
ejde-984	180	37	a	a	DET
ejde-984	180	38	ground	ground	NOUN
ejde-984	180	39	state	state	NOUN
ejde-984	180	40	,	,	PUNCT
ejde-984	180	41	then	then	ADV
ejde-984	180	42	the	the	DET
ejde-984	180	43	associated	associated	ADJ
ejde-984	180	44	lagrange	lagrange	PROPN
ejde-984	180	45	multiplier	multipli	ADJ
ejde-984	180	46	λ	λ	PROPN
ejde-984	180	47	is	be	AUX
ejde-984	180	48	negative	negative	ADJ
ejde-984	180	49	.	.	PUNCT
ejde-984	181	1	3	3	X
ejde-984	181	2	.	.	PUNCT
ejde-984	181	3	subcritical	subcritical	ADJ
ejde-984	181	4	case	case	NOUN
ejde-984	181	5	in	in	ADP
ejde-984	181	6	this	this	DET
ejde-984	181	7	part	part	NOUN
ejde-984	181	8	we	we	PRON
ejde-984	181	9	prove	prove	VERB
ejde-984	181	10	theorems	theorem	NOUN
ejde-984	181	11	2.8	2.8	NUM
ejde-984	181	12	,	,	PUNCT
ejde-984	181	13	2.9	2.9	NUM
ejde-984	181	14	and	and	CCONJ
ejde-984	181	15	2.10	2.10	NUM
ejde-984	181	16	.	.	PUNCT
ejde-984	182	1	first	first	ADV
ejde-984	182	2	we	we	PRON
ejde-984	182	3	list	list	VERB
ejde-984	182	4	some	some	DET
ejde-984	182	5	lemmas	lemma	NOUN
ejde-984	182	6	.	.	PUNCT
ejde-984	183	1	since	since	SCONJ
ejde-984	183	2	some	some	PRON
ejde-984	183	3	of	of	ADP
ejde-984	183	4	these	these	DET
ejde-984	183	5	lemmas	lemma	NOUN
ejde-984	183	6	appear	appear	VERB
ejde-984	183	7	in	in	ADP
ejde-984	183	8	the	the	DET
ejde-984	183	9	references	reference	NOUN
ejde-984	183	10	,	,	PUNCT
ejde-984	183	11	we	we	PRON
ejde-984	183	12	omit	omit	VERB
ejde-984	183	13	their	their	PRON
ejde-984	183	14	proof	proof	NOUN
ejde-984	183	15	here	here	ADV
ejde-984	183	16	.	.	PUNCT
ejde-984	184	1	lemma	lemma	PROPN
ejde-984	184	2	3.1	3.1	NUM
ejde-984	184	3	(	(	PUNCT
ejde-984	184	4	jeanjean	jeanjean	PROPN
ejde-984	184	5	[	[	X
ejde-984	184	6	13	13	NUM
ejde-984	184	7	]	]	NUM
ejde-984	184	8	)	)	PUNCT
ejde-984	184	9	.	.	PUNCT
ejde-984	185	1	for	for	ADP
ejde-984	185	2	u	u	PROPN
ejde-984	185	3	∈	∈	PROPN
ejde-984	185	4	sc	sc	PROPN
ejde-984	185	5	and	and	CCONJ
ejde-984	185	6	s	s	PROPN
ejde-984	185	7	∈	∈	PROPN
ejde-984	185	8	r	r	NOUN
ejde-984	185	9	,	,	PUNCT
ejde-984	185	10	the	the	DET
ejde-984	185	11	map	map	NOUN
ejde-984	185	12	ϕ	ϕ	PROPN
ejde-984	185	13	7→	7→	NUM
ejde-984	185	14	s	s	NOUN
ejde-984	185	15	∗	∗	NOUN
ejde-984	185	16	ϕ	ϕ	NOUN
ejde-984	185	17	from	from	ADP
ejde-984	185	18	tusc	tusc	NOUN
ejde-984	185	19	to	to	ADP
ejde-984	185	20	ts∗usc	ts∗usc	PROPN
ejde-984	185	21	is	be	AUX
ejde-984	185	22	a	a	DET
ejde-984	185	23	linear	linear	ADJ
ejde-984	185	24	isomorphism	isomorphism	NOUN
ejde-984	185	25	with	with	ADP
ejde-984	185	26	inverse	inverse	NOUN
ejde-984	185	27	ψ	ψ	X
ejde-984	185	28	7→	7→	NUM
ejde-984	185	29	(	(	PUNCT
ejde-984	185	30	−s	−s	NOUN
ejde-984	185	31	)	)	PUNCT
ejde-984	185	32	∗	∗	NOUN
ejde-984	185	33	ψ	ψ	NOUN
ejde-984	185	34	,	,	PUNCT
ejde-984	185	35	where	where	SCONJ
ejde-984	185	36	tusc	tusc	NOUN
ejde-984	185	37	=	=	X
ejde-984	185	38	{	{	PUNCT
ejde-984	185	39	ϕ	ϕ	PROPN
ejde-984	185	40	∈	∈	PROPN
ejde-984	185	41	sc	sc	PROPN
ejde-984	185	42	:	:	PUNCT
ejde-984	185	43	∫	∫	PROPN
ejde-984	185	44	rn	rn	NOUN
ejde-984	185	45	uϕdx	uϕdx	PROPN
ejde-984	185	46	=	=	PUNCT
ejde-984	185	47	0	0	NUM
ejde-984	185	48	}	}	PUNCT
ejde-984	185	49	.	.	PUNCT
ejde-984	186	1	next	next	ADV
ejde-984	186	2	,	,	PUNCT
ejde-984	186	3	we	we	PRON
ejde-984	186	4	will	will	AUX
ejde-984	186	5	discuss	discuss	VERB
ejde-984	186	6	the	the	DET
ejde-984	186	7	convergence	convergence	NOUN
ejde-984	186	8	of	of	ADP
ejde-984	186	9	a	a	DET
ejde-984	186	10	class	class	NOUN
ejde-984	186	11	of	of	ADP
ejde-984	186	12	special	special	ADJ
ejde-984	186	13	ps	ps	NOUN
ejde-984	186	14	sequences	sequence	NOUN
ejde-984	186	15	satisfying	satisfy	VERB
ejde-984	186	16	appropriate	appropriate	ADJ
ejde-984	186	17	additional	additional	ADJ
ejde-984	186	18	assumptions	assumption	NOUN
ejde-984	186	19	.	.	PUNCT
ejde-984	187	1	the	the	DET
ejde-984	187	2	idea	idea	NOUN
ejde-984	187	3	used	use	VERB
ejde-984	187	4	in	in	ADP
ejde-984	187	5	the	the	DET
ejde-984	187	6	proof	proof	NOUN
ejde-984	187	7	was	be	AUX
ejde-984	187	8	first	first	ADV
ejde-984	187	9	introduced	introduce	VERB
ejde-984	187	10	by	by	ADP
ejde-984	187	11	jeanjean	jeanjean	PROPN
ejde-984	187	12	[	[	X
ejde-984	187	13	13	13	NUM
ejde-984	187	14	]	]	PUNCT
ejde-984	187	15	.	.	PUNCT
ejde-984	188	1	then	then	ADV
ejde-984	188	2	soave	soave	PROPN
ejde-984	188	3	[	[	X
ejde-984	188	4	24	24	NUM
ejde-984	188	5	]	]	PUNCT
ejde-984	188	6	applied	apply	VERB
ejde-984	188	7	this	this	DET
ejde-984	188	8	idea	idea	NOUN
ejde-984	188	9	to	to	PART
ejde-984	188	10	study	study	VERB
ejde-984	188	11	the	the	DET
ejde-984	188	12	normalized	normalize	VERB
ejde-984	188	13	solutions	solution	NOUN
ejde-984	188	14	to	to	ADP
ejde-984	188	15	the	the	DET
ejde-984	188	16	nonlinear	nonlinear	ADJ
ejde-984	188	17	schrödinger	schrödinger	NOUN
ejde-984	188	18	equation	equation	NOUN
ejde-984	188	19	with	with	ADP
ejde-984	188	20	mixed	mixed	ADJ
ejde-984	188	21	nonlinearities	nonlinearitie	NOUN
ejde-984	188	22	.	.	PUNCT
ejde-984	189	1	lemma	lemma	PROPN
ejde-984	189	2	3.2	3.2	NUM
ejde-984	189	3	(	(	PUNCT
ejde-984	189	4	compactness	compactness	NOUN
ejde-984	189	5	of	of	ADP
ejde-984	189	6	ps	ps	PROPN
ejde-984	189	7	sequences	sequence	NOUN
ejde-984	189	8	)	)	PUNCT
ejde-984	189	9	.	.	PUNCT
ejde-984	190	1	let	let	VERB
ejde-984	190	2	2	2	NUM
ejde-984	190	3	<	<	X
ejde-984	190	4	q	q	X
ejde-984	190	5	≤	≤	NUM
ejde-984	190	6	2	2	NUM
ejde-984	190	7	+	+	NUM
ejde-984	190	8	8s	8s	NUM
ejde-984	190	9	3	3	NUM
ejde-984	190	10	<	<	X
ejde-984	190	11	p	p	X
ejde-984	190	12	<	<	X
ejde-984	190	13	6	6	NUM
ejde-984	190	14	3−2s	3−2s	NUM
ejde-984	190	15	be	be	AUX
ejde-984	190	16	given	give	VERB
ejde-984	190	17	constants	constant	NOUN
ejde-984	190	18	.	.	PUNCT
ejde-984	191	1	we	we	PRON
ejde-984	191	2	suppose	suppose	VERB
ejde-984	191	3	that	that	SCONJ
ejde-984	191	4	{	{	PUNCT
ejde-984	191	5	un}n∈n	un}n∈n	NUM
ejde-984	191	6	⊂	⊂	PROPN
ejde-984	191	7	sc	sc	PROPN
ejde-984	191	8	is	be	AUX
ejde-984	191	9	a	a	DET
ejde-984	191	10	ps	ps	NOUN
ejde-984	191	11	sequence	sequence	NOUN
ejde-984	191	12	for	for	ADP
ejde-984	191	13	eµ|sc	eµ|sc	X
ejde-984	191	14	at	at	ADP
ejde-984	191	15	level	level	NOUN
ejde-984	191	16	c	c	PROPN
ejde-984	191	17	̸=	̸=	PROPN
ejde-984	191	18	0	0	PUNCT
ejde-984	191	19	and	and	CCONJ
ejde-984	191	20	it	it	PRON
ejde-984	191	21	holds	hold	VERB
ejde-984	191	22	(	(	PUNCT
ejde-984	191	23	i	i	NOUN
ejde-984	191	24	)	)	PUNCT
ejde-984	191	25	pµ(un	pµ(un	PROPN
ejde-984	191	26	)	)	PUNCT
ejde-984	191	27	→	→	SYM
ejde-984	191	28	0	0	NUM
ejde-984	191	29	as	as	ADP
ejde-984	191	30	n→	n→	PROPN
ejde-984	191	31	∞.	∞.	PROPN
ejde-984	191	32	(	(	PUNCT
ejde-984	191	33	ii	ii	PROPN
ejde-984	191	34	)	)	PUNCT
ejde-984	191	35	µ	µ	X
ejde-984	191	36	<	<	X
ejde-984	191	37	0	0	PUNCT
ejde-984	192	1	and	and	CCONJ
ejde-984	192	2	(	(	PUNCT
ejde-984	192	3	2.9	2.9	NUM
ejde-984	192	4	)	)	PUNCT
ejde-984	192	5	holds	hold	VERB
ejde-984	192	6	.	.	PUNCT
ejde-984	193	1	then	then	ADV
ejde-984	193	2	,	,	PUNCT
ejde-984	193	3	going	go	VERB
ejde-984	193	4	to	to	ADP
ejde-984	193	5	a	a	DET
ejde-984	193	6	subsequence	subsequence	NOUN
ejde-984	193	7	,	,	PUNCT
ejde-984	193	8	un	un	PROPN
ejde-984	193	9	→	→	SYM
ejde-984	193	10	u	u	NOUN
ejde-984	193	11	strongly	strongly	ADV
ejde-984	193	12	in	in	ADP
ejde-984	193	13	hs(r3	hs(r3	NUM
ejde-984	193	14	)	)	PUNCT
ejde-984	193	15	,	,	PUNCT
ejde-984	193	16	and	and	CCONJ
ejde-984	193	17	u	u	PROPN
ejde-984	193	18	∈	∈	PROPN
ejde-984	193	19	sc	sc	PROPN
ejde-984	193	20	is	be	AUX
ejde-984	193	21	a	a	DET
ejde-984	193	22	radial	radial	ADJ
ejde-984	193	23	solution	solution	NOUN
ejde-984	193	24	to	to	ADP
ejde-984	193	25	(	(	PUNCT
ejde-984	193	26	1.1	1.1	NUM
ejde-984	193	27	)	)	PUNCT
ejde-984	193	28	for	for	ADP
ejde-984	193	29	some	some	DET
ejde-984	193	30	λ	λ	NOUN
ejde-984	193	31	<	<	X
ejde-984	193	32	0	0	NUM
ejde-984	193	33	.	.	PUNCT
ejde-984	193	34	proof	proof	NOUN
ejde-984	193	35	.	.	PUNCT
ejde-984	194	1	in	in	ADP
ejde-984	194	2	this	this	DET
ejde-984	194	3	lemma	lemma	PROPN
ejde-984	194	4	,	,	PUNCT
ejde-984	194	5	we	we	PRON
ejde-984	194	6	argue	argue	VERB
ejde-984	194	7	directly	directly	ADV
ejde-984	194	8	.	.	PUNCT
ejde-984	195	1	since	since	SCONJ
ejde-984	195	2	pµ(un	pµ(un	PROPN
ejde-984	195	3	)	)	PUNCT
ejde-984	195	4	→	→	SYM
ejde-984	195	5	0	0	NUM
ejde-984	195	6	as	as	ADP
ejde-984	195	7	n→	n→	PROPN
ejde-984	195	8	∞	∞	PROPN
ejde-984	195	9	,	,	PUNCT
ejde-984	195	10	we	we	PRON
ejde-984	195	11	have	have	VERB
ejde-984	195	12	that	that	DET
ejde-984	195	13	a|(−∆)s/2un|22	a|(−∆)s/2un|22	PROPN
ejde-984	195	14	+	+	CCONJ
ejde-984	195	15	b|(−∆)s/2un|42	b|(−∆)s/2un|42	PROPN
ejde-984	195	16	−	−	PROPN
ejde-984	195	17	µδs	µδ	NOUN
ejde-984	195	18	,	,	PUNCT
ejde-984	195	19	q|un|qq	q|un|qq	NOUN
ejde-984	195	20	−	−	PROPN
ejde-984	195	21	δs	δs	NOUN
ejde-984	195	22	,	,	PUNCT
ejde-984	195	23	p|un|pp	p|un|pp	NOUN
ejde-984	195	24	=	=	SYM
ejde-984	195	25	o(1	o(1	PROPN
ejde-984	195	26	)	)	PUNCT
ejde-984	195	27	,	,	PUNCT
ejde-984	195	28	(	(	PUNCT
ejde-984	195	29	3.1	3.1	NUM
ejde-984	195	30	)	)	PUNCT
ejde-984	195	31	as	as	ADP
ejde-984	195	32	n→	n→	PROPN
ejde-984	195	33	∞.	∞.	PROPN
ejde-984	195	34	therefore	therefore	ADV
ejde-984	195	35	,	,	PUNCT
ejde-984	195	36	c+	c+	VERB
ejde-984	195	37	1	1	NUM
ejde-984	195	38	≥	≥	NOUN
ejde-984	195	39	eµ(un	eµ(un	PROPN
ejde-984	195	40	)	)	PUNCT
ejde-984	195	41	=	=	PUNCT
ejde-984	196	1	a	a	DET
ejde-984	196	2	2	2	NUM
ejde-984	196	3	∫	∫	NOUN
ejde-984	196	4	r3	r3	PROPN
ejde-984	196	5	|(−∆)s/2un|2dx+	|(−∆)s/2un|2dx+	PROPN
ejde-984	196	6	b	b	PROPN
ejde-984	196	7	4	4	NUM
ejde-984	196	8	(	(	PUNCT
ejde-984	196	9	∫	∫	PROPN
ejde-984	196	10	r3	r3	PROPN
ejde-984	196	11	|(−∆)s/2un|2dx	|(−∆)s/2un|2dx	NUM
ejde-984	196	12	)	)	PUNCT
ejde-984	196	13	2	2	NUM
ejde-984	196	14	−	−	PROPN
ejde-984	196	15	µ	µ	PRON
ejde-984	196	16	q	q	NOUN
ejde-984	196	17	∫	∫	PROPN
ejde-984	196	18	r3	r3	PROPN
ejde-984	196	19	|un|qdx−	|un|qdx−	ADP
ejde-984	196	20	1	1	NUM
ejde-984	196	21	p	p	NOUN
ejde-984	196	22	∫	∫	PROPN
ejde-984	196	23	r3	r3	PROPN
ejde-984	196	24	|un|pdx	|un|pdx	NOUN
ejde-984	196	25	=	=	PUNCT
ejde-984	196	26	a	a	DET
ejde-984	196	27	(	(	PUNCT
ejde-984	196	28	1	1	NUM
ejde-984	196	29	2	2	NUM
ejde-984	196	30	−	−	NOUN
ejde-984	196	31	1	1	NUM
ejde-984	196	32	pδs	pδs	NOUN
ejde-984	196	33	,	,	PUNCT
ejde-984	196	34	p	p	NOUN
ejde-984	196	35	)	)	PUNCT
ejde-984	196	36	∫	∫	PROPN
ejde-984	196	37	r3	r3	PROPN
ejde-984	196	38	|(−∆)s/2un|2dx+	|(−∆)s/2un|2dx+	PROPN
ejde-984	196	39	b	b	PROPN
ejde-984	196	40	(	(	PUNCT
ejde-984	196	41	1	1	NUM
ejde-984	196	42	4	4	NUM
ejde-984	196	43	−	−	NUM
ejde-984	196	44	1	1	NUM
ejde-984	196	45	pδs	pδs	NOUN
ejde-984	196	46	,	,	PUNCT
ejde-984	196	47	p	p	NOUN
ejde-984	196	48	)	)	PUNCT
ejde-984	196	49	(	(	PUNCT
ejde-984	196	50	∫	∫	PROPN
ejde-984	196	51	r3	r3	PROPN
ejde-984	196	52	|(−∆)s/2un|2dx	|(−∆)s/2un|2dx	NUM
ejde-984	196	53	)	)	PUNCT
ejde-984	196	54	2	2	NUM
ejde-984	196	55	−	−	PROPN
ejde-984	196	56	µ	µ	X
ejde-984	196	57	q	q	X
ejde-984	197	1	(	(	PUNCT
ejde-984	197	2	1−	1−	NUM
ejde-984	197	3	qδs	qδs	NOUN
ejde-984	197	4	,	,	PUNCT
ejde-984	197	5	q	q	PROPN
ejde-984	197	6	pδs	pδs	PROPN
ejde-984	197	7	,	,	PUNCT
ejde-984	197	8	p	p	NOUN
ejde-984	197	9	)	)	PUNCT
ejde-984	197	10	∫	∫	PROPN
ejde-984	197	11	r3	r3	PROPN
ejde-984	197	12	|un|qdx+	|un|qdx+	NOUN
ejde-984	197	13	o(1	o(1	NOUN
ejde-984	197	14	)	)	PUNCT
ejde-984	197	15	≥	≥	NOUN
ejde-984	197	16	a	a	DET
ejde-984	197	17	(	(	PUNCT
ejde-984	197	18	1	1	NUM
ejde-984	197	19	2	2	NUM
ejde-984	197	20	−	−	NOUN
ejde-984	197	21	1	1	NUM
ejde-984	197	22	pδs	pδs	NOUN
ejde-984	197	23	,	,	PUNCT
ejde-984	197	24	p	p	NOUN
ejde-984	197	25	)	)	PUNCT
ejde-984	197	26	∫	∫	PROPN
ejde-984	197	27	r3	r3	PROPN
ejde-984	197	28	|(−∆)s/2un|2dx+	|(−∆)s/2un|2dx+	PROPN
ejde-984	197	29	b	b	PROPN
ejde-984	197	30	(	(	PUNCT
ejde-984	197	31	1	1	NUM
ejde-984	197	32	4	4	NUM
ejde-984	197	33	−	−	NUM
ejde-984	197	34	1	1	NUM
ejde-984	197	35	pδs	pδs	NOUN
ejde-984	197	36	,	,	PUNCT
ejde-984	197	37	p	p	NOUN
ejde-984	197	38	)	)	PUNCT
ejde-984	197	39	(	(	PUNCT
ejde-984	197	40	∫	∫	PROPN
ejde-984	197	41	r3	r3	PROPN
ejde-984	197	42	|(−∆)s/2un|2dx	|(−∆)s/2un|2dx	NUM
ejde-984	197	43	)	)	PUNCT
ejde-984	197	44	2	2	NUM
ejde-984	197	45	+	+	NUM
ejde-984	197	46	o(1	o(1	NOUN
ejde-984	197	47	)	)	PUNCT
ejde-984	197	48	,	,	PUNCT
ejde-984	197	49	as	as	ADP
ejde-984	197	50	n	n	X
ejde-984	197	51	→	→	SYM
ejde-984	197	52	∞	∞	PROPN
ejde-984	197	53	,	,	PUNCT
ejde-984	197	54	where	where	SCONJ
ejde-984	197	55	we	we	PRON
ejde-984	197	56	used	use	VERB
ejde-984	197	57	the	the	DET
ejde-984	197	58	fact	fact	NOUN
ejde-984	197	59	that	that	SCONJ
ejde-984	197	60	eµ(un	eµ(un	PROPN
ejde-984	197	61	)	)	PUNCT
ejde-984	197	62	→	→	SYM
ejde-984	198	1	c	c	X
ejde-984	198	2	,	,	PUNCT
ejde-984	198	3	the	the	DET
ejde-984	198	4	equivalent	equivalent	ADJ
ejde-984	198	5	deformation	deformation	NOUN
ejde-984	198	6	of	of	ADP
ejde-984	198	7	(	(	PUNCT
ejde-984	198	8	3.1	3.1	NUM
ejde-984	198	9	)	)	PUNCT
ejde-984	198	10	and	and	CCONJ
ejde-984	198	11	−µ	−µ	ADJ
ejde-984	198	12	q	q	X
ejde-984	198	13	(	(	PUNCT
ejde-984	198	14	1	1	NUM
ejde-984	198	15	−	−	NOUN
ejde-984	198	16	qδs	qδs	PROPN
ejde-984	198	17	,	,	PUNCT
ejde-984	198	18	q	q	NOUN
ejde-984	198	19	pδs	pδs	PROPN
ejde-984	198	20	,	,	PUNCT
ejde-984	198	21	p	p	NOUN
ejde-984	198	22	)	)	PUNCT
ejde-984	198	23	>	>	X
ejde-984	198	24	0	0	PUNCT
ejde-984	199	1	(	(	PUNCT
ejde-984	199	2	since	since	SCONJ
ejde-984	199	3	µ	µ	X
ejde-984	199	4	<	<	X
ejde-984	199	5	0	0	NUM
ejde-984	199	6	,	,	PUNCT
ejde-984	199	7	0	0	NUM
ejde-984	199	8	<	<	X
ejde-984	199	9	qδs	qδs	PROPN
ejde-984	199	10	,	,	PUNCT
ejde-984	199	11	q	q	X
ejde-984	199	12	<	<	X
ejde-984	199	13	pδs	pδs	PROPN
ejde-984	199	14	,	,	PUNCT
ejde-984	199	15	p	p	NOUN
ejde-984	199	16	)	)	PUNCT
ejde-984	199	17	.	.	PUNCT
ejde-984	200	1	from	from	ADP
ejde-984	200	2	the	the	DET
ejde-984	200	3	above	above	ADJ
ejde-984	200	4	inequality	inequality	NOUN
ejde-984	200	5	and	and	CCONJ
ejde-984	200	6	the	the	DET
ejde-984	200	7	fact	fact	NOUN
ejde-984	200	8	that	that	SCONJ
ejde-984	200	9	|un|22	|un|22	AUX
ejde-984	200	10	=	=	SYM
ejde-984	200	11	c2	c2	PROPN
ejde-984	200	12	,	,	PUNCT
ejde-984	200	13	we	we	PRON
ejde-984	200	14	deduce	deduce	VERB
ejde-984	200	15	that	that	SCONJ
ejde-984	200	16	{	{	PUNCT
ejde-984	200	17	un	un	PROPN
ejde-984	200	18	}	}	PUNCT
ejde-984	200	19	is	be	AUX
ejde-984	200	20	a	a	DET
ejde-984	200	21	bounded	bounded	ADJ
ejde-984	200	22	sequence	sequence	NOUN
ejde-984	200	23	in	in	ADP
ejde-984	200	24	hs(r3	hs(r3	NUM
ejde-984	200	25	)	)	PUNCT
ejde-984	200	26	.	.	PUNCT
ejde-984	201	1	besides	besides	ADV
ejde-984	201	2	,	,	PUNCT
ejde-984	201	3	hilbert	hilbert	NOUN
ejde-984	201	4	space	space	NOUN
ejde-984	201	5	hs(r3	hs(r3	ADV
ejde-984	201	6	)	)	PUNCT
ejde-984	201	7	is	be	AUX
ejde-984	201	8	a	a	DET
ejde-984	201	9	reflexive	reflexive	ADJ
ejde-984	201	10	banach	banach	NOUN
ejde-984	201	11	space	space	NOUN
ejde-984	201	12	.	.	PUNCT
ejde-984	202	1	in	in	ADP
ejde-984	202	2	the	the	DET
ejde-984	202	3	reflexive	reflexive	ADJ
ejde-984	202	4	banach	banach	NOUN
ejde-984	202	5	space	space	NOUN
ejde-984	202	6	hs(r3	hs(r3	PUNCT
ejde-984	202	7	)	)	PUNCT
ejde-984	202	8	,	,	PUNCT
ejde-984	202	9	bounded	bound	VERB
ejde-984	202	10	sequence	sequence	NOUN
ejde-984	202	11	{	{	PUNCT
ejde-984	202	12	un	un	PROPN
ejde-984	202	13	}	}	PUNCT
ejde-984	202	14	has	have	VERB
ejde-984	202	15	weakly	weakly	ADJ
ejde-984	202	16	convergent	convergent	ADJ
ejde-984	202	17	subsequence	subsequence	NOUN
ejde-984	202	18	{	{	PUNCT
ejde-984	202	19	un	un	PROPN
ejde-984	202	20	}	}	PUNCT
ejde-984	202	21	(	(	PUNCT
ejde-984	202	22	for	for	ADP
ejde-984	202	23	the	the	DET
ejde-984	202	24	sake	sake	NOUN
ejde-984	202	25	of	of	ADP
ejde-984	202	26	brevity	brevity	NOUN
ejde-984	202	27	,	,	PUNCT
ejde-984	202	28	the	the	DET
ejde-984	202	29	subsequence	subsequence	NOUN
ejde-984	202	30	of	of	ADP
ejde-984	202	31	{	{	PUNCT
ejde-984	202	32	un	un	PROPN
ejde-984	202	33	}	}	PUNCT
ejde-984	202	34	is	be	AUX
ejde-984	202	35	still	still	ADV
ejde-984	202	36	represented	represent	VERB
ejde-984	202	37	by	by	ADP
ejde-984	202	38	{	{	PUNCT
ejde-984	202	39	un	un	PROPN
ejde-984	202	40	}	}	PUNCT
ejde-984	202	41	)	)	PUNCT
ejde-984	202	42	.	.	PUNCT
ejde-984	203	1	according	accord	VERB
ejde-984	203	2	to	to	ADP
ejde-984	203	3	lemma	lemma	PROPN
ejde-984	203	4	2.6	2.6	NUM
ejde-984	203	5	,	,	PUNCT
ejde-984	203	6	hs	hs	INTJ
ejde-984	203	7	r	r	PROPN
ejde-984	203	8	(	(	PUNCT
ejde-984	203	9	r3	r3	PROPN
ejde-984	203	10	)	)	PUNCT
ejde-984	203	11	↪	↪	PROPN
ejde-984	203	12	→	→	SYM
ejde-984	203	13	lp(r3	lp(r3	ADV
ejde-984	203	14	)	)	PUNCT
ejde-984	203	15	compactly	compactly	ADV
ejde-984	203	16	for	for	ADP
ejde-984	203	17	p	p	PROPN
ejde-984	203	18	∈	∈	PROPN
ejde-984	203	19	(	(	PUNCT
ejde-984	203	20	2	2	NUM
ejde-984	203	21	,	,	PUNCT
ejde-984	203	22	2∗s	2∗s	NUM
ejde-984	203	23	)	)	PUNCT
ejde-984	203	24	,	,	PUNCT
ejde-984	203	25	there	there	PRON
ejde-984	203	26	exists	exist	VERB
ejde-984	203	27	u	u	PROPN
ejde-984	203	28	∈	∈	PROPN
ejde-984	203	29	hs	hs	INTJ
ejde-984	203	30	r	r	PROPN
ejde-984	203	31	(	(	PUNCT
ejde-984	203	32	r3	r3	PROPN
ejde-984	203	33	)	)	PUNCT
ejde-984	203	34	such	such	ADJ
ejde-984	203	35	that	that	SCONJ
ejde-984	203	36	un	un	PROPN
ejde-984	203	37	⇀	⇀	PROPN
ejde-984	203	38	u	u	PROPN
ejde-984	203	39	in	in	ADP
ejde-984	203	40	hs	hs	PROPN
ejde-984	203	41	r	r	PROPN
ejde-984	203	42	(	(	PUNCT
ejde-984	203	43	r3	r3	PROPN
ejde-984	203	44	)	)	PUNCT
ejde-984	203	45	,	,	PUNCT
ejde-984	203	46	un	un	PROPN
ejde-984	203	47	→	→	SYM
ejde-984	203	48	u	u	PROPN
ejde-984	203	49	in	in	ADP
ejde-984	203	50	lp(r3	lp(r3	NOUN
ejde-984	203	51	)	)	PUNCT
ejde-984	203	52	,	,	PUNCT
ejde-984	203	53	un(x	un(x	X
ejde-984	203	54	)	)	PUNCT
ejde-984	203	55	→	→	SYM
ejde-984	203	56	u(x	u(x	PROPN
ejde-984	203	57	)	)	PUNCT
ejde-984	203	58	a.e	a.e	PROPN
ejde-984	203	59	.	.	PROPN
ejde-984	203	60	in	in	ADP
ejde-984	203	61	r3	r3	PROPN
ejde-984	203	62	,	,	PUNCT
ejde-984	203	63	(	(	PUNCT
ejde-984	203	64	3.2	3.2	NUM
ejde-984	203	65	)	)	PUNCT
ejde-984	203	66	as	as	ADP
ejde-984	203	67	n→	n→	PUNCT
ejde-984	203	68	∞.	∞.	PROPN
ejde-984	203	69	8	8	NUM
ejde-984	203	70	z.	z.	PROPN
ejde-984	203	71	guo	guo	PROPN
ejde-984	203	72	,	,	PUNCT
ejde-984	203	73	t.	t.	PROPN
ejde-984	203	74	zhang	zhang	PROPN
ejde-984	203	75	ejde-2025/75	ejde-2025/75	NOUN
ejde-984	203	76	since	since	SCONJ
ejde-984	203	77	{	{	PUNCT
ejde-984	203	78	un	un	ADJ
ejde-984	203	79	}	}	PUNCT
ejde-984	203	80	is	be	AUX
ejde-984	203	81	a	a	DET
ejde-984	203	82	bounded	bounded	ADJ
ejde-984	203	83	ps	ps	NOUN
ejde-984	203	84	sequence	sequence	NOUN
ejde-984	203	85	of	of	ADP
ejde-984	203	86	eµ|sc	eµ|sc	X
ejde-984	203	87	,	,	PUNCT
ejde-984	203	88	by	by	ADP
ejde-984	203	89	applying	apply	VERB
ejde-984	203	90	the	the	DET
ejde-984	203	91	lagrange	lagrange	NOUN
ejde-984	203	92	multipliers	multiplier	NOUN
ejde-984	203	93	rule	rule	NOUN
ejde-984	203	94	,	,	PUNCT
ejde-984	203	95	we	we	PRON
ejde-984	203	96	conclude	conclude	VERB
ejde-984	203	97	that	that	SCONJ
ejde-984	203	98	there	there	PRON
ejde-984	203	99	exists	exist	VERB
ejde-984	203	100	λn	λn	PROPN
ejde-984	203	101	∈	∈	PROPN
ejde-984	203	102	r	r	NOUN
ejde-984	203	103	such	such	ADJ
ejde-984	203	104	that	that	SCONJ
ejde-984	203	105	a	a	DET
ejde-984	203	106	∫	∫	PROPN
ejde-984	203	107	r3	r3	PROPN
ejde-984	203	108	(	(	PUNCT
ejde-984	203	109	−∆)s/2un(−∆)s/2ϕdx+	−∆)s/2un(−∆)s/2ϕdx+	NUM
ejde-984	203	110	b|(−∆)s/2un|22	b|(−∆)s/2un|22	NOUN
ejde-984	203	111	∫	∫	PROPN
ejde-984	203	112	r3	r3	PROPN
ejde-984	203	113	(	(	PUNCT
ejde-984	203	114	−∆)s/2un(−∆)s/2ϕdx	−∆)s/2un(−∆)s/2ϕdx	ADV
ejde-984	203	115	−	−	PROPN
ejde-984	203	116	µ	µ	PROPN
ejde-984	203	117	∫	∫	PROPN
ejde-984	203	118	r3	r3	PROPN
ejde-984	203	119	|un|q−2unϕdx−	|un|q−2unϕdx−	NOUN
ejde-984	203	120	∫	∫	PROPN
ejde-984	203	121	r3	r3	PROPN
ejde-984	203	122	|un|p−2unϕdx−	|un|p−2unϕdx−	PROPN
ejde-984	203	123	λn	λn	PROPN
ejde-984	203	124	∫	∫	PROPN
ejde-984	203	125	r3	r3	PROPN
ejde-984	203	126	unϕdx	unϕdx	PROPN
ejde-984	203	127	=	=	SYM
ejde-984	204	1	o(1)∥ϕ∥hs	o(1)∥ϕ∥hs	PROPN
ejde-984	204	2	,	,	PUNCT
ejde-984	204	3	(	(	PUNCT
ejde-984	204	4	3.3	3.3	NUM
ejde-984	204	5	)	)	PUNCT
ejde-984	204	6	for	for	ADP
ejde-984	204	7	all	all	DET
ejde-984	204	8	ϕ	ϕ	PROPN
ejde-984	204	9	∈	∈	PROPN
ejde-984	204	10	hs(r3	hs(r3	NUM
ejde-984	204	11	)	)	PUNCT
ejde-984	204	12	.	.	PUNCT
ejde-984	205	1	letting	let	VERB
ejde-984	205	2	ϕ	ϕ	X
ejde-984	205	3	=	=	SYM
ejde-984	205	4	un	un	PROPN
ejde-984	205	5	,	,	PUNCT
ejde-984	205	6	we	we	PRON
ejde-984	205	7	have	have	VERB
ejde-984	205	8	that	that	DET
ejde-984	205	9	a|(−∆)s/2un|22	a|(−∆)s/2un|22	PROPN
ejde-984	205	10	+	+	CCONJ
ejde-984	206	1	b|(−∆)s/2un|42	b|(−∆)s/2un|42	PROPN
ejde-984	206	2	−	−	NOUN
ejde-984	206	3	µ|un|qq	µ|un|qq	NOUN
ejde-984	206	4	−	−	NOUN
ejde-984	206	5	|un|pp	|un|pp	PUNCT
ejde-984	206	6	−	−	X
ejde-984	206	7	λn|un|22	λn|un|22	X
ejde-984	207	1	=	=	SYM
ejde-984	207	2	o(1)∥un∥hs	o(1)∥un∥hs	PROPN
ejde-984	207	3	.	.	PUNCT
ejde-984	208	1	(	(	PUNCT
ejde-984	208	2	3.4	3.4	NUM
ejde-984	208	3	)	)	PUNCT
ejde-984	208	4	therefore	therefore	ADV
ejde-984	208	5	,	,	PUNCT
ejde-984	208	6	λn	λn	PROPN
ejde-984	208	7	=	=	SYM
ejde-984	208	8	1	1	NUM
ejde-984	208	9	c2	c2	PROPN
ejde-984	208	10	(	(	PUNCT
ejde-984	208	11	a|(−∆)s/2un|22	a|(−∆)s/2un|22	PROPN
ejde-984	208	12	+	+	CCONJ
ejde-984	208	13	b|(−∆)s/2un|42	b|(−∆)s/2un|42	PROPN
ejde-984	208	14	−	−	NOUN
ejde-984	208	15	µ|un|qq	µ|un|qq	NOUN
ejde-984	208	16	−	−	NOUN
ejde-984	208	17	|un|pp	|un|pp	PUNCT
ejde-984	208	18	)	)	PUNCT
ejde-984	209	1	+	+	CCONJ
ejde-984	209	2	o(1)∥un∥hs	o(1)∥un∥hs	PROPN
ejde-984	209	3	.	.	PUNCT
ejde-984	210	1	(	(	PUNCT
ejde-984	210	2	3.5	3.5	NUM
ejde-984	210	3	)	)	PUNCT
ejde-984	210	4	as	as	SCONJ
ejde-984	210	5	{	{	PUNCT
ejde-984	210	6	un	un	PROPN
ejde-984	210	7	}	}	PUNCT
ejde-984	210	8	is	be	AUX
ejde-984	210	9	a	a	DET
ejde-984	210	10	bounded	bounded	ADJ
ejde-984	210	11	sequence	sequence	NOUN
ejde-984	210	12	in	in	ADP
ejde-984	210	13	hs(r3)∩lp(r3)∩lq(r3	hs(r3)∩lp(r3)∩lq(r3	PROPN
ejde-984	210	14	)	)	PUNCT
ejde-984	210	15	,	,	PUNCT
ejde-984	210	16	from	from	ADP
ejde-984	210	17	the	the	DET
ejde-984	210	18	above	above	ADJ
ejde-984	210	19	equation	equation	NOUN
ejde-984	210	20	we	we	PRON
ejde-984	210	21	obtain	obtain	VERB
ejde-984	210	22	that	that	SCONJ
ejde-984	210	23	{	{	PUNCT
ejde-984	210	24	λn	λn	NOUN
ejde-984	210	25	}	}	PUNCT
ejde-984	210	26	is	be	AUX
ejde-984	210	27	a	a	DET
ejde-984	210	28	bounded	bounded	ADJ
ejde-984	210	29	sequence	sequence	NOUN
ejde-984	210	30	.	.	PUNCT
ejde-984	211	1	thus	thus	ADV
ejde-984	211	2	,	,	PUNCT
ejde-984	211	3	going	go	VERB
ejde-984	211	4	if	if	SCONJ
ejde-984	211	5	necessary	necessary	ADJ
ejde-984	211	6	to	to	ADP
ejde-984	211	7	a	a	DET
ejde-984	211	8	subsequence	subsequence	NOUN
ejde-984	211	9	,	,	PUNCT
ejde-984	211	10	there	there	PRON
ejde-984	211	11	exists	exist	VERB
ejde-984	211	12	λ	λ	X
ejde-984	211	13	∈	∈	NOUN
ejde-984	211	14	r	r	NOUN
ejde-984	211	15	such	such	ADJ
ejde-984	211	16	that	that	DET
ejde-984	211	17	λn	λn	NOUN
ejde-984	211	18	→	→	SYM
ejde-984	211	19	λ	λ	PROPN
ejde-984	211	20	(	(	PUNCT
ejde-984	211	21	3.6	3.6	NUM
ejde-984	211	22	)	)	PUNCT
ejde-984	211	23	as	as	ADP
ejde-984	211	24	n→	n→	PUNCT
ejde-984	211	25	∞.	∞.	PROPN
ejde-984	211	26	in	in	ADP
ejde-984	211	27	the	the	DET
ejde-984	211	28	remaining	remain	VERB
ejde-984	211	29	of	of	ADP
ejde-984	211	30	this	this	DET
ejde-984	211	31	proof	proof	NOUN
ejde-984	211	32	,	,	PUNCT
ejde-984	211	33	we	we	PRON
ejde-984	211	34	prove	prove	VERB
ejde-984	211	35	that	that	SCONJ
ejde-984	211	36	λ	λ	PROPN
ejde-984	211	37	<	<	X
ejde-984	211	38	0	0	NUM
ejde-984	211	39	.	.	PUNCT
ejde-984	211	40	from	from	ADP
ejde-984	211	41	pµ(un	pµ(un	PROPN
ejde-984	211	42	)	)	PUNCT
ejde-984	211	43	→	→	SYM
ejde-984	211	44	0	0	NUM
ejde-984	211	45	as	as	ADP
ejde-984	211	46	n→	n→	PROPN
ejde-984	211	47	∞	∞	PROPN
ejde-984	211	48	,	,	PUNCT
ejde-984	211	49	we	we	PRON
ejde-984	211	50	obtain	obtain	VERB
ejde-984	211	51	that	that	SCONJ
ejde-984	211	52	a|(−∆)s/2un|22	a|(−∆)s/2un|22	PROPN
ejde-984	211	53	+	+	CCONJ
ejde-984	211	54	b|(−∆)s/2un|42	b|(−∆)s/2un|42	NOUN
ejde-984	211	55	=	=	PUNCT
ejde-984	211	56	µδs	µδ	NOUN
ejde-984	211	57	,	,	PUNCT
ejde-984	211	58	q|un|qq	q|un|qq	NOUN
ejde-984	211	59	+	+	CCONJ
ejde-984	211	60	δs	δs	NOUN
ejde-984	211	61	,	,	PUNCT
ejde-984	211	62	p|un|pp	p|un|pp	ADV
ejde-984	211	63	+	+	CCONJ
ejde-984	211	64	o(1	o(1	NOUN
ejde-984	211	65	)	)	PUNCT
ejde-984	211	66	≤	≤	NOUN
ejde-984	211	67	δs	δs	NOUN
ejde-984	211	68	,	,	PUNCT
ejde-984	211	69	p|un|pp	p|un|pp	ADV
ejde-984	211	70	+	+	CCONJ
ejde-984	211	71	o(1	o(1	NOUN
ejde-984	211	72	)	)	PUNCT
ejde-984	211	73	.	.	PUNCT
ejde-984	212	1	(	(	PUNCT
ejde-984	212	2	3.7	3.7	NUM
ejde-984	212	3	)	)	PUNCT
ejde-984	212	4	employing	employ	VERB
ejde-984	212	5	the	the	DET
ejde-984	212	6	fractional	fractional	ADJ
ejde-984	212	7	gagliardo	gagliardo	NOUN
ejde-984	212	8	-	-	PUNCT
ejde-984	212	9	nirenberg	nirenberg	NOUN
ejde-984	212	10	inequality	inequality	NOUN
ejde-984	212	11	(	(	PUNCT
ejde-984	212	12	2.2	2.2	NUM
ejde-984	212	13	)	)	PUNCT
ejde-984	212	14	,	,	PUNCT
ejde-984	212	15	we	we	PRON
ejde-984	212	16	derive	derive	VERB
ejde-984	212	17	that	that	SCONJ
ejde-984	212	18	a|(−∆)s/2un|22	a|(−∆)s/2un|22	PROPN
ejde-984	212	19	≤	≤	NUM
ejde-984	212	20	a|(−∆)s/2un|22	a|(−∆)s/2un|22	PROPN
ejde-984	212	21	+	+	CCONJ
ejde-984	212	22	b|(−∆)s/2un|42	b|(−∆)s/2un|42	PROPN
ejde-984	212	23	≤	≤	NUM
ejde-984	212	24	δs	δs	NOUN
ejde-984	212	25	,	,	PUNCT
ejde-984	212	26	p|un|pp	p|un|pp	ADV
ejde-984	212	27	+	+	CCONJ
ejde-984	212	28	o(1	o(1	NOUN
ejde-984	212	29	)	)	PUNCT
ejde-984	212	30	≤	≤	NOUN
ejde-984	212	31	δs	δs	NOUN
ejde-984	212	32	,	,	PUNCT
ejde-984	212	33	pc(s	pc(s	X
ejde-984	212	34	,	,	PUNCT
ejde-984	212	35	p	p	X
ejde-984	212	36	)	)	PUNCT
ejde-984	212	37	p|(−∆)s/2un|	p|(−∆)s/2un|	NOUN
ejde-984	212	38	pδs	pδs	NOUN
ejde-984	212	39	,	,	PUNCT
ejde-984	212	40	p	p	PROPN
ejde-984	212	41	2	2	NUM
ejde-984	212	42	|un|	|un|	NOUN
ejde-984	212	43	p(1−δs	p(1−δs	NOUN
ejde-984	212	44	,	,	PUNCT
ejde-984	212	45	p	p	NOUN
ejde-984	212	46	)	)	PUNCT
ejde-984	212	47	2	2	NUM
ejde-984	212	48	+	+	NUM
ejde-984	212	49	o(1	o(1	NOUN
ejde-984	212	50	)	)	PUNCT
ejde-984	212	51	.	.	PUNCT
ejde-984	213	1	(	(	PUNCT
ejde-984	213	2	3.8	3.8	NUM
ejde-984	213	3	)	)	PUNCT
ejde-984	213	4	it	it	PRON
ejde-984	213	5	is	be	AUX
ejde-984	213	6	easy	easy	ADJ
ejde-984	213	7	to	to	PART
ejde-984	213	8	obtain	obtain	VERB
ejde-984	213	9	that	that	SCONJ
ejde-984	213	10	u	u	NOUN
ejde-984	213	11	̸≡	̸≡	NOUN
ejde-984	213	12	0	0	NUM
ejde-984	213	13	:	:	PUNCT
ejde-984	213	14	assuming	assume	VERB
ejde-984	213	15	by	by	ADP
ejde-984	213	16	contradiction	contradiction	NOUN
ejde-984	213	17	that	that	PRON
ejde-984	213	18	u	u	PRON
ejde-984	213	19	≡	≡	PROPN
ejde-984	213	20	0	0	NUM
ejde-984	213	21	,	,	PUNCT
ejde-984	213	22	then	then	ADV
ejde-984	213	23	we	we	PRON
ejde-984	213	24	obtain	obtain	VERB
ejde-984	213	25	that	that	SCONJ
ejde-984	213	26	limn→∞	limn→∞	ADJ
ejde-984	213	27	|un|qq	|un|qq	NOUN
ejde-984	213	28	=	=	SYM
ejde-984	213	29	limn→∞	limn→∞	X
ejde-984	213	30	|un|pp	|un|pp	X
ejde-984	213	31	=	=	SYM
ejde-984	213	32	0	0	X
ejde-984	213	33	.	.	PUNCT
ejde-984	213	34	using	use	VERB
ejde-984	213	35	that	that	DET
ejde-984	213	36	pµ(un	pµ(un	PROPN
ejde-984	213	37	)	)	PUNCT
ejde-984	214	1	→	→	SYM
ejde-984	214	2	0	0	NUM
ejde-984	214	3	we	we	PRON
ejde-984	214	4	deduce	deduce	VERB
ejde-984	214	5	that	that	SCONJ
ejde-984	214	6	eµ(un	eµ(un	PROPN
ejde-984	214	7	)	)	PUNCT
ejde-984	214	8	→	→	SYM
ejde-984	214	9	0	0	NUM
ejde-984	214	10	,	,	PUNCT
ejde-984	214	11	while	while	SCONJ
ejde-984	214	12	this	this	PRON
ejde-984	214	13	contradicts	contradict	VERB
ejde-984	214	14	the	the	DET
ejde-984	214	15	assumption	assumption	NOUN
ejde-984	214	16	that	that	SCONJ
ejde-984	214	17	eµ(un	eµ(un	PROPN
ejde-984	214	18	)	)	PUNCT
ejde-984	214	19	→	→	PUNCT
ejde-984	215	1	c	c	PROPN
ejde-984	215	2	̸=	̸=	PROPN
ejde-984	215	3	0	0	NUM
ejde-984	215	4	.	.	PUNCT
ejde-984	216	1	thus	thus	ADV
ejde-984	216	2	we	we	PRON
ejde-984	216	3	have	have	VERB
ejde-984	216	4	u	u	NOUN
ejde-984	216	5	̸≡	̸≡	PROPN
ejde-984	216	6	0	0	NUM
ejde-984	216	7	.	.	PUNCT
ejde-984	217	1	since	since	SCONJ
ejde-984	217	2	un	un	PROPN
ejde-984	217	3	∈	∈	PROPN
ejde-984	217	4	sc	sc	PROPN
ejde-984	217	5	and	and	CCONJ
ejde-984	217	6	the	the	DET
ejde-984	217	7	weak	weak	ADJ
ejde-984	217	8	lower	low	ADJ
ejde-984	217	9	semi	semi	NOUN
ejde-984	217	10	-	-	NOUN
ejde-984	217	11	continuity	continuity	NOUN
ejde-984	217	12	of	of	ADP
ejde-984	217	13	the	the	DET
ejde-984	217	14	norm	norm	NOUN
ejde-984	217	15	,	,	PUNCT
ejde-984	217	16	it	it	PRON
ejde-984	217	17	follows	follow	VERB
ejde-984	217	18	that	that	SCONJ
ejde-984	217	19	|u|2	|u|2	PROPN
ejde-984	217	20	≤	≤	PROPN
ejde-984	217	21	c	c	NOUN
ejde-984	217	22	,	,	PUNCT
ejde-984	217	23	then	then	ADV
ejde-984	217	24	we	we	PRON
ejde-984	217	25	have	have	VERB
ejde-984	217	26	that	that	DET
ejde-984	217	27	c0	c0	NOUN
ejde-984	217	28	=	=	PUNCT
ejde-984	217	29	(	(	PUNCT
ejde-984	217	30	a	a	DET
ejde-984	217	31	δs	δs	NOUN
ejde-984	217	32	,	,	PUNCT
ejde-984	217	33	pc(s	pc(s	NUM
ejde-984	217	34	,	,	PUNCT
ejde-984	217	35	p)pcp(1−δs	p)pcp(1−δs	ADJ
ejde-984	217	36	,	,	PUNCT
ejde-984	217	37	p	p	NOUN
ejde-984	217	38	)	)	PUNCT
ejde-984	217	39	)	)	PUNCT
ejde-984	217	40	1	1	NUM
ejde-984	217	41	pδs	pδs	NOUN
ejde-984	217	42	,	,	PUNCT
ejde-984	217	43	p−2	p−2	PROPN
ejde-984	217	44	≤	≤	NUM
ejde-984	217	45	b	b	NOUN
ejde-984	217	46	,	,	PUNCT
ejde-984	217	47	(	(	PUNCT
ejde-984	217	48	3.9	3.9	NUM
ejde-984	217	49	)	)	PUNCT
ejde-984	217	50	where	where	SCONJ
ejde-984	217	51	b	b	NOUN
ejde-984	217	52	=	=	SYM
ejde-984	217	53	limn→∞	limn→∞	PROPN
ejde-984	217	54	|(−∆)s/2un|2	|(−∆)s/2un|2	NOUN
ejde-984	217	55	.	.	PUNCT
ejde-984	218	1	that	that	PRON
ejde-984	218	2	is	be	AUX
ejde-984	218	3	to	to	PART
ejde-984	218	4	say	say	VERB
ejde-984	218	5	,	,	PUNCT
ejde-984	218	6	for	for	ADP
ejde-984	218	7	n	n	CCONJ
ejde-984	218	8	large	large	ADJ
ejde-984	218	9	enough	enough	ADV
ejde-984	218	10	,	,	PUNCT
ejde-984	218	11	we	we	PRON
ejde-984	218	12	obtain	obtain	VERB
ejde-984	218	13	that	that	SCONJ
ejde-984	218	14	|(−∆)s/2un|2	|(−∆)s/2un|2	PROPN
ejde-984	218	15	≥	≥	PROPN
ejde-984	218	16	c0	c0	PROPN
ejde-984	218	17	.	.	PUNCT
ejde-984	219	1	(	(	PUNCT
ejde-984	219	2	3.10	3.10	NUM
ejde-984	219	3	)	)	PUNCT
ejde-984	219	4	inserting	insert	VERB
ejde-984	219	5	(	(	PUNCT
ejde-984	219	6	3.1	3.1	NUM
ejde-984	219	7	)	)	PUNCT
ejde-984	219	8	into	into	ADP
ejde-984	219	9	(	(	PUNCT
ejde-984	219	10	3.5	3.5	NUM
ejde-984	219	11	)	)	PUNCT
ejde-984	219	12	,	,	PUNCT
ejde-984	219	13	we	we	PRON
ejde-984	219	14	have	have	VERB
ejde-984	219	15	that	that	DET
ejde-984	219	16	λn	λn	NOUN
ejde-984	219	17	=	=	SYM
ejde-984	219	18	1	1	NUM
ejde-984	219	19	c2	c2	PROPN
ejde-984	219	20	[	[	X
ejde-984	219	21	(	(	PUNCT
ejde-984	219	22	1−	1−	NUM
ejde-984	219	23	1	1	NUM
ejde-984	219	24	δs	δs	NOUN
ejde-984	219	25	,	,	PUNCT
ejde-984	219	26	p	p	NOUN
ejde-984	219	27	)	)	PUNCT
ejde-984	219	28	(	(	PUNCT
ejde-984	219	29	a+	a+	PUNCT
ejde-984	219	30	b|(−∆)s/2un|22	b|(−∆)s/2un|22	NOUN
ejde-984	219	31	)	)	PUNCT
ejde-984	219	32	|(−∆)s/2un|22	|(−∆)s/2un|22	NOUN
ejde-984	219	33	+	+	X
ejde-984	219	34	µ	µ	X
ejde-984	219	35	(	(	PUNCT
ejde-984	219	36	δs	δs	NOUN
ejde-984	219	37	,	,	PUNCT
ejde-984	219	38	q	q	NOUN
ejde-984	219	39	δs	δs	NOUN
ejde-984	219	40	,	,	PUNCT
ejde-984	219	41	p	p	NOUN
ejde-984	219	42	−	−	PROPN
ejde-984	219	43	1	1	NUM
ejde-984	219	44	)	)	PUNCT
ejde-984	219	45	|un|qq	|un|qq	NOUN
ejde-984	219	46	]	]	PUNCT
ejde-984	220	1	+	+	NUM
ejde-984	220	2	o(1	o(1	NOUN
ejde-984	220	3	)	)	PUNCT
ejde-984	220	4	.	.	PUNCT
ejde-984	221	1	(	(	PUNCT
ejde-984	221	2	3.11	3.11	NUM
ejde-984	221	3	)	)	PUNCT
ejde-984	221	4	according	accord	VERB
ejde-984	221	5	to	to	ADP
ejde-984	221	6	the	the	DET
ejde-984	221	7	fractional	fractional	ADJ
ejde-984	221	8	gagliardo	gagliardo	NOUN
ejde-984	221	9	-	-	PUNCT
ejde-984	221	10	nirenberg	nirenberg	NOUN
ejde-984	221	11	inequality	inequality	NOUN
ejde-984	221	12	(	(	PUNCT
ejde-984	221	13	2.2	2.2	NUM
ejde-984	221	14	)	)	PUNCT
ejde-984	221	15	,	,	PUNCT
ejde-984	221	16	we	we	PRON
ejde-984	221	17	obtain	obtain	VERB
ejde-984	221	18	that	that	SCONJ
ejde-984	221	19	|un|qq	|un|qq	NOUN
ejde-984	221	20	≤	≤	ADV
ejde-984	221	21	c(s	c(	NOUN
ejde-984	221	22	,	,	PUNCT
ejde-984	221	23	q)q|(−∆)s/2un|	q)q|(−∆)s/2un|	PROPN
ejde-984	221	24	qδs	qδs	PROPN
ejde-984	221	25	,	,	PUNCT
ejde-984	221	26	q	q	PROPN
ejde-984	221	27	2	2	NUM
ejde-984	221	28	|un|	|un|	NOUN
ejde-984	221	29	q(1−δs	q(1−δs	NOUN
ejde-984	221	30	,	,	PUNCT
ejde-984	221	31	q	q	NOUN
ejde-984	221	32	)	)	PUNCT
ejde-984	221	33	2	2	NUM
ejde-984	221	34	.	.	PUNCT
ejde-984	222	1	(	(	PUNCT
ejde-984	222	2	3.12	3.12	NUM
ejde-984	222	3	)	)	PUNCT
ejde-984	222	4	by	by	ADP
ejde-984	222	5	un	un	PROPN
ejde-984	222	6	∈	∈	PROPN
ejde-984	222	7	sc	sc	PROPN
ejde-984	222	8	,	,	PUNCT
ejde-984	222	9	we	we	PRON
ejde-984	222	10	have	have	VERB
ejde-984	222	11	that	that	SCONJ
ejde-984	222	12	|un|qq	|un|qq	NOUN
ejde-984	222	13	≤	≤	ADV
ejde-984	222	14	c(s	c(	NOUN
ejde-984	222	15	,	,	PUNCT
ejde-984	222	16	q)q|(−∆)s/2un|	q)q|(−∆)s/2un|	PROPN
ejde-984	222	17	qδs	qδs	PROPN
ejde-984	222	18	,	,	PUNCT
ejde-984	222	19	q	q	PROPN
ejde-984	222	20	2	2	NUM
ejde-984	222	21	cq(1−δs	cq(1−δs	NOUN
ejde-984	222	22	,	,	PUNCT
ejde-984	222	23	q	q	NOUN
ejde-984	222	24	)	)	PUNCT
ejde-984	222	25	.	.	PUNCT
ejde-984	223	1	(	(	PUNCT
ejde-984	223	2	3.13	3.13	NUM
ejde-984	223	3	)	)	PUNCT
ejde-984	223	4	further	far	ADV
ejde-984	223	5	,	,	PUNCT
ejde-984	223	6	by	by	ADP
ejde-984	223	7	1	1	NUM
ejde-984	223	8	−	−	PROPN
ejde-984	223	9	1	1	NUM
ejde-984	223	10	δs	δs	NOUN
ejde-984	223	11	,	,	PUNCT
ejde-984	223	12	p	p	X
ejde-984	223	13	<	<	X
ejde-984	223	14	0	0	PUNCT
ejde-984	223	15	(	(	PUNCT
ejde-984	223	16	since	since	SCONJ
ejde-984	223	17	0	0	NUM
ejde-984	223	18	<	<	X
ejde-984	223	19	δs	δs	NOUN
ejde-984	223	20	,	,	PUNCT
ejde-984	223	21	p	p	X
ejde-984	223	22	<	<	X
ejde-984	223	23	1	1	NUM
ejde-984	223	24	)	)	PUNCT
ejde-984	223	25	,	,	PUNCT
ejde-984	223	26	combining	combine	VERB
ejde-984	223	27	(	(	PUNCT
ejde-984	223	28	3.9	3.9	NUM
ejde-984	223	29	)	)	PUNCT
ejde-984	223	30	,	,	PUNCT
ejde-984	223	31	(	(	PUNCT
ejde-984	223	32	3.11	3.11	NUM
ejde-984	223	33	)	)	PUNCT
ejde-984	223	34	with	with	ADP
ejde-984	223	35	(	(	PUNCT
ejde-984	223	36	3.13	3.13	NUM
ejde-984	223	37	)	)	PUNCT
ejde-984	223	38	,	,	PUNCT
ejde-984	223	39	we	we	PRON
ejde-984	223	40	can	can	AUX
ejde-984	223	41	deduce	deduce	VERB
ejde-984	223	42	that	that	PRON
ejde-984	223	43	λn	λn	PROPN
ejde-984	223	44	≤	≤	NOUN
ejde-984	223	45	1	1	NUM
ejde-984	223	46	c2	c2	PROPN
ejde-984	223	47	|(−∆)s/2un|	|(−∆)s/2un|	PROPN
ejde-984	223	48	qδs	qδs	PROPN
ejde-984	223	49	,	,	PUNCT
ejde-984	223	50	q	q	PROPN
ejde-984	223	51	2	2	NUM
ejde-984	223	52	[	[	X
ejde-984	223	53	(	(	PUNCT
ejde-984	223	54	1−	1−	NUM
ejde-984	223	55	1	1	NUM
ejde-984	223	56	δs	δs	NOUN
ejde-984	223	57	,	,	PUNCT
ejde-984	223	58	p	p	NOUN
ejde-984	223	59	)	)	PUNCT
ejde-984	223	60	(	(	PUNCT
ejde-984	223	61	a+bc2	a+bc2	NOUN
ejde-984	223	62	0	0	NUM
ejde-984	223	63	)	)	PUNCT
ejde-984	223	64	c	c	NOUN
ejde-984	223	65	2−qδs	2−qδs	NOUN
ejde-984	223	66	,	,	PUNCT
ejde-984	223	67	q	q	NOUN
ejde-984	223	68	0	0	NUM
ejde-984	223	69	+	+	NOUN
ejde-984	223	70	µ	µ	X
ejde-984	223	71	(	(	PUNCT
ejde-984	223	72	δs	δs	NOUN
ejde-984	223	73	,	,	PUNCT
ejde-984	223	74	q	q	NOUN
ejde-984	223	75	δs	δs	NOUN
ejde-984	223	76	,	,	PUNCT
ejde-984	223	77	p	p	NOUN
ejde-984	223	78	−1	−1	NOUN
ejde-984	223	79	)	)	PUNCT
ejde-984	223	80	c(s	c(	NOUN
ejde-984	223	81	,	,	PUNCT
ejde-984	223	82	q)qcq(1−δs	q)qcq(1−δs	PROPN
ejde-984	223	83	,	,	PUNCT
ejde-984	223	84	q	q	NOUN
ejde-984	223	85	)	)	PUNCT
ejde-984	223	86	]	]	PUNCT
ejde-984	224	1	+	+	PUNCT
ejde-984	224	2	o(1	o(1	NOUN
ejde-984	224	3	)	)	PUNCT
ejde-984	224	4	.	.	PUNCT
ejde-984	225	1	(	(	PUNCT
ejde-984	225	2	3.14	3.14	NUM
ejde-984	225	3	)	)	PUNCT
ejde-984	225	4	ejde-2025/75	ejde-2025/75	NOUN
ejde-984	225	5	normalized	normalize	VERB
ejde-984	225	6	solutions	solution	NOUN
ejde-984	225	7	of	of	ADP
ejde-984	225	8	fractional	fractional	PROPN
ejde-984	225	9	kirchhoff	kirchhoff	NOUN
ejde-984	225	10	equations	equation	NOUN
ejde-984	225	11	9	9	NUM
ejde-984	225	12	by	by	ADP
ejde-984	225	13	(	(	PUNCT
ejde-984	225	14	3.10	3.10	NUM
ejde-984	225	15	)	)	PUNCT
ejde-984	225	16	and	and	CCONJ
ejde-984	225	17	(	(	PUNCT
ejde-984	225	18	2.9	2.9	NUM
ejde-984	225	19	)	)	PUNCT
ejde-984	225	20	,	,	PUNCT
ejde-984	225	21	we	we	PRON
ejde-984	225	22	derive	derive	VERB
ejde-984	225	23	that	that	SCONJ
ejde-984	225	24	λn	λn	PROPN
ejde-984	225	25	≤	≤	ADV
ejde-984	225	26	1	1	NUM
ejde-984	225	27	c2	c2	PROPN
ejde-984	225	28	c	c	PROPN
ejde-984	225	29	qδs	qδs	PROPN
ejde-984	225	30	,	,	PUNCT
ejde-984	225	31	q	q	NOUN
ejde-984	225	32	0	0	PUNCT
ejde-984	226	1	[	[	X
ejde-984	226	2	(	(	PUNCT
ejde-984	226	3	1−	1−	NUM
ejde-984	226	4	1	1	NUM
ejde-984	226	5	δs	δs	NOUN
ejde-984	226	6	,	,	PUNCT
ejde-984	226	7	p	p	NOUN
ejde-984	226	8	)	)	PUNCT
ejde-984	226	9	(	(	PUNCT
ejde-984	226	10	a+	a+	X
ejde-984	226	11	bc2	bc2	NOUN
ejde-984	226	12	0	0	NUM
ejde-984	226	13	)	)	PUNCT
ejde-984	226	14	c	c	NOUN
ejde-984	226	15	2−qδs	2−qδs	NOUN
ejde-984	226	16	,	,	PUNCT
ejde-984	226	17	q	q	NOUN
ejde-984	226	18	0	0	NUM
ejde-984	226	19	+	+	CCONJ
ejde-984	226	20	µ	µ	X
ejde-984	226	21	(	(	PUNCT
ejde-984	226	22	δs	δs	NOUN
ejde-984	226	23	,	,	PUNCT
ejde-984	226	24	q	q	NOUN
ejde-984	226	25	δs	δs	NOUN
ejde-984	226	26	,	,	PUNCT
ejde-984	226	27	p	p	NOUN
ejde-984	226	28	−	−	PROPN
ejde-984	226	29	1	1	NUM
ejde-984	226	30	)	)	PUNCT
ejde-984	226	31	c(s	c(	NOUN
ejde-984	226	32	,	,	PUNCT
ejde-984	226	33	q)qcq(1−δs	q)qcq(1−δs	PROPN
ejde-984	226	34	,	,	PUNCT
ejde-984	226	35	q	q	NOUN
ejde-984	226	36	)	)	PUNCT
ejde-984	226	37	]	]	PUNCT
ejde-984	227	1	+	+	PUNCT
ejde-984	227	2	o(1	o(1	NOUN
ejde-984	227	3	)	)	PUNCT
ejde-984	227	4	.	.	PUNCT
ejde-984	228	1	(	(	PUNCT
ejde-984	228	2	3.15	3.15	NUM
ejde-984	228	3	)	)	PUNCT
ejde-984	228	4	therefore	therefore	ADV
ejde-984	228	5	,	,	PUNCT
ejde-984	228	6	considering	consider	VERB
ejde-984	228	7	condition	condition	NOUN
ejde-984	228	8	(	(	PUNCT
ejde-984	228	9	2.9	2.9	NUM
ejde-984	228	10	)	)	PUNCT
ejde-984	228	11	,	,	PUNCT
ejde-984	228	12	we	we	PRON
ejde-984	228	13	obtain	obtain	VERB
ejde-984	228	14	that	that	SCONJ
ejde-984	228	15	λn	λn	PROPN
ejde-984	228	16	≤	≤	ADV
ejde-984	228	17	1	1	NUM
ejde-984	228	18	c2	c2	PROPN
ejde-984	228	19	c	c	PROPN
ejde-984	228	20	qδs	qδs	PROPN
ejde-984	228	21	,	,	PUNCT
ejde-984	228	22	q	q	PROPN
ejde-984	228	23	0	0	NUM
ejde-984	228	24	ϵ0	ϵ0	ADJ
ejde-984	228	25	+	+	CCONJ
ejde-984	228	26	o(1	o(1	NOUN
ejde-984	228	27	)	)	PUNCT
ejde-984	228	28	,	,	PUNCT
ejde-984	228	29	(	(	PUNCT
ejde-984	228	30	3.16	3.16	NUM
ejde-984	228	31	)	)	PUNCT
ejde-984	228	32	for	for	ADP
ejde-984	228	33	n	n	CCONJ
ejde-984	228	34	adequately	adequately	ADV
ejde-984	228	35	large	large	ADJ
ejde-984	228	36	.	.	PUNCT
ejde-984	229	1	taking	take	VERB
ejde-984	229	2	the	the	DET
ejde-984	229	3	limit	limit	NOUN
ejde-984	229	4	of	of	ADP
ejde-984	229	5	the	the	DET
ejde-984	229	6	above	above	ADJ
ejde-984	229	7	formula	formula	NOUN
ejde-984	229	8	as	as	ADP
ejde-984	229	9	n→	n→	PROPN
ejde-984	229	10	∞	∞	PROPN
ejde-984	229	11	,	,	PUNCT
ejde-984	229	12	we	we	PRON
ejde-984	229	13	have	have	VERB
ejde-984	229	14	that	that	PRON
ejde-984	230	1	λ	λ	PROPN
ejde-984	230	2	≤	≤	NOUN
ejde-984	230	3	1	1	NUM
ejde-984	230	4	c2	c2	PROPN
ejde-984	230	5	c	c	PROPN
ejde-984	230	6	qδs	qδs	PROPN
ejde-984	230	7	,	,	PUNCT
ejde-984	230	8	q	q	PROPN
ejde-984	230	9	0	0	NUM
ejde-984	230	10	ϵ0	ϵ0	NOUN
ejde-984	230	11	<	<	X
ejde-984	230	12	0	0	NUM
ejde-984	230	13	.	.	PUNCT
ejde-984	231	1	(	(	PUNCT
ejde-984	231	2	3.17	3.17	NUM
ejde-984	231	3	)	)	PUNCT
ejde-984	231	4	thus	thus	ADV
ejde-984	231	5	λ	λ	X
ejde-984	231	6	<	<	X
ejde-984	231	7	0	0	NUM
ejde-984	231	8	,	,	PUNCT
ejde-984	231	9	and	and	CCONJ
ejde-984	231	10	the	the	DET
ejde-984	231	11	claim	claim	NOUN
ejde-984	231	12	is	be	AUX
ejde-984	231	13	proved	prove	VERB
ejde-984	231	14	.	.	PUNCT
ejde-984	232	1	finally	finally	ADV
ejde-984	232	2	,	,	PUNCT
ejde-984	232	3	we	we	PRON
ejde-984	232	4	prove	prove	VERB
ejde-984	232	5	that	that	SCONJ
ejde-984	232	6	un	un	PROPN
ejde-984	232	7	→	→	SYM
ejde-984	232	8	u	u	NOUN
ejde-984	232	9	strongly	strongly	ADV
ejde-984	232	10	in	in	ADP
ejde-984	232	11	hs(r3	hs(r3	NUM
ejde-984	232	12	)	)	PUNCT
ejde-984	232	13	.	.	PUNCT
ejde-984	233	1	by	by	ADP
ejde-984	233	2	taking	take	VERB
ejde-984	233	3	the	the	DET
ejde-984	233	4	limit	limit	NOUN
ejde-984	233	5	of	of	ADP
ejde-984	233	6	(	(	PUNCT
ejde-984	233	7	3.3	3.3	NUM
ejde-984	233	8	)	)	PUNCT
ejde-984	233	9	,	,	PUNCT
ejde-984	233	10	one	one	PRON
ejde-984	233	11	has	have	VERB
ejde-984	233	12	a	a	DET
ejde-984	233	13	∫	∫	PROPN
ejde-984	233	14	r3	r3	PROPN
ejde-984	233	15	(	(	PUNCT
ejde-984	233	16	−∆)s/2u(−∆)s/2ϕdx+	−∆)s/2u(−∆)s/2ϕdx+	PROPN
ejde-984	233	17	bb2	bb2	PROPN
ejde-984	233	18	∫	∫	PROPN
ejde-984	233	19	r3	r3	PROPN
ejde-984	233	20	(	(	PUNCT
ejde-984	233	21	−∆)s/2u(−∆)s/2ϕdx−	−∆)s/2u(−∆)s/2ϕdx−	PROPN
ejde-984	233	22	µ	µ	PRON
ejde-984	233	23	∫	∫	PROPN
ejde-984	233	24	r3	r3	PROPN
ejde-984	233	25	|u|q−2uϕdx	|u|q−2uϕdx	PUNCT
ejde-984	233	26	−	−	PROPN
ejde-984	234	1	∫	∫	PROPN
ejde-984	235	1	r3	r3	PROPN
ejde-984	236	1	|u|p−2uϕdx−	|u|p−2uϕdx−	PROPN
ejde-984	236	2	λ	λ	PROPN
ejde-984	236	3	∫	∫	PROPN
ejde-984	236	4	r3	r3	PROPN
ejde-984	236	5	uϕdx	uϕdx	ADV
ejde-984	236	6	=	=	SYM
ejde-984	236	7	0	0	NUM
ejde-984	236	8	,	,	PUNCT
ejde-984	236	9	(	(	PUNCT
ejde-984	236	10	3.18	3.18	NUM
ejde-984	236	11	)	)	PUNCT
ejde-984	236	12	for	for	ADP
ejde-984	236	13	all	all	DET
ejde-984	236	14	ϕ	ϕ	PROPN
ejde-984	236	15	∈	∈	PROPN
ejde-984	236	16	hs(r3	hs(r3	NUM
ejde-984	236	17	)	)	PUNCT
ejde-984	236	18	,	,	PUNCT
ejde-984	236	19	that	that	ADV
ejde-984	236	20	is	is	ADV
ejde-984	236	21	,	,	PUNCT
ejde-984	236	22	u	u	NOUN
ejde-984	236	23	satisfies	satisfie	NOUN
ejde-984	236	24	(	(	PUNCT
ejde-984	236	25	a+b2b)(−∆)su	a+b2b)(−∆)su	NOUN
ejde-984	236	26	=	=	PUNCT
ejde-984	236	27	λu+	λu+	NOUN
ejde-984	236	28	µ|u|q−2u+	µ|u|q−2u+	PROPN
ejde-984	236	29	|u|p−2u	|u|p−2u	PROPN
ejde-984	236	30	,	,	PUNCT
ejde-984	236	31	namely	namely	ADV
ejde-984	236	32	,	,	PUNCT
ejde-984	236	33	u	u	NOUN
ejde-984	236	34	solves	solve	NOUN
ejde-984	236	35	(	(	PUNCT
ejde-984	236	36	1.1	1.1	NUM
ejde-984	236	37	)	)	PUNCT
ejde-984	236	38	for	for	ADP
ejde-984	236	39	some	some	DET
ejde-984	236	40	λ	λ	NOUN
ejde-984	236	41	<	<	X
ejde-984	236	42	0	0	NUM
ejde-984	236	43	.	.	PUNCT
ejde-984	236	44	testing	testing	NOUN
ejde-984	236	45	(	(	PUNCT
ejde-984	236	46	3.18	3.18	NUM
ejde-984	236	47	)	)	PUNCT
ejde-984	236	48	,	,	PUNCT
ejde-984	236	49	(	(	PUNCT
ejde-984	236	50	3.3	3.3	NUM
ejde-984	236	51	)	)	PUNCT
ejde-984	236	52	with	with	ADP
ejde-984	236	53	ϕ	ϕ	NOUN
ejde-984	236	54	=	=	SYM
ejde-984	236	55	un	un	PROPN
ejde-984	236	56	−	−	PROPN
ejde-984	236	57	u	u	PROPN
ejde-984	236	58	,	,	PUNCT
ejde-984	236	59	we	we	PRON
ejde-984	236	60	can	can	AUX
ejde-984	236	61	see	see	VERB
ejde-984	236	62	that	that	DET
ejde-984	236	63	(	(	PUNCT
ejde-984	236	64	a+b2b	a+b2b	NOUN
ejde-984	236	65	)	)	PUNCT
ejde-984	236	66	∫	∫	PROPN
ejde-984	236	67	r3	r3	PROPN
ejde-984	236	68	|(−∆)s/2(un	|(−∆)s/2(un	PROPN
ejde-984	237	1	−	−	PROPN
ejde-984	237	2	u)|2dx−	u)|2dx−	NOUN
ejde-984	238	1	λ	λ	PROPN
ejde-984	238	2	∫	∫	PROPN
ejde-984	238	3	r3	r3	PROPN
ejde-984	238	4	|un	|un	ADP
ejde-984	238	5	−	−	PROPN
ejde-984	238	6	u|2dx→	u|2dx→	PROPN
ejde-984	238	7	0	0	NUM
ejde-984	238	8	,	,	PUNCT
ejde-984	238	9	as	as	ADP
ejde-984	238	10	n→	n→	PUNCT
ejde-984	238	11	∞.	∞.	PROPN
ejde-984	238	12	since	since	SCONJ
ejde-984	238	13	λ	λ	PROPN
ejde-984	238	14	<	<	X
ejde-984	238	15	0	0	NUM
ejde-984	238	16	,	,	PUNCT
ejde-984	238	17	we	we	PRON
ejde-984	238	18	deduce	deduce	VERB
ejde-984	238	19	that	that	SCONJ
ejde-984	238	20	{	{	PUNCT
ejde-984	238	21	un	un	PROPN
ejde-984	238	22	}	}	PUNCT
ejde-984	238	23	converges	converge	VERB
ejde-984	238	24	strongly	strongly	ADV
ejde-984	238	25	to	to	ADP
ejde-984	238	26	u	u	NOUN
ejde-984	238	27	in	in	ADP
ejde-984	238	28	hs(r3	hs(r3	NUM
ejde-984	238	29	)	)	PUNCT
ejde-984	238	30	.	.	PUNCT
ejde-984	239	1	□	□	PUNCT
ejde-984	239	2	lemma	lemma	PROPN
ejde-984	239	3	3.3	3.3	NUM
ejde-984	239	4	.	.	PUNCT
ejde-984	240	1	let	let	VERB
ejde-984	240	2	µ	µ	X
ejde-984	240	3	<	<	X
ejde-984	240	4	0	0	NUM
ejde-984	240	5	,	,	PUNCT
ejde-984	240	6	and	and	CCONJ
ejde-984	240	7	2	2	NUM
ejde-984	240	8	<	<	X
ejde-984	240	9	q	q	X
ejde-984	240	10	≤	≤	NUM
ejde-984	240	11	2	2	NUM
ejde-984	240	12	+	+	NUM
ejde-984	240	13	8s	8s	NUM
ejde-984	240	14	3	3	NUM
ejde-984	240	15	<	<	X
ejde-984	240	16	p	p	X
ejde-984	240	17	<	<	X
ejde-984	240	18	2∗s	2∗s	NUM
ejde-984	240	19	be	be	AUX
ejde-984	240	20	given	give	VERB
ejde-984	240	21	constants	constant	NOUN
ejde-984	240	22	.	.	PUNCT
ejde-984	241	1	then	then	ADV
ejde-984	241	2	p0	p0	NOUN
ejde-984	241	3	c,µ	c,µ	NOUN
ejde-984	241	4	=	=	SYM
ejde-984	241	5	∅	∅	NOUN
ejde-984	241	6	and	and	CCONJ
ejde-984	241	7	pc,µ	pc,µ	NOUN
ejde-984	241	8	is	be	AUX
ejde-984	241	9	a	a	DET
ejde-984	241	10	smooth	smooth	ADJ
ejde-984	241	11	manifold	manifold	NOUN
ejde-984	241	12	of	of	ADP
ejde-984	241	13	codimension	codimension	NOUN
ejde-984	241	14	2	2	NUM
ejde-984	241	15	in	in	ADP
ejde-984	241	16	hs(r3	hs(r3	NUM
ejde-984	241	17	)	)	PUNCT
ejde-984	241	18	.	.	PUNCT
ejde-984	242	1	proof	proof	NOUN
ejde-984	242	2	.	.	PUNCT
ejde-984	243	1	suppose	suppose	VERB
ejde-984	243	2	by	by	ADP
ejde-984	243	3	contradiction	contradiction	NOUN
ejde-984	243	4	that	that	SCONJ
ejde-984	243	5	this	this	PRON
ejde-984	243	6	is	be	AUX
ejde-984	243	7	not	not	PART
ejde-984	243	8	the	the	DET
ejde-984	243	9	case	case	NOUN
ejde-984	243	10	,	,	PUNCT
ejde-984	243	11	namely	namely	ADV
ejde-984	243	12	we	we	PRON
ejde-984	243	13	set	set	VERB
ejde-984	243	14	p0	p0	NOUN
ejde-984	243	15	c,µ	c,µ	NOUN
ejde-984	243	16	̸=	̸=	PROPN
ejde-984	243	17	∅	∅	NOUN
ejde-984	243	18	,	,	PUNCT
ejde-984	243	19	then	then	ADV
ejde-984	243	20	from	from	ADP
ejde-984	243	21	the	the	DET
ejde-984	243	22	definition	definition	NOUN
ejde-984	243	23	of	of	ADP
ejde-984	243	24	p0	p0	NOUN
ejde-984	243	25	c,µ	c,µ	NOUN
ejde-984	243	26	,	,	PUNCT
ejde-984	243	27	we	we	PRON
ejde-984	243	28	can	can	AUX
ejde-984	243	29	derive	derive	VERB
ejde-984	243	30	that	that	SCONJ
ejde-984	243	31	there	there	PRON
ejde-984	243	32	exists	exist	VERB
ejde-984	243	33	u	u	PROPN
ejde-984	243	34	∈	∈	PROPN
ejde-984	243	35	sc	sc	PROPN
ejde-984	243	36	such	such	ADJ
ejde-984	243	37	that	that	DET
ejde-984	243	38	pµ(u	pµ(u	NOUN
ejde-984	243	39	)	)	PUNCT
ejde-984	243	40	=	=	SYM
ejde-984	243	41	0	0	PUNCT
ejde-984	244	1	and	and	CCONJ
ejde-984	244	2	(	(	PUNCT
ejde-984	244	3	jµ	jµ	PROPN
ejde-984	244	4	u	u	NOUN
ejde-984	244	5	)	)	PUNCT
ejde-984	244	6	′′	′′	PROPN
ejde-984	244	7	(	(	PUNCT
ejde-984	244	8	0	0	NUM
ejde-984	244	9	)	)	PUNCT
ejde-984	244	10	=	=	SYM
ejde-984	244	11	0	0	X
ejde-984	244	12	.	.	PUNCT
ejde-984	245	1	therefore	therefore	ADV
ejde-984	245	2	,	,	PUNCT
ejde-984	245	3	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	245	4	+	+	CCONJ
ejde-984	245	5	b|(−∆)s/2u|42	b|(−∆)s/2u|42	NOUN
ejde-984	245	6	=	=	SYM
ejde-984	245	7	µδs	µδ	NOUN
ejde-984	245	8	,	,	PUNCT
ejde-984	245	9	q|u|qq	q|u|qq	PROPN
ejde-984	245	10	+	+	CCONJ
ejde-984	245	11	δs	δs	NOUN
ejde-984	245	12	,	,	PUNCT
ejde-984	245	13	p|u|pp	p|u|pp	PROPN
ejde-984	245	14	,	,	PUNCT
ejde-984	245	15	(	(	PUNCT
ejde-984	245	16	3.19	3.19	NUM
ejde-984	245	17	)	)	PUNCT
ejde-984	245	18	2a|(−∆)s/2u|22	2a|(−∆)s/2u|22	PROPN
ejde-984	246	1	+	+	CCONJ
ejde-984	246	2	4b|(−∆)s/2u|42	4b|(−∆)s/2u|42	NUM
ejde-984	246	3	=	=	SYM
ejde-984	246	4	µqδ2s	µqδ2s	NOUN
ejde-984	246	5	,	,	PUNCT
ejde-984	246	6	q|u|qq	q|u|qq	NOUN
ejde-984	246	7	+	+	CCONJ
ejde-984	246	8	pδ2s	pδ2s	NOUN
ejde-984	246	9	,	,	PUNCT
ejde-984	246	10	p|u|pp	p|u|pp	PROPN
ejde-984	246	11	.	.	PUNCT
ejde-984	247	1	(	(	PUNCT
ejde-984	247	2	3.20	3.20	NUM
ejde-984	247	3	)	)	PUNCT
ejde-984	247	4	combining	combine	VERB
ejde-984	247	5	(	(	PUNCT
ejde-984	247	6	3.19	3.19	NUM
ejde-984	247	7	)	)	PUNCT
ejde-984	247	8	with	with	ADP
ejde-984	247	9	(	(	PUNCT
ejde-984	247	10	3.20	3.20	NUM
ejde-984	247	11	)	)	PUNCT
ejde-984	247	12	,	,	PUNCT
ejde-984	247	13	one	one	PRON
ejde-984	247	14	has	have	VERB
ejde-984	247	15	(	(	PUNCT
ejde-984	247	16	pδs	pδs	PROPN
ejde-984	247	17	,	,	PUNCT
ejde-984	247	18	p	p	NOUN
ejde-984	247	19	−	−	PROPN
ejde-984	247	20	2)a|(−∆)s/2u|22	2)a|(−∆)s/2u|22	NOUN
ejde-984	247	21	+	+	CCONJ
ejde-984	247	22	(	(	PUNCT
ejde-984	247	23	pδs	pδs	PROPN
ejde-984	247	24	,	,	PUNCT
ejde-984	247	25	p	p	NOUN
ejde-984	247	26	−	−	PROPN
ejde-984	247	27	4)b|(−∆)s/2u|42	4)b|(−∆)s/2u|42	NUM
ejde-984	247	28	=	=	SYM
ejde-984	247	29	µδs	µδ	NOUN
ejde-984	247	30	,	,	PUNCT
ejde-984	247	31	q(pδs	q(pδ	NOUN
ejde-984	247	32	,	,	PUNCT
ejde-984	247	33	p	p	NOUN
ejde-984	247	34	−	−	PROPN
ejde-984	247	35	qδs	qδs	PROPN
ejde-984	247	36	,	,	PUNCT
ejde-984	247	37	q)|u|qq	q)|u|qq	NOUN
ejde-984	247	38	≤	≤	ADJ
ejde-984	247	39	0	0	NUM
ejde-984	247	40	,	,	PUNCT
ejde-984	247	41	(	(	PUNCT
ejde-984	247	42	3.21	3.21	NUM
ejde-984	247	43	)	)	PUNCT
ejde-984	247	44	where	where	SCONJ
ejde-984	247	45	pδs	pδs	PROPN
ejde-984	247	46	,	,	PUNCT
ejde-984	247	47	p	p	X
ejde-984	247	48	>	>	X
ejde-984	247	49	4	4	NUM
ejde-984	247	50	≥	≥	NOUN
ejde-984	247	51	qδs	qδs	PROPN
ejde-984	247	52	,	,	PUNCT
ejde-984	247	53	q	q	NOUN
ejde-984	247	54	by	by	ADP
ejde-984	247	55	2	2	NUM
ejde-984	247	56	<	<	X
ejde-984	247	57	q	q	X
ejde-984	247	58	≤	≤	NUM
ejde-984	247	59	2	2	NUM
ejde-984	247	60	+	+	NUM
ejde-984	247	61	8s	8s	NUM
ejde-984	247	62	3	3	NUM
ejde-984	247	63	<	<	X
ejde-984	247	64	p	p	X
ejde-984	247	65	<	<	X
ejde-984	247	66	2∗s	2∗s	NUM
ejde-984	247	67	,	,	PUNCT
ejde-984	247	68	δs	δs	NOUN
ejde-984	247	69	,	,	PUNCT
ejde-984	247	70	q	q	PROPN
ejde-984	247	71	>	>	X
ejde-984	247	72	0	0	NUM
ejde-984	247	73	by	by	ADP
ejde-984	247	74	q	q	PROPN
ejde-984	247	75	>	>	X
ejde-984	247	76	2	2	X
ejde-984	247	77	.	.	PUNCT
ejde-984	248	1	then	then	ADV
ejde-984	248	2	,	,	PUNCT
ejde-984	248	3	it	it	PRON
ejde-984	248	4	follows	follow	VERB
ejde-984	248	5	that	that	SCONJ
ejde-984	248	6	|(−∆)s/2u|2	|(−∆)s/2u|2	PROPN
ejde-984	248	7	=	=	SYM
ejde-984	248	8	0	0	PROPN
ejde-984	248	9	.	.	PUNCT
ejde-984	249	1	(	(	PUNCT
ejde-984	249	2	3.22	3.22	NUM
ejde-984	249	3	)	)	PUNCT
ejde-984	249	4	further	far	ADV
ejde-984	249	5	,	,	PUNCT
ejde-984	249	6	from	from	ADP
ejde-984	249	7	(	(	PUNCT
ejde-984	249	8	3.19	3.19	NUM
ejde-984	249	9	)	)	PUNCT
ejde-984	249	10	,	,	PUNCT
ejde-984	249	11	(	(	PUNCT
ejde-984	249	12	3.21	3.21	NUM
ejde-984	249	13	)	)	PUNCT
ejde-984	249	14	and	and	CCONJ
ejde-984	249	15	the	the	DET
ejde-984	249	16	fractional	fractional	ADJ
ejde-984	249	17	gagliardo	gagliardo	NOUN
ejde-984	249	18	-	-	PUNCT
ejde-984	249	19	nirenberg	nirenberg	NOUN
ejde-984	249	20	inequality	inequality	NOUN
ejde-984	249	21	(	(	PUNCT
ejde-984	249	22	2.2	2.2	NUM
ejde-984	249	23	)	)	PUNCT
ejde-984	249	24	,	,	PUNCT
ejde-984	249	25	we	we	PRON
ejde-984	249	26	obtain	obtain	VERB
ejde-984	249	27	that	that	DET
ejde-984	249	28	|u|q	|u|q	PROPN
ejde-984	249	29	=	=	SYM
ejde-984	249	30	0	0	NUM
ejde-984	250	1	and	and	CCONJ
ejde-984	250	2	|u|p	|u|p	PROPN
ejde-984	250	3	=	=	SYM
ejde-984	250	4	0	0	X
ejde-984	250	5	.	.	PUNCT
ejde-984	251	1	thus	thus	ADV
ejde-984	251	2	,	,	PUNCT
ejde-984	251	3	we	we	PRON
ejde-984	251	4	deduce	deduce	VERB
ejde-984	251	5	that	that	SCONJ
ejde-984	251	6	u	u	PROPN
ejde-984	251	7	≡	≡	PROPN
ejde-984	251	8	0	0	NUM
ejde-984	251	9	,	,	PUNCT
ejde-984	251	10	which	which	PRON
ejde-984	251	11	is	be	AUX
ejde-984	251	12	in	in	ADP
ejde-984	251	13	contradiction	contradiction	NOUN
ejde-984	251	14	with	with	ADP
ejde-984	251	15	u	u	PROPN
ejde-984	251	16	∈	∈	PROPN
ejde-984	251	17	sc	sc	PROPN
ejde-984	251	18	.	.	PUNCT
ejde-984	252	1	so	so	ADV
ejde-984	252	2	we	we	PRON
ejde-984	252	3	obtain	obtain	VERB
ejde-984	252	4	p0	p0	NOUN
ejde-984	252	5	c,µ	c,µ	NOUN
ejde-984	252	6	=	=	PUNCT
ejde-984	252	7	∅.	∅.	NOUN
ejde-984	252	8	to	to	ADP
ejde-984	252	9	proof	proof	NOUN
ejde-984	252	10	that	that	SCONJ
ejde-984	252	11	pc,µ	pc,µ	NOUN
ejde-984	252	12	is	be	AUX
ejde-984	252	13	a	a	DET
ejde-984	252	14	smooth	smooth	ADJ
ejde-984	252	15	manifold	manifold	NOUN
ejde-984	252	16	of	of	ADP
ejde-984	252	17	codimension	codimension	NOUN
ejde-984	252	18	2	2	NUM
ejde-984	252	19	in	in	ADP
ejde-984	252	20	hs(r3	hs(r3	NUM
ejde-984	252	21	)	)	PUNCT
ejde-984	252	22	is	be	AUX
ejde-984	252	23	very	very	ADV
ejde-984	252	24	similar	similar	ADJ
ejde-984	252	25	to	to	ADP
ejde-984	252	26	the	the	DET
ejde-984	252	27	one	one	NUM
ejde-984	252	28	of	of	ADP
ejde-984	252	29	[	[	X
ejde-984	252	30	24	24	NUM
ejde-984	252	31	,	,	PUNCT
ejde-984	252	32	lemma	lemma	PROPN
ejde-984	252	33	5.2	5.2	NUM
ejde-984	252	34	]	]	PUNCT
ejde-984	252	35	,	,	PUNCT
ejde-984	252	36	therefore	therefore	ADV
ejde-984	252	37	we	we	PRON
ejde-984	252	38	omit	omit	VERB
ejde-984	252	39	it	it	PRON
ejde-984	252	40	here	here	ADV
ejde-984	252	41	.	.	PUNCT
ejde-984	253	1	□	□	PUNCT
ejde-984	253	2	since	since	SCONJ
ejde-984	253	3	p0	p0	NOUN
ejde-984	253	4	c,µ	c,µ	NOUN
ejde-984	253	5	=	=	SYM
ejde-984	253	6	∅	∅	NOUN
ejde-984	253	7	by	by	ADP
ejde-984	253	8	lemma	lemma	PROPN
ejde-984	253	9	3.3	3.3	NUM
ejde-984	253	10	,	,	PUNCT
ejde-984	253	11	we	we	PRON
ejde-984	253	12	observe	observe	VERB
ejde-984	253	13	that	that	SCONJ
ejde-984	253	14	pc,µ	pc,µ	NOUN
ejde-984	253	15	is	be	AUX
ejde-984	253	16	a	a	DET
ejde-984	253	17	natural	natural	ADJ
ejde-984	253	18	constraint	constraint	NOUN
ejde-984	253	19	in	in	ADP
ejde-984	253	20	the	the	DET
ejde-984	253	21	following	follow	VERB
ejde-984	253	22	sense	sense	NOUN
ejde-984	253	23	.	.	PUNCT
ejde-984	254	1	lemma	lemma	PROPN
ejde-984	254	2	3.4	3.4	NUM
ejde-984	254	3	.	.	PUNCT
ejde-984	255	1	let	let	VERB
ejde-984	255	2	µ	µ	X
ejde-984	255	3	<	<	X
ejde-984	255	4	0	0	NUM
ejde-984	255	5	and	and	CCONJ
ejde-984	255	6	2	2	NUM
ejde-984	255	7	<	<	X
ejde-984	255	8	q	q	X
ejde-984	255	9	≤	≤	NUM
ejde-984	255	10	2	2	NUM
ejde-984	255	11	+	+	NUM
ejde-984	255	12	8s	8s	NUM
ejde-984	255	13	3	3	NUM
ejde-984	255	14	<	<	X
ejde-984	255	15	p	p	X
ejde-984	255	16	<	<	X
ejde-984	255	17	2∗s	2∗s	NUM
ejde-984	255	18	be	be	AUX
ejde-984	255	19	given	give	VERB
ejde-984	255	20	constants	constant	NOUN
ejde-984	255	21	.	.	PUNCT
ejde-984	256	1	if	if	SCONJ
ejde-984	256	2	u	u	PROPN
ejde-984	256	3	∈	∈	PROPN
ejde-984	256	4	pc,µ	pc,µ	NOUN
ejde-984	256	5	is	be	AUX
ejde-984	256	6	a	a	DET
ejde-984	256	7	critical	critical	ADJ
ejde-984	256	8	point	point	NOUN
ejde-984	256	9	for	for	ADP
ejde-984	256	10	eµ|pc,µ	eµ|pc,µ	NOUN
ejde-984	256	11	,	,	PUNCT
ejde-984	256	12	then	then	ADV
ejde-984	256	13	u	u	NOUN
ejde-984	256	14	is	be	AUX
ejde-984	256	15	a	a	DET
ejde-984	256	16	critical	critical	ADJ
ejde-984	256	17	point	point	NOUN
ejde-984	256	18	for	for	ADP
ejde-984	256	19	eµ|sc	eµ|sc	X
ejde-984	256	20	.	.	PUNCT
ejde-984	257	1	10	10	NUM
ejde-984	257	2	z.	z.	PROPN
ejde-984	257	3	guo	guo	PROPN
ejde-984	257	4	,	,	PUNCT
ejde-984	257	5	t.	t.	PROPN
ejde-984	257	6	zhang	zhang	PROPN
ejde-984	257	7	ejde-2025/75	ejde-2025/75	ADJ
ejde-984	257	8	proof	proof	NOUN
ejde-984	257	9	.	.	PUNCT
ejde-984	258	1	from	from	ADP
ejde-984	258	2	lemma	lemma	PROPN
ejde-984	258	3	3.3	3.3	NUM
ejde-984	258	4	,	,	PUNCT
ejde-984	258	5	we	we	PRON
ejde-984	258	6	see	see	VERB
ejde-984	258	7	that	that	DET
ejde-984	258	8	pc,µ	pc,µ	NOUN
ejde-984	258	9	is	be	AUX
ejde-984	258	10	a	a	DET
ejde-984	258	11	smooth	smooth	ADJ
ejde-984	258	12	manifold	manifold	NOUN
ejde-984	258	13	of	of	ADP
ejde-984	258	14	codimension	codimension	NOUN
ejde-984	258	15	2	2	NUM
ejde-984	258	16	in	in	ADP
ejde-984	258	17	hs(r3	hs(r3	NUM
ejde-984	258	18	)	)	PUNCT
ejde-984	258	19	and	and	CCONJ
ejde-984	258	20	p0	p0	NOUN
ejde-984	258	21	c,µ	c,µ	NOUN
ejde-984	258	22	=	=	PUNCT
ejde-984	258	23	∅.	∅.	NOUN
ejde-984	258	24	if	if	SCONJ
ejde-984	258	25	u	u	PROPN
ejde-984	258	26	∈	∈	PROPN
ejde-984	258	27	pc,µ	pc,µ	NOUN
ejde-984	258	28	is	be	AUX
ejde-984	258	29	a	a	DET
ejde-984	258	30	critical	critical	ADJ
ejde-984	258	31	point	point	NOUN
ejde-984	258	32	for	for	ADP
ejde-984	258	33	eµ|pc,µ	eµ|pc,µ	NOUN
ejde-984	258	34	,	,	PUNCT
ejde-984	258	35	then	then	ADV
ejde-984	258	36	by	by	ADP
ejde-984	258	37	the	the	DET
ejde-984	258	38	lagrange	lagrange	PROPN
ejde-984	258	39	multipliers	multiplier	NOUN
ejde-984	258	40	rule	rule	NOUN
ejde-984	258	41	,	,	PUNCT
ejde-984	258	42	one	one	PRON
ejde-984	258	43	gets	get	VERB
ejde-984	258	44	that	that	SCONJ
ejde-984	258	45	there	there	PRON
ejde-984	258	46	exist	exist	VERB
ejde-984	258	47	λ	λ	NOUN
ejde-984	258	48	,	,	PUNCT
ejde-984	258	49	ν	ν	PROPN
ejde-984	258	50	∈	∈	NOUN
ejde-984	258	51	r	r	NOUN
ejde-984	258	52	such	such	ADJ
ejde-984	258	53	that	that	DET
ejde-984	258	54	⟨e′	⟨e′	PROPN
ejde-984	258	55	µ(u	µ(u	NOUN
ejde-984	258	56	)	)	PUNCT
ejde-984	258	57	,	,	PUNCT
ejde-984	258	58	ϕ⟩	ϕ⟩	ADP
ejde-984	258	59	−	−	PROPN
ejde-984	259	1	λ	λ	INTJ
ejde-984	259	2	∫	∫	PROPN
ejde-984	259	3	r3	r3	PROPN
ejde-984	259	4	uϕdx−	uϕdx−	PROPN
ejde-984	260	1	ν⟨p	ν⟨p	ADP
ejde-984	260	2	′	′	NUM
ejde-984	260	3	µ(u	µ(u	NOUN
ejde-984	260	4	)	)	PUNCT
ejde-984	260	5	,	,	PUNCT
ejde-984	260	6	ϕ⟩	ϕ⟩	PUNCT
ejde-984	260	7	=	=	SYM
ejde-984	260	8	0	0	NUM
ejde-984	260	9	for	for	ADP
ejde-984	260	10	any	any	DET
ejde-984	260	11	ϕ	ϕ	PROPN
ejde-984	260	12	∈	∈	PROPN
ejde-984	260	13	hs(r3	hs(r3	NUM
ejde-984	260	14	)	)	PUNCT
ejde-984	260	15	,	,	PUNCT
ejde-984	260	16	that	that	ADV
ejde-984	260	17	is	is	ADV
ejde-984	260	18	,	,	PUNCT
ejde-984	260	19	u	u	NOUN
ejde-984	260	20	solves	solve	NOUN
ejde-984	260	21	[	[	PUNCT
ejde-984	260	22	(	(	PUNCT
ejde-984	260	23	1−	1−	NUM
ejde-984	260	24	2ν)a+	2ν)a+	NUM
ejde-984	260	25	(	(	PUNCT
ejde-984	260	26	1−	1−	NUM
ejde-984	260	27	4ν)b	4ν)b	NUM
ejde-984	260	28	∫	∫	PROPN
ejde-984	260	29	r3	r3	PROPN
ejde-984	260	30	|(−∆)s/2u|2dx	|(−∆)s/2u|2dx	VERB
ejde-984	260	31	]	]	PUNCT
ejde-984	260	32	(	(	PUNCT
ejde-984	260	33	−∆)su	−∆)su	NOUN
ejde-984	260	34	=	=	PUNCT
ejde-984	260	35	λu+	λu+	PROPN
ejde-984	260	36	µ(1−	µ(1−	PROPN
ejde-984	260	37	νqδs	νqδs	PROPN
ejde-984	260	38	,	,	PUNCT
ejde-984	260	39	q)|u|q−2u+	q)|u|q−2u+	PROPN
ejde-984	260	40	(	(	PUNCT
ejde-984	260	41	1−	1−	NUM
ejde-984	260	42	νpδs	νpδs	NOUN
ejde-984	260	43	,	,	PUNCT
ejde-984	260	44	p)|u|p−2u	p)|u|p−2u	NUM
ejde-984	260	45	in	in	ADP
ejde-984	260	46	r3	r3	PROPN
ejde-984	260	47	.	.	PUNCT
ejde-984	261	1	(	(	PUNCT
ejde-984	261	2	3.23	3.23	NUM
ejde-984	261	3	)	)	PUNCT
ejde-984	261	4	by	by	ADP
ejde-984	261	5	lemma	lemma	PROPN
ejde-984	261	6	2.2	2.2	NUM
ejde-984	261	7	,	,	PUNCT
ejde-984	261	8	we	we	PRON
ejde-984	261	9	obtain	obtain	VERB
ejde-984	261	10	that	that	SCONJ
ejde-984	261	11	3−	3−	NUM
ejde-984	261	12	2s	2s	NUM
ejde-984	261	13	2	2	NUM
ejde-984	261	14	(	(	PUNCT
ejde-984	261	15	1−	1−	NUM
ejde-984	261	16	2ν)a	2ν)a	NUM
ejde-984	261	17	∫	∫	PROPN
ejde-984	261	18	r3	r3	PROPN
ejde-984	261	19	|(−∆)s/2u|2dx+	|(−∆)s/2u|2dx+	NUM
ejde-984	261	20	3−	3−	NUM
ejde-984	261	21	2s	2s	NUM
ejde-984	261	22	2	2	NUM
ejde-984	261	23	(	(	PUNCT
ejde-984	261	24	1−	1−	NUM
ejde-984	261	25	4ν)b	4ν)b	NUM
ejde-984	261	26	(	(	PUNCT
ejde-984	261	27	∫	∫	PROPN
ejde-984	261	28	r3	r3	PROPN
ejde-984	261	29	|(−∆)s/2u|2dx	|(−∆)s/2u|2dx	VERB
ejde-984	261	30	)	)	PUNCT
ejde-984	261	31	2	2	NUM
ejde-984	261	32	−	−	NOUN
ejde-984	261	33	3	3	NUM
ejde-984	261	34	2	2	NUM
ejde-984	261	35	λ	λ	PROPN
ejde-984	261	36	∫	∫	PROPN
ejde-984	261	37	r3	r3	PROPN
ejde-984	261	38	u2dx+	u2dx+	PROPN
ejde-984	261	39	3µ(νqδs	3µ(νqδs	NUM
ejde-984	261	40	,	,	PUNCT
ejde-984	261	41	q	q	NOUN
ejde-984	261	42	−	−	PROPN
ejde-984	261	43	1	1	NUM
ejde-984	261	44	)	)	PUNCT
ejde-984	261	45	q	q	NOUN
ejde-984	261	46	∫	∫	PROPN
ejde-984	261	47	r3	r3	PROPN
ejde-984	261	48	uqdx+	uqdx+	PROPN
ejde-984	261	49	3(νpδs	3(νpδs	NUM
ejde-984	261	50	,	,	PUNCT
ejde-984	261	51	p	p	NOUN
ejde-984	261	52	−	−	PROPN
ejde-984	261	53	1	1	NUM
ejde-984	261	54	)	)	PUNCT
ejde-984	261	55	p	p	NOUN
ejde-984	261	56	∫	∫	PROPN
ejde-984	261	57	r3	r3	PROPN
ejde-984	261	58	updx	updx	VERB
ejde-984	261	59	=	=	X
ejde-984	261	60	0	0	NUM
ejde-984	261	61	in	in	ADP
ejde-984	261	62	r3	r3	PROPN
ejde-984	261	63	.	.	PUNCT
ejde-984	262	1	(	(	PUNCT
ejde-984	262	2	3.24	3.24	NUM
ejde-984	262	3	)	)	PUNCT
ejde-984	262	4	multiplying	multiplying	NOUN
ejde-984	262	5	(	(	PUNCT
ejde-984	262	6	3.23	3.23	NUM
ejde-984	262	7	)	)	PUNCT
ejde-984	262	8	by	by	ADP
ejde-984	262	9	u	u	NOUN
ejde-984	262	10	and	and	CCONJ
ejde-984	262	11	integrating	integrating	NOUN
ejde-984	262	12	,	,	PUNCT
ejde-984	262	13	then	then	ADV
ejde-984	262	14	combining	combine	VERB
ejde-984	262	15	it	it	PRON
ejde-984	262	16	with	with	ADP
ejde-984	262	17	(	(	PUNCT
ejde-984	262	18	3.24	3.24	NUM
ejde-984	262	19	)	)	PUNCT
ejde-984	262	20	,	,	PUNCT
ejde-984	262	21	we	we	PRON
ejde-984	262	22	obtain	obtain	VERB
ejde-984	262	23	that	that	SCONJ
ejde-984	262	24	(	(	PUNCT
ejde-984	262	25	1−	1−	NUM
ejde-984	262	26	2ν)a	2ν)a	NUM
ejde-984	262	27	∫	∫	PROPN
ejde-984	262	28	r3	r3	PROPN
ejde-984	262	29	|(−∆)s/2u|2dx+	|(−∆)s/2u|2dx+	PROPN
ejde-984	262	30	(	(	PUNCT
ejde-984	262	31	1−	1−	NUM
ejde-984	262	32	4ν)b	4ν)b	NUM
ejde-984	262	33	(	(	PUNCT
ejde-984	262	34	∫	∫	PROPN
ejde-984	262	35	r3	r3	PROPN
ejde-984	262	36	|(−∆)s/2u|2dx	|(−∆)s/2u|2dx	VERB
ejde-984	262	37	)	)	PUNCT
ejde-984	262	38	2	2	NUM
ejde-984	262	39	+	+	NUM
ejde-984	262	40	µδs	µδ	NOUN
ejde-984	262	41	,	,	PUNCT
ejde-984	262	42	q(νqδs	q(νqδs	NOUN
ejde-984	262	43	,	,	PUNCT
ejde-984	262	44	q	q	NOUN
ejde-984	262	45	−	−	PROPN
ejde-984	262	46	1	1	NUM
ejde-984	262	47	)	)	PUNCT
ejde-984	262	48	∫	∫	PROPN
ejde-984	262	49	r3	r3	PROPN
ejde-984	262	50	uqdx+	uqdx+	NOUN
ejde-984	262	51	δs	δs	NOUN
ejde-984	262	52	,	,	PUNCT
ejde-984	262	53	p(νpδs	p(νpδ	NOUN
ejde-984	262	54	,	,	PUNCT
ejde-984	262	55	p	p	NOUN
ejde-984	262	56	−	−	PROPN
ejde-984	262	57	1	1	NUM
ejde-984	262	58	)	)	PUNCT
ejde-984	262	59	∫	∫	PROPN
ejde-984	262	60	r3	r3	PROPN
ejde-984	262	61	updx	updx	VERB
ejde-984	262	62	=	=	X
ejde-984	262	63	0	0	X
ejde-984	262	64	.	.	PUNCT
ejde-984	263	1	(	(	PUNCT
ejde-984	263	2	3.25	3.25	NUM
ejde-984	263	3	)	)	PUNCT
ejde-984	263	4	by	by	ADP
ejde-984	263	5	(	(	PUNCT
ejde-984	263	6	3.25	3.25	NUM
ejde-984	263	7	)	)	PUNCT
ejde-984	263	8	and	and	CCONJ
ejde-984	263	9	pµ(u	pµ(u	NOUN
ejde-984	263	10	)	)	PUNCT
ejde-984	264	1	=	=	SYM
ejde-984	264	2	0	0	NUM
ejde-984	264	3	,	,	PUNCT
ejde-984	264	4	we	we	PRON
ejde-984	264	5	have	have	VERB
ejde-984	264	6	that	that	DET
ejde-984	264	7	ν	ν	PROPN
ejde-984	264	8	(	(	PUNCT
ejde-984	264	9	2a|(−∆)s/2u|22	2a|(−∆)s/2u|22	NUM
ejde-984	264	10	+	+	NUM
ejde-984	264	11	4b|(−∆)s/2u|42	4b|(−∆)s/2u|42	NUM
ejde-984	264	12	−	−	PROPN
ejde-984	264	13	µqδ2s	µqδ2s	NOUN
ejde-984	264	14	,	,	PUNCT
ejde-984	264	15	q|u|qq	q|u|qq	PROPN
ejde-984	264	16	−	−	PROPN
ejde-984	264	17	pδ2s	pδ2s	NOUN
ejde-984	264	18	,	,	PUNCT
ejde-984	264	19	p|u|pp	p|u|pp	NOUN
ejde-984	264	20	)	)	PUNCT
ejde-984	265	1	=	=	PUNCT
ejde-984	265	2	0	0	NUM
ejde-984	265	3	,	,	PUNCT
ejde-984	265	4	which	which	PRON
ejde-984	265	5	implies	imply	VERB
ejde-984	265	6	that	that	SCONJ
ejde-984	265	7	ν	ν	NOUN
ejde-984	265	8	=	=	SYM
ejde-984	265	9	0	0	PUNCT
ejde-984	265	10	since	since	SCONJ
ejde-984	265	11	u	u	NOUN
ejde-984	265	12	/∈	/∈	PROPN
ejde-984	265	13	p0	p0	NOUN
ejde-984	265	14	c,µ	c,µ	NOUN
ejde-984	265	15	:	:	PUNCT
ejde-984	265	16	from	from	ADP
ejde-984	265	17	lemma	lemma	PROPN
ejde-984	265	18	3.3	3.3	NUM
ejde-984	265	19	,	,	PUNCT
ejde-984	265	20	we	we	PRON
ejde-984	265	21	see	see	VERB
ejde-984	265	22	that	that	DET
ejde-984	265	23	pc,µ	pc,µ	NOUN
ejde-984	265	24	is	be	AUX
ejde-984	265	25	a	a	DET
ejde-984	265	26	smooth	smooth	ADJ
ejde-984	265	27	manifold	manifold	NOUN
ejde-984	265	28	of	of	ADP
ejde-984	265	29	codimension	codimension	NOUN
ejde-984	265	30	2	2	NUM
ejde-984	265	31	in	in	ADP
ejde-984	265	32	hs(r3	hs(r3	NUM
ejde-984	265	33	)	)	PUNCT
ejde-984	265	34	and	and	CCONJ
ejde-984	265	35	p0	p0	NOUN
ejde-984	265	36	c,µ	c,µ	NOUN
ejde-984	265	37	=	=	SYM
ejde-984	265	38	∅	∅	NOUN
ejde-984	265	39	,	,	PUNCT
ejde-984	265	40	namely	namely	ADV
ejde-984	265	41	,	,	PUNCT
ejde-984	265	42	(	(	PUNCT
ejde-984	265	43	2a|(−∆)s/2u|22	2a|(−∆)s/2u|22	NUM
ejde-984	265	44	+	+	NUM
ejde-984	265	45	4b|(−∆)s/2u|42	4b|(−∆)s/2u|42	NUM
ejde-984	265	46	−	−	PROPN
ejde-984	265	47	µqδ2s	µqδ2s	NOUN
ejde-984	265	48	,	,	PUNCT
ejde-984	265	49	q|u|qq	q|u|qq	PROPN
ejde-984	265	50	−	−	PROPN
ejde-984	265	51	pδ2s	pδ2s	NOUN
ejde-984	265	52	,	,	PUNCT
ejde-984	265	53	p|u|pp	p|u|pp	NOUN
ejde-984	265	54	)	)	PUNCT
ejde-984	265	55	̸=	̸=	PROPN
ejde-984	265	56	0	0	NUM
ejde-984	265	57	.	.	PUNCT
ejde-984	266	1	□	□	PUNCT
ejde-984	266	2	lemma	lemma	PROPN
ejde-984	266	3	3.5	3.5	NUM
ejde-984	266	4	.	.	PUNCT
ejde-984	267	1	for	for	ADP
ejde-984	267	2	every	every	DET
ejde-984	267	3	u	u	PROPN
ejde-984	267	4	∈	∈	PROPN
ejde-984	267	5	sc	sc	PROPN
ejde-984	267	6	,	,	PUNCT
ejde-984	267	7	there	there	PRON
ejde-984	267	8	exists	exist	VERB
ejde-984	267	9	a	a	DET
ejde-984	267	10	unique	unique	ADJ
ejde-984	267	11	tu	tu	PROPN
ejde-984	267	12	∈	∈	PROPN
ejde-984	267	13	r	r	NOUN
ejde-984	267	14	such	such	ADJ
ejde-984	267	15	that	that	SCONJ
ejde-984	267	16	tu	tu	PROPN
ejde-984	267	17	∗u	∗u	PROPN
ejde-984	267	18	∈	∈	PROPN
ejde-984	267	19	pc,µ.	pc,µ.	NOUN
ejde-984	267	20	moreover	moreover	ADV
ejde-984	267	21	,	,	PUNCT
ejde-984	267	22	tu	tu	PROPN
ejde-984	267	23	is	be	AUX
ejde-984	267	24	the	the	DET
ejde-984	267	25	unique	unique	ADJ
ejde-984	267	26	critical	critical	ADJ
ejde-984	267	27	point	point	NOUN
ejde-984	267	28	of	of	ADP
ejde-984	267	29	jµ	jµ	PROPN
ejde-984	267	30	u	u	NOUN
ejde-984	267	31	and	and	CCONJ
ejde-984	267	32	it	it	PRON
ejde-984	267	33	is	be	AUX
ejde-984	267	34	a	a	DET
ejde-984	267	35	strict	strict	ADJ
ejde-984	267	36	maximum	maximum	ADJ
ejde-984	267	37	point	point	NOUN
ejde-984	267	38	at	at	ADP
ejde-984	267	39	the	the	DET
ejde-984	267	40	positive	positive	ADJ
ejde-984	267	41	level	level	NOUN
ejde-984	267	42	.	.	PUNCT
ejde-984	268	1	moreover	moreover	ADV
ejde-984	268	2	,	,	PUNCT
ejde-984	268	3	(	(	PUNCT
ejde-984	268	4	i	i	NOUN
ejde-984	268	5	)	)	PUNCT
ejde-984	268	6	pc,µ	pc,µ	NOUN
ejde-984	269	1	=	=	NOUN
ejde-984	269	2	p−	p−	PRON
ejde-984	269	3	c,µ.	c,µ.	NOUN
ejde-984	269	4	(	(	PUNCT
ejde-984	269	5	ii	ii	NOUN
ejde-984	269	6	)	)	PUNCT
ejde-984	269	7	jµ	jµ	PROPN
ejde-984	269	8	u	u	NOUN
ejde-984	269	9	is	be	AUX
ejde-984	269	10	strictly	strictly	ADV
ejde-984	269	11	decreasing	decrease	VERB
ejde-984	269	12	and	and	CCONJ
ejde-984	269	13	concave	concave	VERB
ejde-984	269	14	on	on	ADP
ejde-984	269	15	(	(	PUNCT
ejde-984	269	16	tu,+∞	tu,+∞	NOUN
ejde-984	269	17	)	)	PUNCT
ejde-984	269	18	and	and	CCONJ
ejde-984	269	19	tu	tu	X
ejde-984	269	20	<	<	X
ejde-984	269	21	0	0	PUNCT
ejde-984	269	22	implies	imply	VERB
ejde-984	269	23	that	that	SCONJ
ejde-984	269	24	pµ(u	pµ(u	NOUN
ejde-984	269	25	)	)	PUNCT
ejde-984	269	26	<	<	X
ejde-984	269	27	0	0	X
ejde-984	269	28	.	.	PUNCT
ejde-984	270	1	(	(	PUNCT
ejde-984	270	2	iii	iii	X
ejde-984	270	3	)	)	PUNCT
ejde-984	270	4	the	the	DET
ejde-984	270	5	function	function	NOUN
ejde-984	270	6	u	u	PROPN
ejde-984	270	7	∈	∈	PROPN
ejde-984	270	8	sc	sc	PROPN
ejde-984	270	9	7→	7→	PROPN
ejde-984	270	10	tu	tu	PROPN
ejde-984	270	11	is	be	AUX
ejde-984	270	12	of	of	ADP
ejde-984	270	13	class	class	NOUN
ejde-984	270	14	c1	c1	NOUN
ejde-984	270	15	.	.	PUNCT
ejde-984	271	1	(	(	PUNCT
ejde-984	271	2	iv	iv	X
ejde-984	271	3	)	)	PUNCT
ejde-984	271	4	if	if	SCONJ
ejde-984	271	5	pµ(u	pµ(u	NOUN
ejde-984	271	6	)	)	PUNCT
ejde-984	271	7	<	<	X
ejde-984	271	8	0	0	NUM
ejde-984	271	9	,	,	PUNCT
ejde-984	271	10	then	then	ADV
ejde-984	271	11	tu	tu	X
ejde-984	271	12	<	<	X
ejde-984	271	13	0	0	X
ejde-984	271	14	.	.	PUNCT
ejde-984	272	1	proof	proof	NOUN
ejde-984	272	2	.	.	PUNCT
ejde-984	273	1	for	for	ADP
ejde-984	273	2	every	every	DET
ejde-984	273	3	u	u	PROPN
ejde-984	273	4	∈	∈	PROPN
ejde-984	273	5	sc	sc	PROPN
ejde-984	273	6	,	,	PUNCT
ejde-984	273	7	according	accord	VERB
ejde-984	273	8	to	to	ADP
ejde-984	273	9	the	the	DET
ejde-984	273	10	definition	definition	NOUN
ejde-984	273	11	of	of	ADP
ejde-984	273	12	jµ	jµ	PROPN
ejde-984	273	13	u	u	NOUN
ejde-984	273	14	(	(	PUNCT
ejde-984	273	15	τ	τ	PROPN
ejde-984	273	16	)	)	PUNCT
ejde-984	273	17	,	,	PUNCT
ejde-984	273	18	we	we	PRON
ejde-984	273	19	obtain	obtain	VERB
ejde-984	273	20	lim	lim	NOUN
ejde-984	273	21	τ→−∞	τ→−∞	X
ejde-984	274	1	jµ	jµ	PROPN
ejde-984	274	2	u	u	NOUN
ejde-984	274	3	(	(	PUNCT
ejde-984	274	4	τ	τ	X
ejde-984	274	5	)	)	PUNCT
ejde-984	274	6	=	=	PUNCT
ejde-984	274	7	0	0	PUNCT
ejde-984	274	8	+	+	NUM
ejde-984	274	9	and	and	CCONJ
ejde-984	274	10	lim	lim	PROPN
ejde-984	274	11	τ→+∞	τ→+∞	PROPN
ejde-984	274	12	jµ	jµ	PROPN
ejde-984	274	13	u	u	NOUN
ejde-984	274	14	(	(	PUNCT
ejde-984	274	15	τ	τ	X
ejde-984	274	16	)	)	PUNCT
ejde-984	274	17	=	=	SYM
ejde-984	274	18	−∞.	−∞.	PROPN
ejde-984	274	19	thus	thus	ADV
ejde-984	274	20	,	,	PUNCT
ejde-984	274	21	jµ	jµ	PRON
ejde-984	274	22	u	u	NOUN
ejde-984	274	23	has	have	VERB
ejde-984	274	24	at	at	ADV
ejde-984	274	25	least	least	ADJ
ejde-984	274	26	one	one	NUM
ejde-984	274	27	global	global	ADJ
ejde-984	274	28	maximum	maximum	ADJ
ejde-984	274	29	point	point	NOUN
ejde-984	274	30	tu	tu	PROPN
ejde-984	274	31	at	at	ADP
ejde-984	274	32	positive	positive	ADJ
ejde-984	274	33	level	level	NOUN
ejde-984	274	34	.	.	PUNCT
ejde-984	275	1	besides	besides	SCONJ
ejde-984	275	2	,	,	PUNCT
ejde-984	275	3	this	this	PRON
ejde-984	275	4	is	be	AUX
ejde-984	275	5	the	the	DET
ejde-984	275	6	one	one	NUM
ejde-984	275	7	and	and	CCONJ
ejde-984	275	8	only	only	ADV
ejde-984	275	9	critical	critical	ADJ
ejde-984	275	10	point	point	NOUN
ejde-984	275	11	of	of	ADP
ejde-984	275	12	jµ	jµ	PROPN
ejde-984	275	13	u	u	NOUN
ejde-984	275	14	.	.	PUNCT
ejde-984	276	1	for	for	ADP
ejde-984	276	2	each	each	DET
ejde-984	276	3	u	u	PROPN
ejde-984	276	4	∈	∈	PROPN
ejde-984	276	5	sc	sc	PROPN
ejde-984	276	6	,	,	PUNCT
ejde-984	276	7	we	we	PRON
ejde-984	276	8	define	define	VERB
ejde-984	276	9	h(t	h(t	PRON
ejde-984	276	10	)	)	PUNCT
ejde-984	276	11	=	=	SYM
ejde-984	277	1	at2s	at2s	NOUN
ejde-984	277	2	2	2	NUM
ejde-984	277	3	|(−∆)s/2u|22	|(−∆)s/2u|22	NOUN
ejde-984	277	4	+	+	CCONJ
ejde-984	277	5	bt4s	bt4s	PROPN
ejde-984	277	6	4	4	NUM
ejde-984	277	7	|(−∆)s/2u|42	|(−∆)s/2u|42	NUM
ejde-984	277	8	−	−	NOUN
ejde-984	277	9	tpδs	tpδs	NOUN
ejde-984	277	10	,	,	PUNCT
ejde-984	277	11	ps	ps	PROPN
ejde-984	277	12	p	p	PROPN
ejde-984	277	13	|u|pp	|u|pp	PROPN
ejde-984	278	1	−	−	PROPN
ejde-984	278	2	µ	µ	DET
ejde-984	278	3	tqδs	tqδs	NOUN
ejde-984	278	4	,	,	PUNCT
ejde-984	278	5	qs	qs	ADP
ejde-984	278	6	q	q	NOUN
ejde-984	278	7	|u|qq	|u|qq	NOUN
ejde-984	278	8	.	.	PUNCT
ejde-984	279	1	indeed	indeed	ADV
ejde-984	279	2	,	,	PUNCT
ejde-984	279	3	jµ	jµ	PROPN
ejde-984	279	4	u	u	NOUN
ejde-984	279	5	(	(	PUNCT
ejde-984	279	6	τ	τ	X
ejde-984	279	7	)	)	PUNCT
ejde-984	279	8	=	=	SYM
ejde-984	279	9	h(eτ	h(eτ	PROPN
ejde-984	279	10	)	)	PUNCT
ejde-984	279	11	,	,	PUNCT
ejde-984	279	12	then	then	ADV
ejde-984	279	13	we	we	PRON
ejde-984	279	14	obtain	obtain	VERB
ejde-984	279	15	jµ	jµ	PROPN
ejde-984	279	16	u	u	NOUN
ejde-984	279	17	′(τ	′(τ	NOUN
ejde-984	279	18	)	)	PUNCT
ejde-984	280	1	=	=	SYM
ejde-984	280	2	h′(eτ	h′(eτ	PROPN
ejde-984	280	3	)	)	PUNCT
ejde-984	280	4	eτ	eτ	NOUN
ejde-984	280	5	.	.	PUNCT
ejde-984	281	1	thus	thus	ADV
ejde-984	281	2	,	,	PUNCT
ejde-984	281	3	it	it	PRON
ejde-984	281	4	is	be	AUX
ejde-984	281	5	sufficient	sufficient	ADJ
ejde-984	281	6	to	to	PART
ejde-984	281	7	study	study	VERB
ejde-984	281	8	the	the	DET
ejde-984	281	9	function	function	NOUN
ejde-984	281	10	h.	h.	PROPN
ejde-984	282	1	the	the	DET
ejde-984	282	2	derivative	derivative	NOUN
ejde-984	282	3	of	of	ADP
ejde-984	282	4	h	h	NOUN
ejde-984	282	5	can	can	AUX
ejde-984	282	6	be	be	AUX
ejde-984	282	7	written	write	VERB
ejde-984	282	8	as	as	ADP
ejde-984	282	9	h′(t	h′(t	NOUN
ejde-984	282	10	)	)	PUNCT
ejde-984	283	1	=	=	SYM
ejde-984	283	2	t4s−1η(t	t4s−1η(t	NOUN
ejde-984	283	3	)	)	PUNCT
ejde-984	283	4	,	,	PUNCT
ejde-984	283	5	where	where	SCONJ
ejde-984	283	6	η(t	η(t	NOUN
ejde-984	283	7	)	)	PUNCT
ejde-984	283	8	=	=	SYM
ejde-984	283	9	ast−2s|(−∆)s/2u|22	ast−2s|(−∆)s/2u|22	X
ejde-984	284	1	+	+	NUM
ejde-984	284	2	bs|(−∆)s/2u|42	bs|(−∆)s/2u|42	NOUN
ejde-984	284	3	−	−	NOUN
ejde-984	284	4	δs	δs	NOUN
ejde-984	284	5	,	,	PUNCT
ejde-984	284	6	pst	pst	NOUN
ejde-984	284	7	(	(	PUNCT
ejde-984	284	8	pδs	pδs	PROPN
ejde-984	284	9	,	,	PUNCT
ejde-984	284	10	p−4)s|u|pp	p−4)s|u|pp	PRON
ejde-984	284	11	−	−	PROPN
ejde-984	284	12	µsδs	µsδ	NOUN
ejde-984	284	13	,	,	PUNCT
ejde-984	284	14	qt	qt	NOUN
ejde-984	284	15	(	(	PUNCT
ejde-984	284	16	qδs	qδs	PROPN
ejde-984	284	17	,	,	PUNCT
ejde-984	284	18	q−4)s|u|qq	q−4)s|u|qq	NOUN
ejde-984	284	19	,	,	PUNCT
ejde-984	284	20	and	and	CCONJ
ejde-984	284	21	η	η	PROPN
ejde-984	284	22	satisfies	satisfie	NOUN
ejde-984	284	23	lim	lim	PROPN
ejde-984	284	24	t→0	t→0	AUX
ejde-984	284	25	+	+	CCONJ
ejde-984	284	26	η(t	η(t	NOUN
ejde-984	284	27	)	)	PUNCT
ejde-984	284	28	=	=	PUNCT
ejde-984	285	1	+	+	NUM
ejde-984	285	2	∞	∞	PROPN
ejde-984	285	3	,	,	PUNCT
ejde-984	285	4	lim	lim	PROPN
ejde-984	285	5	t→+∞	t→+∞	PROPN
ejde-984	285	6	η(t	η(t	NOUN
ejde-984	285	7	)	)	PUNCT
ejde-984	285	8	=	=	PUNCT
ejde-984	285	9	−∞	−∞	PROPN
ejde-984	285	10	,	,	PUNCT
ejde-984	285	11	η′(t	η′(t	PROPN
ejde-984	285	12	)	)	PUNCT
ejde-984	285	13	<	<	X
ejde-984	285	14	0	0	NUM
ejde-984	285	15	,	,	PUNCT
ejde-984	285	16	for	for	ADP
ejde-984	285	17	all	all	DET
ejde-984	285	18	t	t	PROPN
ejde-984	285	19	>	>	X
ejde-984	285	20	0	0	X
ejde-984	285	21	.	.	X
ejde-984	285	22	ejde-2025/75	ejde-2025/75	ADJ
ejde-984	285	23	normalized	normalize	VERB
ejde-984	285	24	solutions	solution	NOUN
ejde-984	285	25	of	of	ADP
ejde-984	285	26	fractional	fractional	PROPN
ejde-984	285	27	kirchhoff	kirchhoff	NOUN
ejde-984	285	28	equations	equation	NOUN
ejde-984	285	29	11	11	NUM
ejde-984	285	30	from	from	ADP
ejde-984	285	31	the	the	DET
ejde-984	285	32	above	above	ADJ
ejde-984	285	33	analysis	analysis	NOUN
ejde-984	285	34	for	for	ADP
ejde-984	285	35	the	the	DET
ejde-984	285	36	function	function	NOUN
ejde-984	285	37	η	η	PROPN
ejde-984	285	38	,	,	PUNCT
ejde-984	285	39	we	we	PRON
ejde-984	285	40	obtain	obtain	VERB
ejde-984	285	41	it	it	PRON
ejde-984	285	42	has	have	VERB
ejde-984	285	43	a	a	DET
ejde-984	285	44	unique	unique	ADJ
ejde-984	285	45	zero	zero	NUM
ejde-984	285	46	point	point	NOUN
ejde-984	285	47	t̃	t̃	PROPN
ejde-984	285	48	in	in	ADP
ejde-984	285	49	(	(	PUNCT
ejde-984	285	50	0,+∞	0,+∞	NUM
ejde-984	285	51	)	)	PUNCT
ejde-984	285	52	.	.	PUNCT
ejde-984	286	1	by	by	ADP
ejde-984	286	2	h′(t	h′(t	NOUN
ejde-984	286	3	)	)	PUNCT
ejde-984	286	4	=	=	SYM
ejde-984	286	5	t4s−1η(t	t4s−1η(t	NOUN
ejde-984	286	6	)	)	PUNCT
ejde-984	286	7	,	,	PUNCT
ejde-984	286	8	it	it	PRON
ejde-984	286	9	holds	hold	VERB
ejde-984	286	10	that	that	SCONJ
ejde-984	286	11	the	the	DET
ejde-984	286	12	function	function	NOUN
ejde-984	286	13	h	h	NOUN
ejde-984	286	14	has	have	VERB
ejde-984	286	15	a	a	DET
ejde-984	286	16	unique	unique	ADJ
ejde-984	286	17	critical	critical	ADJ
ejde-984	286	18	point	point	NOUN
ejde-984	286	19	t̃	t̃	PROPN
ejde-984	286	20	and	and	CCONJ
ejde-984	286	21	tu	tu	NOUN
ejde-984	286	22	=	=	PUNCT
ejde-984	286	23	ln	ln	ADJ
ejde-984	286	24	t̃.	t̃.	PROPN
ejde-984	286	25	by	by	ADP
ejde-984	286	26	lim	lim	PROPN
ejde-984	286	27	t→0	t→0	PROPN
ejde-984	286	28	+	+	CCONJ
ejde-984	286	29	h(t	h(t	PROPN
ejde-984	286	30	)	)	PUNCT
ejde-984	286	31	=	=	PUNCT
ejde-984	287	1	0	0	PUNCT
ejde-984	287	2	+	+	ADJ
ejde-984	287	3	,	,	PUNCT
ejde-984	287	4	and	and	CCONJ
ejde-984	287	5	lim	lim	PROPN
ejde-984	287	6	t→+∞	t→+∞	PROPN
ejde-984	287	7	h(t	h(t	PROPN
ejde-984	287	8	)	)	PUNCT
ejde-984	287	9	=	=	SYM
ejde-984	287	10	−∞	−∞	NOUN
ejde-984	287	11	,	,	PUNCT
ejde-984	287	12	we	we	PRON
ejde-984	287	13	deduce	deduce	VERB
ejde-984	287	14	that	that	SCONJ
ejde-984	287	15	h(t̃	h(t̃	VERB
ejde-984	287	16	)	)	PUNCT
ejde-984	287	17	>	>	X
ejde-984	288	1	0	0	X
ejde-984	288	2	.	.	PUNCT
ejde-984	289	1	from	from	ADP
ejde-984	289	2	the	the	DET
ejde-984	289	3	above	above	ADJ
ejde-984	289	4	arguments	argument	NOUN
ejde-984	289	5	and	and	CCONJ
ejde-984	289	6	spµ(tu	spµ(tu	PROPN
ejde-984	289	7	∗	∗	X
ejde-984	289	8	u	u	NOUN
ejde-984	289	9	)	)	PUNCT
ejde-984	289	10	=	=	SYM
ejde-984	289	11	(	(	PUNCT
ejde-984	289	12	jµ	jµ	PROPN
ejde-984	289	13	u	u	NOUN
ejde-984	289	14	)	)	PUNCT
ejde-984	289	15	′(tu	′(tu	PROPN
ejde-984	289	16	)	)	PUNCT
ejde-984	289	17	,	,	PUNCT
ejde-984	289	18	we	we	PRON
ejde-984	289	19	deduce	deduce	VERB
ejde-984	289	20	that	that	SCONJ
ejde-984	289	21	for	for	ADP
ejde-984	289	22	each	each	DET
ejde-984	289	23	u	u	PROPN
ejde-984	289	24	∈	∈	PROPN
ejde-984	289	25	sc	sc	PROPN
ejde-984	289	26	,	,	PUNCT
ejde-984	289	27	there	there	PRON
ejde-984	289	28	exists	exist	VERB
ejde-984	289	29	a	a	DET
ejde-984	289	30	unique	unique	ADJ
ejde-984	289	31	tu	tu	PROPN
ejde-984	289	32	∈	∈	PROPN
ejde-984	289	33	r	r	NOUN
ejde-984	289	34	such	such	ADJ
ejde-984	289	35	that	that	DET
ejde-984	289	36	pµ(tu	pµ(tu	PROPN
ejde-984	289	37	∗	∗	NOUN
ejde-984	289	38	u	u	NOUN
ejde-984	289	39	)	)	PUNCT
ejde-984	289	40	=	=	SYM
ejde-984	289	41	0	0	NUM
ejde-984	289	42	,	,	PUNCT
ejde-984	289	43	namely	namely	ADV
ejde-984	289	44	,	,	PUNCT
ejde-984	289	45	tu	tu	PROPN
ejde-984	289	46	∗	∗	NOUN
ejde-984	289	47	u	u	NOUN
ejde-984	289	48	∈	∈	PROPN
ejde-984	289	49	pc,µ.	pc,µ.	NOUN
ejde-984	289	50	let	let	VERB
ejde-984	289	51	u	u	PRON
ejde-984	289	52	∈	∈	NOUN
ejde-984	289	53	pc,µ	pc,µ	NOUN
ejde-984	289	54	,	,	PUNCT
ejde-984	289	55	then	then	ADV
ejde-984	289	56	we	we	PRON
ejde-984	289	57	obtain	obtain	VERB
ejde-984	289	58	tu	tu	PROPN
ejde-984	289	59	=	=	SYM
ejde-984	289	60	0	0	PUNCT
ejde-984	290	1	and	and	CCONJ
ejde-984	290	2	as	as	SCONJ
ejde-984	290	3	tu	tu	PROPN
ejde-984	290	4	is	be	AUX
ejde-984	290	5	a	a	DET
ejde-984	290	6	maximum	maximum	ADJ
ejde-984	290	7	point	point	NOUN
ejde-984	290	8	of	of	ADP
ejde-984	290	9	jµ	jµ	PROPN
ejde-984	290	10	u	u	NOUN
ejde-984	290	11	,	,	PUNCT
ejde-984	290	12	we	we	PRON
ejde-984	290	13	obtain	obtain	VERB
ejde-984	290	14	that	that	PRON
ejde-984	290	15	(	(	PUNCT
ejde-984	290	16	jµ	jµ	PROPN
ejde-984	290	17	u	u	NOUN
ejde-984	290	18	)	)	PUNCT
ejde-984	290	19	′′	′′	PROPN
ejde-984	290	20	(	(	PUNCT
ejde-984	290	21	0	0	NUM
ejde-984	290	22	)	)	PUNCT
ejde-984	290	23	≤	≤	NOUN
ejde-984	290	24	0	0	NUM
ejde-984	290	25	.	.	PUNCT
ejde-984	291	1	since	since	SCONJ
ejde-984	291	2	p0	p0	NOUN
ejde-984	291	3	c,µ	c,µ	NOUN
ejde-984	291	4	=	=	SYM
ejde-984	291	5	∅	∅	NOUN
ejde-984	291	6	,	,	PUNCT
ejde-984	291	7	we	we	PRON
ejde-984	291	8	conclude	conclude	VERB
ejde-984	291	9	that	that	SCONJ
ejde-984	291	10	(	(	PUNCT
ejde-984	291	11	jµ	jµ	PROPN
ejde-984	291	12	u	u	NOUN
ejde-984	291	13	)	)	PUNCT
ejde-984	292	1	′′	′′	PROPN
ejde-984	292	2	(	(	PUNCT
ejde-984	292	3	0	0	NUM
ejde-984	292	4	)	)	PUNCT
ejde-984	292	5	<	<	X
ejde-984	292	6	0	0	X
ejde-984	292	7	.	.	PUNCT
ejde-984	293	1	thus	thus	ADV
ejde-984	293	2	,	,	PUNCT
ejde-984	293	3	pc,µ	pc,µ	NOUN
ejde-984	293	4	=	=	NOUN
ejde-984	293	5	p−	p−	VERB
ejde-984	293	6	c,µ.	c,µ.	NOUN
ejde-984	293	7	from	from	ADP
ejde-984	293	8	the	the	DET
ejde-984	293	9	calculus	calculus	NOUN
ejde-984	293	10	above	above	ADV
ejde-984	293	11	,	,	PUNCT
ejde-984	293	12	we	we	PRON
ejde-984	293	13	can	can	AUX
ejde-984	293	14	also	also	ADV
ejde-984	293	15	deduce	deduce	VERB
ejde-984	293	16	that	that	SCONJ
ejde-984	293	17	jµ	jµ	NUM
ejde-984	293	18	u	u	NOUN
ejde-984	293	19	is	be	AUX
ejde-984	293	20	strictly	strictly	ADV
ejde-984	293	21	decreasing	decrease	VERB
ejde-984	293	22	and	and	CCONJ
ejde-984	293	23	concave	concave	VERB
ejde-984	293	24	on	on	ADP
ejde-984	293	25	(	(	PUNCT
ejde-984	293	26	tu,+∞	tu,+∞	NOUN
ejde-984	293	27	)	)	PUNCT
ejde-984	293	28	.	.	PUNCT
ejde-984	294	1	since	since	SCONJ
ejde-984	294	2	(	(	PUNCT
ejde-984	294	3	jµ	jµ	PROPN
ejde-984	294	4	u	u	NOUN
ejde-984	294	5	)	)	PUNCT
ejde-984	294	6	′(t	′(t	PROPN
ejde-984	294	7	)	)	PUNCT
ejde-984	294	8	<	<	X
ejde-984	294	9	0	0	PUNCT
ejde-984	295	1	if	if	SCONJ
ejde-984	295	2	and	and	CCONJ
ejde-984	295	3	only	only	ADV
ejde-984	295	4	if	if	SCONJ
ejde-984	295	5	t	t	PROPN
ejde-984	295	6	>	>	X
ejde-984	295	7	tu	tu	PROPN
ejde-984	295	8	,	,	PUNCT
ejde-984	295	9	we	we	PRON
ejde-984	295	10	conclude	conclude	VERB
ejde-984	295	11	that	that	PRON
ejde-984	295	12	pµ(u	pµ(u	VERB
ejde-984	295	13	)	)	PUNCT
ejde-984	295	14	=	=	SYM
ejde-984	295	15	1	1	NUM
ejde-984	295	16	s	s	X
ejde-984	295	17	(	(	PUNCT
ejde-984	295	18	j	j	PROPN
ejde-984	295	19	µ	µ	X
ejde-984	295	20	u	u	NOUN
ejde-984	295	21	)	)	PUNCT
ejde-984	295	22	′(0	′(0	PROPN
ejde-984	295	23	)	)	PUNCT
ejde-984	295	24	<	<	X
ejde-984	295	25	0	0	PUNCT
ejde-984	296	1	if	if	SCONJ
ejde-984	296	2	and	and	CCONJ
ejde-984	296	3	only	only	ADV
ejde-984	296	4	if	if	SCONJ
ejde-984	296	5	tu	tu	PROPN
ejde-984	296	6	<	<	X
ejde-984	296	7	0	0	PROPN
ejde-984	296	8	.	.	PUNCT
ejde-984	296	9	item	item	NOUN
ejde-984	296	10	(	(	PUNCT
ejde-984	296	11	iii	iii	NOUN
ejde-984	296	12	)	)	PUNCT
ejde-984	296	13	holds	hold	VERB
ejde-984	296	14	when	when	SCONJ
ejde-984	296	15	applying	apply	VERB
ejde-984	296	16	the	the	DET
ejde-984	296	17	implicit	implicit	ADJ
ejde-984	296	18	function	function	NOUN
ejde-984	296	19	theorem	theorem	VERB
ejde-984	296	20	to	to	ADP
ejde-984	296	21	the	the	DET
ejde-984	296	22	function	function	NOUN
ejde-984	296	23	φ(τ	φ(τ	NOUN
ejde-984	296	24	,	,	PUNCT
ejde-984	296	25	u	u	NOUN
ejde-984	296	26	)	)	PUNCT
ejde-984	296	27	=	=	SYM
ejde-984	296	28	(	(	PUNCT
ejde-984	296	29	jµ	jµ	PROPN
ejde-984	296	30	u	u	NOUN
ejde-984	296	31	)	)	PUNCT
ejde-984	296	32	′(τ	′(τ	NUM
ejde-984	296	33	)	)	PUNCT
ejde-984	296	34	.	.	PUNCT
ejde-984	297	1	we	we	PRON
ejde-984	297	2	use	use	VERB
ejde-984	297	3	that	that	DET
ejde-984	297	4	φ(tu	φ(tu	NOUN
ejde-984	297	5	,	,	PUNCT
ejde-984	297	6	u	u	NOUN
ejde-984	297	7	)	)	PUNCT
ejde-984	297	8	=	=	SYM
ejde-984	297	9	(	(	PUNCT
ejde-984	297	10	jµ	jµ	PROPN
ejde-984	297	11	u	u	NOUN
ejde-984	297	12	)	)	PUNCT
ejde-984	297	13	′(tu	′(tu	PROPN
ejde-984	297	14	)	)	PUNCT
ejde-984	297	15	=	=	SYM
ejde-984	297	16	0	0	NUM
ejde-984	297	17	,	,	PUNCT
ejde-984	297	18	that	that	SCONJ
ejde-984	297	19	∂τφ(tu	∂τφ(tu	ADJ
ejde-984	297	20	,	,	PUNCT
ejde-984	297	21	u	u	NOUN
ejde-984	297	22	)	)	PUNCT
ejde-984	297	23	=	=	SYM
ejde-984	297	24	(	(	PUNCT
ejde-984	297	25	jµ	jµ	PROPN
ejde-984	297	26	u	u	NOUN
ejde-984	297	27	)	)	PUNCT
ejde-984	297	28	′′	′′	PROPN
ejde-984	297	29	(	(	PUNCT
ejde-984	297	30	tu	tu	PROPN
ejde-984	297	31	)	)	PUNCT
ejde-984	297	32	<	<	X
ejde-984	297	33	0	0	NUM
ejde-984	297	34	,	,	PUNCT
ejde-984	297	35	and	and	CCONJ
ejde-984	297	36	the	the	DET
ejde-984	297	37	fact	fact	NOUN
ejde-984	297	38	that	that	SCONJ
ejde-984	297	39	it	it	PRON
ejde-984	297	40	is	be	AUX
ejde-984	297	41	not	not	PART
ejde-984	297	42	possible	possible	ADJ
ejde-984	297	43	to	to	PART
ejde-984	297	44	pass	pass	VERB
ejde-984	297	45	with	with	ADP
ejde-984	297	46	continuity	continuity	NOUN
ejde-984	297	47	from	from	ADP
ejde-984	297	48	p+	p+	NOUN
ejde-984	297	49	c,µ	c,µ	NOUN
ejde-984	297	50	to	to	ADP
ejde-984	297	51	p−	p−	NOUN
ejde-984	297	52	c,µ	c,µ	NOUN
ejde-984	297	53	(	(	PUNCT
ejde-984	297	54	since	since	SCONJ
ejde-984	297	55	p0	p0	NOUN
ejde-984	297	56	c,µ	c,µ	NOUN
ejde-984	297	57	=	=	SYM
ejde-984	297	58	∅	∅	NOUN
ejde-984	297	59	)	)	PUNCT
ejde-984	297	60	,	,	PUNCT
ejde-984	297	61	therefore	therefore	ADV
ejde-984	297	62	we	we	PRON
ejde-984	297	63	obtain	obtain	VERB
ejde-984	297	64	u	u	PROPN
ejde-984	297	65	7→	7→	PROPN
ejde-984	297	66	tu	tu	PROPN
ejde-984	297	67	is	be	AUX
ejde-984	297	68	c1	c1	NOUN
ejde-984	297	69	.	.	PUNCT
ejde-984	298	1	□	□	PUNCT
ejde-984	298	2	lemma	lemma	PROPN
ejde-984	298	3	3.6	3.6	NUM
ejde-984	298	4	.	.	PUNCT
ejde-984	299	1	it	it	PRON
ejde-984	299	2	holds	hold	VERB
ejde-984	299	3	that	that	SCONJ
ejde-984	299	4	m(c	m(c	PROPN
ejde-984	299	5	,	,	PUNCT
ejde-984	299	6	µ	µ	NOUN
ejde-984	299	7	)	)	PUNCT
ejde-984	299	8	=	=	SYM
ejde-984	299	9	inf	inf	PROPN
ejde-984	299	10	u∈pc,µ	u∈pc,µ	NOUN
ejde-984	299	11	eµ(u	eµ(u	NOUN
ejde-984	299	12	)	)	PUNCT
ejde-984	299	13	>	>	X
ejde-984	299	14	0	0	X
ejde-984	299	15	.	.	PUNCT
ejde-984	299	16	proof	proof	NOUN
ejde-984	299	17	.	.	PUNCT
ejde-984	300	1	setting	set	VERB
ejde-984	300	2	u	u	PRON
ejde-984	300	3	∈	∈	PROPN
ejde-984	300	4	pc,µ	pc,µ	NOUN
ejde-984	300	5	,	,	PUNCT
ejde-984	300	6	by	by	ADP
ejde-984	300	7	fractional	fractional	ADJ
ejde-984	300	8	gagliardo	gagliardo	NOUN
ejde-984	300	9	-	-	PUNCT
ejde-984	300	10	nirenberg	nirenberg	NOUN
ejde-984	300	11	inequality	inequality	NOUN
ejde-984	300	12	and	and	CCONJ
ejde-984	300	13	µ	µ	X
ejde-984	300	14	<	<	X
ejde-984	300	15	0	0	NUM
ejde-984	300	16	we	we	PRON
ejde-984	300	17	obtain	obtain	VERB
ejde-984	300	18	that	that	DET
ejde-984	300	19	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	300	20	≤	≤	NUM
ejde-984	300	21	δs	δs	NOUN
ejde-984	300	22	,	,	PUNCT
ejde-984	300	23	p|u|pp	p|u|pp	PROPN
ejde-984	300	24	≤	≤	NUM
ejde-984	300	25	δs	δs	NOUN
ejde-984	300	26	,	,	PUNCT
ejde-984	300	27	pc(s	pc(s	X
ejde-984	300	28	,	,	PUNCT
ejde-984	300	29	p	p	NOUN
ejde-984	300	30	)	)	PUNCT
ejde-984	300	31	p|	p|	ADV
ejde-984	300	32	−∆s/2u|pδs	−∆s/2u|pδs	NUM
ejde-984	300	33	,	,	PUNCT
ejde-984	300	34	p2	p2	X
ejde-984	300	35	|u|p(1−δs	|u|p(1−δs	NOUN
ejde-984	300	36	,	,	PUNCT
ejde-984	300	37	p	p	NOUN
ejde-984	300	38	)	)	PUNCT
ejde-984	300	39	2	2	NUM
ejde-984	300	40	.	.	PUNCT
ejde-984	301	1	(	(	PUNCT
ejde-984	301	2	3.26	3.26	NUM
ejde-984	301	3	)	)	PUNCT
ejde-984	301	4	from	from	ADP
ejde-984	301	5	the	the	DET
ejde-984	301	6	definition	definition	NOUN
ejde-984	301	7	of	of	ADP
ejde-984	301	8	pc,µ	pc,µ	NOUN
ejde-984	301	9	,	,	PUNCT
ejde-984	301	10	we	we	PRON
ejde-984	301	11	obtain	obtain	VERB
ejde-984	301	12	u	u	PROPN
ejde-984	301	13	∈	∈	PROPN
ejde-984	301	14	sc	sc	PROPN
ejde-984	301	15	,	,	PUNCT
ejde-984	301	16	namely	namely	ADV
ejde-984	301	17	,	,	PUNCT
ejde-984	301	18	|u|2	|u|2	PROPN
ejde-984	301	19	=	=	PROPN
ejde-984	301	20	c.	c.	PROPN
ejde-984	301	21	therefore	therefore	ADV
ejde-984	301	22	,	,	PUNCT
ejde-984	301	23	|(−∆)s/2u|2	|(−∆)s/2u|2	PROPN
ejde-984	301	24	≥	≥	PRON
ejde-984	301	25	(	(	PUNCT
ejde-984	301	26	a	a	DET
ejde-984	301	27	δs	δs	NOUN
ejde-984	301	28	,	,	PUNCT
ejde-984	301	29	pc(s	pc(s	NUM
ejde-984	301	30	,	,	PUNCT
ejde-984	301	31	p)pcp(1−δs	p)pcp(1−δs	ADJ
ejde-984	301	32	,	,	PUNCT
ejde-984	301	33	p	p	NOUN
ejde-984	301	34	)	)	PUNCT
ejde-984	301	35	)	)	PUNCT
ejde-984	301	36	1	1	NUM
ejde-984	301	37	pδs	pδs	NOUN
ejde-984	301	38	,	,	PUNCT
ejde-984	301	39	p−2	p−2	PROPN
ejde-984	301	40	⇒	⇒	PROPN
ejde-984	301	41	inf	inf	PROPN
ejde-984	301	42	pc,µ	pc,µ	NOUN
ejde-984	301	43	|(−∆)s/2u|2	|(−∆)s/2u|2	X
ejde-984	301	44	>	>	X
ejde-984	301	45	0	0	PROPN
ejde-984	301	46	.	.	PUNCT
ejde-984	302	1	(	(	PUNCT
ejde-984	302	2	3.27	3.27	NUM
ejde-984	302	3	)	)	PUNCT
ejde-984	302	4	for	for	ADP
ejde-984	302	5	every	every	DET
ejde-984	302	6	u	u	PROPN
ejde-984	302	7	∈	∈	PROPN
ejde-984	302	8	pc,µ	pc,µ	NOUN
ejde-984	302	9	,	,	PUNCT
ejde-984	302	10	we	we	PRON
ejde-984	302	11	obtain	obtain	VERB
ejde-984	302	12	that	that	PRON
ejde-984	302	13	eµ(u	eµ(u	NOUN
ejde-984	302	14	)	)	PUNCT
ejde-984	302	15	=	=	SYM
ejde-984	303	1	(	(	PUNCT
ejde-984	303	2	1	1	NUM
ejde-984	303	3	2	2	NUM
ejde-984	303	4	−	−	NOUN
ejde-984	303	5	1	1	NUM
ejde-984	303	6	pδs	pδs	NOUN
ejde-984	303	7	,	,	PUNCT
ejde-984	303	8	p	p	NOUN
ejde-984	303	9	)	)	PUNCT
ejde-984	303	10	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	304	1	+	+	CCONJ
ejde-984	304	2	(	(	PUNCT
ejde-984	304	3	1	1	NUM
ejde-984	304	4	4	4	NUM
ejde-984	304	5	−	−	NUM
ejde-984	304	6	1	1	NUM
ejde-984	304	7	pδs	pδs	NOUN
ejde-984	304	8	,	,	PUNCT
ejde-984	304	9	p	p	NOUN
ejde-984	304	10	)	)	PUNCT
ejde-984	304	11	b|(−∆)s/2u|42	b|(−∆)s/2u|42	NOUN
ejde-984	304	12	−	−	PROPN
ejde-984	304	13	µ	µ	X
ejde-984	304	14	q	q	X
ejde-984	304	15	(	(	PUNCT
ejde-984	304	16	1−	1−	NUM
ejde-984	304	17	qδs	qδs	NOUN
ejde-984	304	18	,	,	PUNCT
ejde-984	304	19	q	q	PROPN
ejde-984	304	20	pδs	pδs	PROPN
ejde-984	304	21	,	,	PUNCT
ejde-984	304	22	p	p	NOUN
ejde-984	304	23	)	)	PUNCT
ejde-984	304	24	|u|qq	|u|qq	NOUN
ejde-984	304	25	≥	≥	NUM
ejde-984	304	26	(	(	PUNCT
ejde-984	304	27	1	1	NUM
ejde-984	304	28	2	2	NUM
ejde-984	304	29	−	−	NOUN
ejde-984	304	30	1	1	NUM
ejde-984	304	31	pδs	pδs	NOUN
ejde-984	304	32	,	,	PUNCT
ejde-984	304	33	p	p	NOUN
ejde-984	304	34	)	)	PUNCT
ejde-984	304	35	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	304	36	.	.	PUNCT
ejde-984	305	1	then	then	ADV
ejde-984	305	2	pδs	pδs	PROPN
ejde-984	305	3	,	,	PUNCT
ejde-984	305	4	p	p	X
ejde-984	305	5	>	>	X
ejde-984	305	6	2	2	NUM
ejde-984	305	7	and	and	CCONJ
ejde-984	305	8	(	(	PUNCT
ejde-984	305	9	3.27	3.27	NUM
ejde-984	305	10	)	)	PUNCT
ejde-984	305	11	imply	imply	VERB
ejde-984	305	12	that	that	SCONJ
ejde-984	305	13	m(c	m(c	PROPN
ejde-984	305	14	,	,	PUNCT
ejde-984	305	15	µ	µ	NOUN
ejde-984	305	16	)	)	PUNCT
ejde-984	305	17	=	=	SYM
ejde-984	305	18	inf	inf	PROPN
ejde-984	305	19	pc,µ	pc,µ	NOUN
ejde-984	305	20	eµ(u	eµ(u	NOUN
ejde-984	305	21	)	)	PUNCT
ejde-984	305	22	>	>	X
ejde-984	305	23	0	0	X
ejde-984	305	24	.	.	PUNCT
ejde-984	305	25	□	□	PUNCT
ejde-984	305	26	lemma	lemma	PROPN
ejde-984	305	27	3.7	3.7	NUM
ejde-984	305	28	.	.	PUNCT
ejde-984	306	1	for	for	ADP
ejde-984	306	2	2	2	NUM
ejde-984	306	3	<	<	X
ejde-984	306	4	q	q	PROPN
ejde-984	306	5	≤	≤	NUM
ejde-984	306	6	2	2	NUM
ejde-984	306	7	+	+	NUM
ejde-984	306	8	8s	8s	NUM
ejde-984	306	9	3	3	NUM
ejde-984	306	10	<	<	X
ejde-984	306	11	p	p	X
ejde-984	306	12	<	<	X
ejde-984	306	13	2∗s	2∗s	NUM
ejde-984	306	14	and	and	CCONJ
ejde-984	306	15	µ	µ	X
ejde-984	306	16	<	<	X
ejde-984	306	17	0	0	NUM
ejde-984	306	18	,	,	PUNCT
ejde-984	306	19	there	there	PRON
ejde-984	306	20	exists	exist	VERB
ejde-984	307	1	k	k	PROPN
ejde-984	307	2	>	>	X
ejde-984	307	3	0	0	PUNCT
ejde-984	307	4	small	small	ADJ
ejde-984	307	5	enough	enough	ADV
ejde-984	307	6	such	such	ADJ
ejde-984	307	7	that	that	SCONJ
ejde-984	307	8	0	0	NUM
ejde-984	307	9	<	<	X
ejde-984	307	10	sup	sup	PROPN
ejde-984	307	11	āk	āk	PROPN
ejde-984	307	12	eµ(u	eµ(u	NOUN
ejde-984	307	13	)	)	PUNCT
ejde-984	307	14	<	<	X
ejde-984	307	15	m(c	m(c	PROPN
ejde-984	307	16	,	,	PUNCT
ejde-984	307	17	µ	µ	NOUN
ejde-984	307	18	)	)	PUNCT
ejde-984	307	19	and	and	CCONJ
ejde-984	307	20	u	u	PROPN
ejde-984	307	21	∈	∈	PROPN
ejde-984	307	22	āk	āk	PROPN
ejde-984	307	23	⇒	⇒	PROPN
ejde-984	307	24	eµ(u	eµ(u	NOUN
ejde-984	307	25	)	)	PUNCT
ejde-984	307	26	,	,	PUNCT
ejde-984	307	27	pµ(u	pµ(u	NOUN
ejde-984	307	28	)	)	PUNCT
ejde-984	307	29	>	>	X
ejde-984	307	30	0	0	NUM
ejde-984	307	31	,	,	PUNCT
ejde-984	307	32	where	where	SCONJ
ejde-984	307	33	āk	āk	PROPN
ejde-984	307	34	=	=	PRON
ejde-984	307	35	{	{	PUNCT
ejde-984	307	36	u	u	NOUN
ejde-984	307	37	∈	∈	PROPN
ejde-984	307	38	sc	sc	PROPN
ejde-984	307	39	:	:	PUNCT
ejde-984	307	40	|(−∆)s/2u|22	|(−∆)s/2u|22	X
ejde-984	307	41	≤	≤	NUM
ejde-984	307	42	k	k	X
ejde-984	307	43	}	}	PUNCT
ejde-984	307	44	.	.	PUNCT
ejde-984	308	1	proof	proof	NOUN
ejde-984	308	2	.	.	PUNCT
ejde-984	309	1	for	for	ADP
ejde-984	309	2	u	u	PROPN
ejde-984	309	3	∈	∈	PROPN
ejde-984	309	4	āk	āk	PROPN
ejde-984	309	5	with	with	ADP
ejde-984	309	6	k	k	PROPN
ejde-984	309	7	small	small	ADJ
ejde-984	309	8	enough	enough	ADV
ejde-984	309	9	,	,	PUNCT
ejde-984	309	10	thanks	thank	NOUN
ejde-984	309	11	to	to	ADP
ejde-984	309	12	the	the	DET
ejde-984	309	13	fractional	fractional	ADJ
ejde-984	309	14	gagliardo	gagliardo	NOUN
ejde-984	309	15	-	-	PUNCT
ejde-984	309	16	nirenberg	nirenberg	NOUN
ejde-984	309	17	inequality	inequality	NOUN
ejde-984	309	18	and	and	CCONJ
ejde-984	309	19	the	the	DET
ejde-984	309	20	fact	fact	NOUN
ejde-984	309	21	that	that	SCONJ
ejde-984	309	22	pδs	pδs	PROPN
ejde-984	309	23	,	,	PUNCT
ejde-984	309	24	p	p	X
ejde-984	309	25	>	>	X
ejde-984	309	26	4	4	NUM
ejde-984	309	27	(	(	PUNCT
ejde-984	309	28	because	because	SCONJ
ejde-984	309	29	of	of	ADP
ejde-984	309	30	p	p	X
ejde-984	309	31	>	>	X
ejde-984	309	32	2	2	NUM
ejde-984	309	33	+	+	CCONJ
ejde-984	309	34	8s	8s	NUM
ejde-984	309	35	3	3	NUM
ejde-984	309	36	)	)	PUNCT
ejde-984	309	37	,	,	PUNCT
ejde-984	309	38	we	we	PRON
ejde-984	309	39	have	have	VERB
ejde-984	309	40	that	that	PRON
ejde-984	309	41	eµ(u	eµ(u	NOUN
ejde-984	309	42	)	)	PUNCT
ejde-984	309	43	=	=	PUNCT
ejde-984	310	1	a	a	DET
ejde-984	310	2	2	2	NUM
ejde-984	310	3	|(−∆	|(−∆	NOUN
ejde-984	310	4	)	)	PUNCT
ejde-984	310	5	s	s	PART
ejde-984	310	6	2u|22	2u|22	NOUN
ejde-984	311	1	+	+	CCONJ
ejde-984	311	2	b	b	SYM
ejde-984	311	3	4	4	NUM
ejde-984	311	4	|(−∆	|(−∆	NOUN
ejde-984	311	5	)	)	PUNCT
ejde-984	311	6	s	s	PART
ejde-984	311	7	2u|42	2u|42	NOUN
ejde-984	312	1	−	−	PROPN
ejde-984	312	2	µ	µ	NOUN
ejde-984	312	3	q	q	NOUN
ejde-984	312	4	|u|qq	|u|qq	NOUN
ejde-984	312	5	−	−	ADP
ejde-984	312	6	1	1	NUM
ejde-984	312	7	p	p	PROPN
ejde-984	312	8	|u|pp	|u|pp	PROPN
ejde-984	312	9	≥	≥	NOUN
ejde-984	312	10	a	a	DET
ejde-984	312	11	2	2	NUM
ejde-984	312	12	|(−∆	|(−∆	NOUN
ejde-984	312	13	)	)	PUNCT
ejde-984	312	14	s	s	PART
ejde-984	312	15	2u|22	2u|22	NOUN
ejde-984	312	16	−	−	NOUN
ejde-984	312	17	1	1	NUM
ejde-984	313	1	p	p	NOUN
ejde-984	313	2	c(s	c(	NOUN
ejde-984	313	3	,	,	PUNCT
ejde-984	313	4	p)pcp(1−δs	p)pcp(1−δs	ADJ
ejde-984	313	5	,	,	PUNCT
ejde-984	313	6	p)|(−∆	p)|(−∆	NUM
ejde-984	313	7	)	)	PUNCT
ejde-984	313	8	s	s	PART
ejde-984	313	9	2u|pδs	2u|pδs	NUM
ejde-984	313	10	,	,	PUNCT
ejde-984	313	11	p2	p2	X
ejde-984	313	12	>	>	X
ejde-984	313	13	0	0	PUNCT
ejde-984	314	1	and	and	CCONJ
ejde-984	314	2	that	that	PRON
ejde-984	314	3	pµ(u	pµ(u	NOUN
ejde-984	314	4	)	)	PUNCT
ejde-984	314	5	=	=	SYM
ejde-984	314	6	a|(−∆	a|(−∆	PROPN
ejde-984	314	7	)	)	PUNCT
ejde-984	314	8	s	s	PART
ejde-984	314	9	2u|22	2u|22	NOUN
ejde-984	314	10	+	+	CCONJ
ejde-984	314	11	b|(−∆	b|(−∆	NOUN
ejde-984	314	12	)	)	PUNCT
ejde-984	314	13	s	s	PART
ejde-984	314	14	2u|42	2u|42	NOUN
ejde-984	314	15	−	−	NOUN
ejde-984	314	16	µδs	µδ	NOUN
ejde-984	314	17	,	,	PUNCT
ejde-984	314	18	q|u|qq	q|u|qq	NOUN
ejde-984	314	19	−	−	NOUN
ejde-984	314	20	δs	δs	NOUN
ejde-984	314	21	,	,	PUNCT
ejde-984	314	22	p|u|pp	p|u|pp	PROPN
ejde-984	314	23	≥	≥	NUM
ejde-984	314	24	a|(−∆	a|(−∆	PROPN
ejde-984	314	25	)	)	PUNCT
ejde-984	314	26	s	s	PART
ejde-984	314	27	2u|22	2u|22	NOUN
ejde-984	314	28	−	−	ADP
ejde-984	314	29	c(s	c(	NOUN
ejde-984	314	30	,	,	PUNCT
ejde-984	314	31	p)pcp(1−δs	p)pcp(1−δs	ADJ
ejde-984	314	32	,	,	PUNCT
ejde-984	314	33	p)|(−∆	p)|(−∆	NUM
ejde-984	314	34	)	)	PUNCT
ejde-984	314	35	s	s	PART
ejde-984	314	36	2u|pδs	2u|pδs	NUM
ejde-984	314	37	,	,	PUNCT
ejde-984	314	38	p2	p2	PROPN
ejde-984	314	39	δs	δs	NOUN
ejde-984	314	40	,	,	PUNCT
ejde-984	314	41	p	p	X
ejde-984	314	42	>	>	X
ejde-984	314	43	0	0	NUM
ejde-984	314	44	.	.	PROPN
ejde-984	314	45	12	12	NUM
ejde-984	314	46	z.	z.	PROPN
ejde-984	314	47	guo	guo	PROPN
ejde-984	314	48	,	,	PUNCT
ejde-984	314	49	t.	t.	PROPN
ejde-984	314	50	zhang	zhang	PROPN
ejde-984	315	1	ejde-2025/75	ejde-2025/75	ADJ
ejde-984	315	2	then	then	ADV
ejde-984	315	3	,	,	PUNCT
ejde-984	315	4	if	if	SCONJ
ejde-984	315	5	u	u	PROPN
ejde-984	315	6	∈	∈	PROPN
ejde-984	315	7	āk	āk	PROPN
ejde-984	315	8	,	,	PUNCT
ejde-984	315	9	we	we	PRON
ejde-984	315	10	have	have	VERB
ejde-984	315	11	that	that	PRON
ejde-984	315	12	supāk	supāk	ADP
ejde-984	315	13	eµ(u	eµ(u	NOUN
ejde-984	315	14	)	)	PUNCT
ejde-984	315	15	≥	≥	NOUN
ejde-984	315	16	eµ(u	eµ(u	NOUN
ejde-984	315	17	)	)	PUNCT
ejde-984	315	18	>	>	X
ejde-984	315	19	0	0	PUNCT
ejde-984	315	20	and	and	CCONJ
ejde-984	315	21	pµ(u	pµ(u	NOUN
ejde-984	315	22	)	)	PUNCT
ejde-984	315	23	>	>	X
ejde-984	316	1	0	0	X
ejde-984	316	2	.	.	PUNCT
ejde-984	317	1	now	now	ADV
ejde-984	317	2	,	,	PUNCT
ejde-984	317	3	replacing	replace	VERB
ejde-984	317	4	k	k	PROPN
ejde-984	317	5	with	with	ADP
ejde-984	317	6	a	a	DET
ejde-984	317	7	smaller	small	ADJ
ejde-984	317	8	quantity	quantity	NOUN
ejde-984	317	9	,	,	PUNCT
ejde-984	317	10	recalling	recall	VERB
ejde-984	317	11	that	that	SCONJ
ejde-984	317	12	m(c	m(c	PROPN
ejde-984	317	13	,	,	PUNCT
ejde-984	317	14	µ	µ	NOUN
ejde-984	317	15	)	)	PUNCT
ejde-984	317	16	>	>	X
ejde-984	317	17	0	0	PUNCT
ejde-984	317	18	by	by	ADP
ejde-984	317	19	lemma	lemma	PROPN
ejde-984	317	20	3.6	3.6	NUM
ejde-984	317	21	and	and	CCONJ
ejde-984	317	22	applying	apply	VERB
ejde-984	317	23	fractional	fractional	ADJ
ejde-984	317	24	gagliardonirenberg	gagliardonirenberg	NOUN
ejde-984	317	25	inequality	inequality	NOUN
ejde-984	317	26	,	,	PUNCT
ejde-984	317	27	we	we	PRON
ejde-984	317	28	conclude	conclude	VERB
ejde-984	317	29	that	that	PRON
ejde-984	317	30	eµ(u	eµ(u	NOUN
ejde-984	317	31	)	)	PUNCT
ejde-984	317	32	≤	≤	NOUN
ejde-984	317	33	a	a	DET
ejde-984	317	34	2	2	NUM
ejde-984	317	35	|(−∆	|(−∆	NOUN
ejde-984	317	36	)	)	PUNCT
ejde-984	317	37	s	s	PART
ejde-984	317	38	2u|22	2u|22	NOUN
ejde-984	318	1	+	+	CCONJ
ejde-984	318	2	b	b	SYM
ejde-984	318	3	4	4	NUM
ejde-984	318	4	|(−∆	|(−∆	NOUN
ejde-984	318	5	)	)	PUNCT
ejde-984	318	6	s	s	PART
ejde-984	318	7	2u|42	2u|42	NOUN
ejde-984	319	1	+	+	CCONJ
ejde-984	319	2	|µ|	|µ|	PROPN
ejde-984	319	3	q	q	NOUN
ejde-984	319	4	c(s	c(	NOUN
ejde-984	319	5	,	,	PUNCT
ejde-984	319	6	q)qcq(1−δs	q)qcq(1−δs	NOUN
ejde-984	319	7	,	,	PUNCT
ejde-984	319	8	q)|(−∆	q)|(−∆	NOUN
ejde-984	319	9	)	)	PUNCT
ejde-984	319	10	s	s	VERB
ejde-984	319	11	2u|qδs	2u|qδs	NUM
ejde-984	319	12	,	,	PUNCT
ejde-984	319	13	q2	q2	NOUN
ejde-984	319	14	<	<	X
ejde-984	319	15	m(c	m(c	PROPN
ejde-984	319	16	,	,	PUNCT
ejde-984	319	17	µ	µ	NOUN
ejde-984	319	18	)	)	PUNCT
ejde-984	319	19	.	.	PUNCT
ejde-984	320	1	thus	thus	ADV
ejde-984	320	2	supāk	supāk	X
ejde-984	320	3	eµ(u	eµ(u	NOUN
ejde-984	320	4	)	)	PUNCT
ejde-984	320	5	<	<	X
ejde-984	321	1	m(c	m(c	PROPN
ejde-984	321	2	,	,	PUNCT
ejde-984	321	3	µ	µ	NOUN
ejde-984	321	4	)	)	PUNCT
ejde-984	321	5	.	.	PUNCT
ejde-984	322	1	□	□	PUNCT
ejde-984	322	2	now	now	ADV
ejde-984	322	3	,	,	PUNCT
ejde-984	322	4	we	we	PRON
ejde-984	322	5	define	define	VERB
ejde-984	322	6	ec	ec	PROPN
ejde-984	322	7	µ	µ	X
ejde-984	322	8	=	=	PUNCT
ejde-984	322	9	{	{	PUNCT
ejde-984	322	10	u	u	NOUN
ejde-984	322	11	∈	∈	PROPN
ejde-984	322	12	sc	sc	PROPN
ejde-984	322	13	:	:	PUNCT
ejde-984	322	14	eµ(u	eµ(u	NOUN
ejde-984	322	15	)	)	PUNCT
ejde-984	322	16	≤	≤	NUM
ejde-984	323	1	c	c	X
ejde-984	323	2	}	}	PUNCT
ejde-984	323	3	and	and	CCONJ
ejde-984	323	4	the	the	DET
ejde-984	323	5	minimax	minimax	NOUN
ejde-984	323	6	class	class	NOUN
ejde-984	323	7	τ̃	τ̃	PROPN
ejde-984	323	8	:	:	PUNCT
ejde-984	323	9	=	=	SYM
ejde-984	323	10	{	{	PUNCT
ejde-984	323	11	γ	γ	X
ejde-984	323	12	=	=	SYM
ejde-984	323	13	(	(	PUNCT
ejde-984	323	14	α	α	X
ejde-984	323	15	,	,	PUNCT
ejde-984	323	16	β	β	NOUN
ejde-984	323	17	)	)	PUNCT
ejde-984	323	18	∈	∈	PROPN
ejde-984	323	19	c([0	c([0	PROPN
ejde-984	323	20	,	,	PUNCT
ejde-984	323	21	1],r×	1],r×	NUM
ejde-984	323	22	sr	sr	PROPN
ejde-984	323	23	c	c	PROPN
ejde-984	323	24	)	)	PUNCT
ejde-984	323	25	:	:	PUNCT
ejde-984	323	26	γ(0	γ(0	PROPN
ejde-984	323	27	)	)	PUNCT
ejde-984	323	28	∈	∈	PROPN
ejde-984	323	29	(	(	PUNCT
ejde-984	323	30	0	0	NUM
ejde-984	323	31	,	,	PUNCT
ejde-984	323	32	āk	āk	PROPN
ejde-984	323	33	)	)	PUNCT
ejde-984	323	34	and	and	CCONJ
ejde-984	323	35	γ(1	γ(1	PROPN
ejde-984	323	36	)	)	PUNCT
ejde-984	323	37	∈	∈	PROPN
ejde-984	323	38	(	(	PUNCT
ejde-984	323	39	0	0	NUM
ejde-984	323	40	,	,	PUNCT
ejde-984	323	41	e0	e0	PROPN
ejde-984	323	42	µ	µ	NUM
ejde-984	323	43	)	)	PUNCT
ejde-984	323	44	}	}	PUNCT
ejde-984	323	45	,	,	PUNCT
ejde-984	323	46	where	where	SCONJ
ejde-984	323	47	sr	sr	PROPN
ejde-984	323	48	c	c	PROPN
ejde-984	323	49	=	=	SYM
ejde-984	323	50	sc	sc	PROPN
ejde-984	323	51	∩hs	∩hs	ADJ
ejde-984	323	52	r	r	NOUN
ejde-984	323	53	.	.	PUNCT
ejde-984	324	1	we	we	PRON
ejde-984	324	2	define	define	VERB
ejde-984	324	3	the	the	DET
ejde-984	324	4	minimax	minimax	NOUN
ejde-984	324	5	level	level	NOUN
ejde-984	324	6	as	as	SCONJ
ejde-984	324	7	follows	follow	VERB
ejde-984	324	8	:	:	PUNCT
ejde-984	324	9	σ(c	σ(c	PROPN
ejde-984	324	10	,	,	PUNCT
ejde-984	324	11	µ	µ	NOUN
ejde-984	324	12	)	)	PUNCT
ejde-984	324	13	=	=	SYM
ejde-984	324	14	inf	inf	PROPN
ejde-984	324	15	γ∈τ̃	γ∈τ̃	PROPN
ejde-984	324	16	max	max	PROPN
ejde-984	324	17	(	(	PUNCT
ejde-984	324	18	τ	τ	PROPN
ejde-984	324	19	,	,	PUNCT
ejde-984	324	20	u)∈γ([0,1	u)∈γ([0,1	NOUN
ejde-984	324	21	]	]	PUNCT
ejde-984	324	22	)	)	PUNCT
ejde-984	324	23	ẽµ(τ	ẽµ(τ	PROPN
ejde-984	324	24	,	,	PUNCT
ejde-984	324	25	u	u	NOUN
ejde-984	324	26	)	)	PUNCT
ejde-984	324	27	,	,	PUNCT
ejde-984	324	28	where	where	SCONJ
ejde-984	324	29	ẽµ(τ	ẽµ(τ	NOUN
ejde-984	324	30	,	,	PUNCT
ejde-984	324	31	u	u	NOUN
ejde-984	324	32	)	)	PUNCT
ejde-984	324	33	=	=	SYM
ejde-984	324	34	eµ(τ	eµ(τ	NOUN
ejde-984	324	35	∗	∗	X
ejde-984	324	36	u	u	NOUN
ejde-984	324	37	)	)	PUNCT
ejde-984	324	38	=	=	SYM
ejde-984	325	1	jµ	jµ	PROPN
ejde-984	325	2	u	u	NOUN
ejde-984	325	3	(	(	PUNCT
ejde-984	325	4	τ	τ	X
ejde-984	325	5	)	)	PUNCT
ejde-984	325	6	=	=	NOUN
ejde-984	325	7	ae2sτ	ae2sτ	NUM
ejde-984	325	8	2	2	NUM
ejde-984	325	9	|(−∆)s/2u|22	|(−∆)s/2u|22	NOUN
ejde-984	325	10	+	+	CCONJ
ejde-984	325	11	be4sτ	be4sτ	SYM
ejde-984	325	12	4	4	NUM
ejde-984	325	13	|(−∆)s/2u|42	|(−∆)s/2u|42	NUM
ejde-984	325	14	−	−	PROPN
ejde-984	325	15	µ	µ	DET
ejde-984	325	16	eqδs	eqδs	NOUN
ejde-984	325	17	,	,	PUNCT
ejde-984	325	18	qsτ	qsτ	NOUN
ejde-984	325	19	q	q	NOUN
ejde-984	325	20	|u|qq	|u|qq	NOUN
ejde-984	325	21	−	−	NOUN
ejde-984	325	22	epδs	epδs	ADJ
ejde-984	325	23	,	,	PUNCT
ejde-984	325	24	psτ	psτ	VERB
ejde-984	325	25	p	p	PROPN
ejde-984	325	26	|u|pp	|u|pp	NOUN
ejde-984	325	27	.	.	PUNCT
ejde-984	326	1	proof	proof	NOUN
ejde-984	326	2	of	of	ADP
ejde-984	326	3	theorem	theorem	ADJ
ejde-984	326	4	2.8	2.8	NUM
ejde-984	326	5	.	.	PUNCT
ejde-984	327	1	assume	assume	VERB
ejde-984	327	2	by	by	ADP
ejde-984	327	3	contradiction	contradiction	NOUN
ejde-984	327	4	that	that	SCONJ
ejde-984	327	5	if	if	SCONJ
ejde-984	327	6	there	there	PRON
ejde-984	327	7	exists	exist	VERB
ejde-984	327	8	a	a	DET
ejde-984	327	9	solution	solution	NOUN
ejde-984	327	10	u	u	NOUN
ejde-984	327	11	to	to	ADP
ejde-984	327	12	(	(	PUNCT
ejde-984	327	13	1.1)-(1.2	1.1)-(1.2	NUM
ejde-984	327	14	)	)	PUNCT
ejde-984	327	15	,	,	PUNCT
ejde-984	327	16	then	then	ADV
ejde-984	327	17	by	by	ADP
ejde-984	327	18	the	the	DET
ejde-984	327	19	pohožaev	pohožaev	PROPN
ejde-984	327	20	identity	identity	NOUN
ejde-984	327	21	pµ(u	pµ(u	NOUN
ejde-984	327	22	)	)	PUNCT
ejde-984	327	23	=	=	SYM
ejde-984	328	1	0	0	NUM
ejde-984	328	2	,	,	PUNCT
ejde-984	328	3	we	we	PRON
ejde-984	328	4	obtain	obtain	VERB
ejde-984	328	5	that	that	PRON
ejde-984	328	6	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	328	7	+	+	NUM
ejde-984	328	8	b|(−∆)s/2u|42	b|(−∆)s/2u|42	NOUN
ejde-984	328	9	=	=	SYM
ejde-984	328	10	µδs	µδ	NOUN
ejde-984	328	11	,	,	PUNCT
ejde-984	328	12	q|u|qq	q|u|qq	NOUN
ejde-984	328	13	+	+	CCONJ
ejde-984	328	14	4	4	NUM
ejde-984	328	15	p̄	p̄	NOUN
ejde-984	328	16	|u|p̄p̄	|u|p̄p̄	NOUN
ejde-984	328	17	,	,	PUNCT
ejde-984	328	18	where	where	SCONJ
ejde-984	328	19	δs	δs	NOUN
ejde-984	328	20	,	,	PUNCT
ejde-984	328	21	p̄	p̄	NOUN
ejde-984	328	22	=	=	NOUN
ejde-984	328	23	4	4	NUM
ejde-984	328	24	p̄	p̄	ADJ
ejde-984	328	25	when	when	SCONJ
ejde-984	328	26	p	p	PROPN
ejde-984	328	27	=	=	SYM
ejde-984	328	28	2	2	NUM
ejde-984	328	29	+	+	NUM
ejde-984	328	30	8s	8s	NUM
ejde-984	328	31	3	3	NUM
ejde-984	328	32	=	=	SYM
ejde-984	328	33	p̄.	p̄.	VERB
ejde-984	328	34	since	since	SCONJ
ejde-984	328	35	b	b	PROPN
ejde-984	328	36	≥	≥	NOUN
ejde-984	328	37	4	4	NUM
ejde-984	328	38	p̄c(s	p̄c(s	ADJ
ejde-984	328	39	,	,	PUNCT
ejde-984	328	40	p̄	p̄	NUM
ejde-984	328	41	)	)	PUNCT
ejde-984	328	42	p̄cp̄−4	p̄cp̄−4	PROPN
ejde-984	328	43	,	,	PUNCT
ejde-984	328	44	by	by	ADP
ejde-984	328	45	the	the	DET
ejde-984	328	46	fractional	fractional	ADJ
ejde-984	328	47	gagliardonirenberg	gagliardonirenberg	NOUN
ejde-984	328	48	inequality	inequality	NOUN
ejde-984	328	49	,	,	PUNCT
ejde-984	328	50	we	we	PRON
ejde-984	328	51	have	have	VERB
ejde-984	328	52	that	that	PRON
ejde-984	328	53	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	329	1	+	+	NUM
ejde-984	329	2	b|(−∆)s/2u|42	b|(−∆)s/2u|42	NOUN
ejde-984	329	3	−	−	NOUN
ejde-984	329	4	4	4	NUM
ejde-984	329	5	p̄	p̄	NOUN
ejde-984	329	6	|u|p̄p̄	|u|p̄p̄	PROPN
ejde-984	329	7	≥	≥	PRON
ejde-984	329	8	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	330	1	+	+	CCONJ
ejde-984	330	2	b|(−∆)s/2u|42	b|(−∆)s/2u|42	NOUN
ejde-984	330	3	−	−	NOUN
ejde-984	330	4	4	4	NUM
ejde-984	330	5	p̄	p̄	NOUN
ejde-984	330	6	c(s	c(	NOUN
ejde-984	330	7	,	,	PUNCT
ejde-984	330	8	p)p̄cp̄−4|(−∆)s/2u|42	p)p̄cp̄−4|(−∆)s/2u|42	NOUN
ejde-984	330	9	≥	≥	NUM
ejde-984	330	10	0	0	NUM
ejde-984	330	11	,	,	PUNCT
ejde-984	330	12	and	and	CCONJ
ejde-984	330	13	hence	hence	ADV
ejde-984	330	14	we	we	PRON
ejde-984	330	15	deduce	deduce	VERB
ejde-984	330	16	that	that	SCONJ
ejde-984	331	1	0	0	NUM
ejde-984	331	2	>	>	SYM
ejde-984	331	3	µδs	µδ	NOUN
ejde-984	331	4	,	,	PUNCT
ejde-984	331	5	q|u|qq	q|u|qq	NOUN
ejde-984	331	6	=	=	SYM
ejde-984	331	7	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	331	8	+	+	NUM
ejde-984	331	9	b|(−∆)s/2u|42	b|(−∆)s/2u|42	NOUN
ejde-984	331	10	−	−	NOUN
ejde-984	331	11	4	4	NUM
ejde-984	331	12	p̄	p̄	NOUN
ejde-984	331	13	|u|p̄p̄	|u|p̄p̄	PROPN
ejde-984	331	14	≥	≥	NUM
ejde-984	331	15	0	0	NUM
ejde-984	331	16	,	,	PUNCT
ejde-984	331	17	which	which	PRON
ejde-984	331	18	is	be	AUX
ejde-984	331	19	a	a	DET
ejde-984	331	20	contradiction	contradiction	NOUN
ejde-984	331	21	.	.	PUNCT
ejde-984	332	1	□	□	PUNCT
ejde-984	332	2	proof	proof	NOUN
ejde-984	332	3	of	of	ADP
ejde-984	332	4	theorem	theorem	NOUN
ejde-984	332	5	2.9	2.9	NUM
ejde-984	332	6	.	.	PUNCT
ejde-984	333	1	through	through	ADP
ejde-984	333	2	simple	simple	ADJ
ejde-984	333	3	analysis	analysis	NOUN
ejde-984	333	4	we	we	PRON
ejde-984	333	5	can	can	AUX
ejde-984	333	6	see	see	VERB
ejde-984	333	7	that	that	SCONJ
ejde-984	333	8	it	it	PRON
ejde-984	333	9	is	be	AUX
ejde-984	333	10	suffice	suffice	NOUN
ejde-984	333	11	to	to	PART
ejde-984	333	12	prove	prove	VERB
ejde-984	333	13	that	that	SCONJ
ejde-984	333	14	there	there	PRON
ejde-984	333	15	exists	exist	VERB
ejde-984	333	16	a	a	DET
ejde-984	333	17	ps	ps	NOUN
ejde-984	333	18	sequence	sequence	NOUN
ejde-984	333	19	such	such	ADJ
ejde-984	333	20	that	that	SCONJ
ejde-984	333	21	conditions	condition	NOUN
ejde-984	333	22	(	(	PUNCT
ejde-984	333	23	i	i	NOUN
ejde-984	333	24	)	)	PUNCT
ejde-984	333	25	and	and	CCONJ
ejde-984	333	26	(	(	PUNCT
ejde-984	333	27	ii	ii	NOUN
ejde-984	333	28	)	)	PUNCT
ejde-984	333	29	of	of	ADP
ejde-984	333	30	lemma	lemma	PROPN
ejde-984	333	31	3.2	3.2	NUM
ejde-984	333	32	hold	hold	NOUN
ejde-984	333	33	.	.	PUNCT
ejde-984	334	1	by	by	ADP
ejde-984	334	2	lim	lim	PROPN
ejde-984	334	3	τ→−∞	τ→−∞	PROPN
ejde-984	335	1	jµ	jµ	PROPN
ejde-984	335	2	u	u	NOUN
ejde-984	335	3	(	(	PUNCT
ejde-984	335	4	τ	τ	X
ejde-984	335	5	)	)	PUNCT
ejde-984	335	6	=	=	PUNCT
ejde-984	335	7	0	0	PUNCT
ejde-984	335	8	+	+	NUM
ejde-984	335	9	and	and	CCONJ
ejde-984	335	10	lim	lim	PROPN
ejde-984	335	11	τ→+∞	τ→+∞	PROPN
ejde-984	335	12	jµ	jµ	PROPN
ejde-984	335	13	u	u	NOUN
ejde-984	335	14	(	(	PUNCT
ejde-984	335	15	τ	τ	X
ejde-984	335	16	)	)	PUNCT
ejde-984	335	17	=	=	SYM
ejde-984	335	18	−∞	−∞	NOUN
ejde-984	335	19	,	,	PUNCT
ejde-984	335	20	it	it	PRON
ejde-984	335	21	follows	follow	VERB
ejde-984	335	22	that	that	SCONJ
ejde-984	335	23	there	there	PRON
ejde-984	335	24	exist	exist	VERB
ejde-984	335	25	τ1	τ1	NOUN
ejde-984	335	26	and	and	CCONJ
ejde-984	335	27	τ2	τ2	NOUN
ejde-984	335	28	∈	∈	NOUN
ejde-984	335	29	r	r	NOUN
ejde-984	335	30	satisfying	satisfy	VERB
ejde-984	335	31	jµ	jµ	PROPN
ejde-984	335	32	u	u	NOUN
ejde-984	335	33	(	(	PUNCT
ejde-984	335	34	τ	τ	X
ejde-984	335	35	)	)	PUNCT
ejde-984	335	36	≤	≤	NOUN
ejde-984	335	37	k	k	NOUN
ejde-984	335	38	for	for	ADP
ejde-984	335	39	all	all	PRON
ejde-984	335	40	τ	τ	X
ejde-984	335	41	<	<	X
ejde-984	335	42	τ1	τ1	PROPN
ejde-984	335	43	(	(	PUNCT
ejde-984	335	44	k	k	X
ejde-984	335	45	>	>	X
ejde-984	335	46	0	0	NUM
ejde-984	335	47	)	)	PUNCT
ejde-984	335	48	,	,	PUNCT
ejde-984	335	49	jµ	jµ	PROPN
ejde-984	335	50	u	u	NOUN
ejde-984	335	51	(	(	PUNCT
ejde-984	335	52	τ	τ	X
ejde-984	335	53	)	)	PUNCT
ejde-984	335	54	≤	≤	NOUN
ejde-984	335	55	0	0	NUM
ejde-984	335	56	for	for	ADP
ejde-984	335	57	all	all	DET
ejde-984	335	58	τ	τ	PROPN
ejde-984	335	59	>	>	X
ejde-984	335	60	τ2	τ2	PROPN
ejde-984	335	61	.	.	PUNCT
ejde-984	336	1	next	next	ADV
ejde-984	336	2	,	,	PUNCT
ejde-984	336	3	we	we	PRON
ejde-984	336	4	define	define	VERB
ejde-984	336	5	γu	γu	INTJ
ejde-984	336	6	:	:	PUNCT
ejde-984	337	1	[	[	X
ejde-984	337	2	0	0	NUM
ejde-984	337	3	,	,	PUNCT
ejde-984	337	4	1	1	NUM
ejde-984	337	5	]	]	PUNCT
ejde-984	337	6	→	→	PUNCT
ejde-984	337	7	r×	r×	NOUN
ejde-984	337	8	sr	sr	PROPN
ejde-984	337	9	c	c	PROPN
ejde-984	337	10	by	by	ADP
ejde-984	337	11	γu(p	γu(p	NOUN
ejde-984	337	12	)	)	PUNCT
ejde-984	337	13	=	=	SYM
ejde-984	337	14	(	(	PUNCT
ejde-984	337	15	0	0	NUM
ejde-984	337	16	,	,	PUNCT
ejde-984	337	17	(	(	PUNCT
ejde-984	337	18	(	(	PUNCT
ejde-984	337	19	1−	1−	NUM
ejde-984	337	20	p)τ1	p)τ1	ADJ
ejde-984	337	21	+	+	CCONJ
ejde-984	337	22	pτ2	pτ2	NOUN
ejde-984	337	23	)	)	PUNCT
ejde-984	337	24	∗	∗	NOUN
ejde-984	337	25	u	u	NOUN
ejde-984	337	26	)	)	PUNCT
ejde-984	337	27	,	,	PUNCT
ejde-984	337	28	(	(	PUNCT
ejde-984	337	29	3.28	3.28	NUM
ejde-984	337	30	)	)	PUNCT
ejde-984	337	31	which	which	PRON
ejde-984	337	32	is	be	AUX
ejde-984	337	33	a	a	DET
ejde-984	337	34	path	path	NOUN
ejde-984	337	35	in	in	ADP
ejde-984	337	36	τ̃	τ̃	PROPN
ejde-984	337	37	,	,	PUNCT
ejde-984	337	38	therefore	therefore	ADV
ejde-984	337	39	σ(c	σ(c	PROPN
ejde-984	337	40	,	,	PUNCT
ejde-984	337	41	µ	µ	NOUN
ejde-984	337	42	)	)	PUNCT
ejde-984	337	43	is	be	AUX
ejde-984	337	44	a	a	DET
ejde-984	337	45	real	real	ADJ
ejde-984	337	46	number	number	NOUN
ejde-984	337	47	.	.	PUNCT
ejde-984	338	1	now	now	ADV
ejde-984	338	2	,	,	PUNCT
ejde-984	338	3	we	we	PRON
ejde-984	338	4	aim	aim	VERB
ejde-984	338	5	to	to	PART
ejde-984	338	6	prove	prove	VERB
ejde-984	338	7	the	the	DET
ejde-984	338	8	claim	claim	NOUN
ejde-984	338	9	that	that	SCONJ
ejde-984	338	10	for	for	ADP
ejde-984	338	11	all	all	DET
ejde-984	338	12	γ	γ	PROPN
ejde-984	338	13	∈	∈	PROPN
ejde-984	338	14	τ̃	τ̃	PROPN
ejde-984	338	15	,	,	PUNCT
ejde-984	338	16	there	there	PRON
ejde-984	338	17	exists	exist	VERB
ejde-984	338	18	τγ	τγ	ADP
ejde-984	338	19	∈	∈	PROPN
ejde-984	338	20	(	(	PUNCT
ejde-984	338	21	0	0	NUM
ejde-984	338	22	,	,	PUNCT
ejde-984	338	23	1	1	NUM
ejde-984	338	24	)	)	PUNCT
ejde-984	338	25	such	such	ADJ
ejde-984	338	26	that	that	SCONJ
ejde-984	338	27	α(τγ	α(τγ	NUM
ejde-984	338	28	)	)	PUNCT
ejde-984	338	29	∗	∗	NOUN
ejde-984	338	30	β(τγ	β(τγ	NUM
ejde-984	338	31	)	)	PUNCT
ejde-984	338	32	∈	∈	PROPN
ejde-984	338	33	p−	p−	NOUN
ejde-984	338	34	c,µ.	c,µ.	NOUN
ejde-984	338	35	(	(	PUNCT
ejde-984	338	36	3.29	3.29	NUM
ejde-984	338	37	)	)	PUNCT
ejde-984	338	38	actually	actually	ADV
ejde-984	338	39	,	,	PUNCT
ejde-984	338	40	γ(0	γ(0	PROPN
ejde-984	338	41	)	)	PUNCT
ejde-984	338	42	=	=	PUNCT
ejde-984	339	1	(	(	PUNCT
ejde-984	339	2	0	0	NUM
ejde-984	339	3	,	,	PUNCT
ejde-984	339	4	β(0	β(0	NOUN
ejde-984	339	5	)	)	PUNCT
ejde-984	339	6	)	)	PUNCT
ejde-984	340	1	∈	∈	PROPN
ejde-984	340	2	(	(	PUNCT
ejde-984	340	3	0	0	NUM
ejde-984	340	4	,	,	PUNCT
ejde-984	340	5	āk	āk	PROPN
ejde-984	340	6	)	)	PUNCT
ejde-984	340	7	and	and	CCONJ
ejde-984	340	8	spµ(t	spµ(t	NOUN
ejde-984	340	9	∗	∗	NOUN
ejde-984	340	10	u	u	NOUN
ejde-984	340	11	)	)	PUNCT
ejde-984	340	12	=	=	SYM
ejde-984	340	13	(	(	PUNCT
ejde-984	340	14	jµ	jµ	PROPN
ejde-984	340	15	u	u	NOUN
ejde-984	340	16	)	)	PUNCT
ejde-984	340	17	′(t	′(t	PROPN
ejde-984	340	18	)	)	PUNCT
ejde-984	340	19	.	.	PUNCT
ejde-984	341	1	according	accord	VERB
ejde-984	341	2	to	to	ADP
ejde-984	341	3	lemma	lemma	PROPN
ejde-984	341	4	3.7	3.7	NUM
ejde-984	341	5	,	,	PUNCT
ejde-984	341	6	we	we	PRON
ejde-984	341	7	obtain	obtain	VERB
ejde-984	341	8	pµ(β(0	pµ(β(0	NOUN
ejde-984	341	9	)	)	PUNCT
ejde-984	341	10	)	)	PUNCT
ejde-984	342	1	=	=	PUNCT
ejde-984	342	2	1	1	NUM
ejde-984	342	3	s	s	X
ejde-984	342	4	(	(	PUNCT
ejde-984	342	5	j	j	PROPN
ejde-984	342	6	µ	µ	X
ejde-984	342	7	u	u	NOUN
ejde-984	342	8	)	)	PUNCT
ejde-984	342	9	′(0	′(0	PROPN
ejde-984	342	10	)	)	PUNCT
ejde-984	342	11	>	>	X
ejde-984	342	12	0	0	NUM
ejde-984	342	13	,	,	PUNCT
ejde-984	342	14	further	far	ADV
ejde-984	342	15	,	,	PUNCT
ejde-984	342	16	by	by	ADP
ejde-984	342	17	lemma	lemma	PROPN
ejde-984	342	18	3.5	3.5	NUM
ejde-984	342	19	,	,	PUNCT
ejde-984	342	20	we	we	PRON
ejde-984	342	21	obtain	obtain	VERB
ejde-984	342	22	t0∗β(0	t0∗β(0	PROPN
ejde-984	342	23	)	)	PUNCT
ejde-984	342	24	=	=	SYM
ejde-984	342	25	tβ(0	tβ(0	PROPN
ejde-984	342	26	)	)	PUNCT
ejde-984	342	27	>	>	X
ejde-984	342	28	0	0	X
ejde-984	342	29	.	.	PUNCT
ejde-984	343	1	in	in	ADP
ejde-984	343	2	addition	addition	NOUN
ejde-984	343	3	,	,	PUNCT
ejde-984	343	4	by	by	ADP
ejde-984	343	5	eµ(β(1	eµ(β(1	NOUN
ejde-984	343	6	)	)	PUNCT
ejde-984	343	7	)	)	PUNCT
ejde-984	343	8	=	=	SYM
ejde-984	343	9	ẽµ(γ(1	ẽµ(γ(1	PROPN
ejde-984	343	10	)	)	PUNCT
ejde-984	343	11	)	)	PUNCT
ejde-984	343	12	≤	≤	ADV
ejde-984	343	13	0	0	NUM
ejde-984	343	14	,	,	PUNCT
ejde-984	343	15	we	we	PRON
ejde-984	343	16	deduce	deduce	VERB
ejde-984	343	17	that	that	SCONJ
ejde-984	343	18	tα(1)∗β(1	tα(1)∗β(1	X
ejde-984	343	19	)	)	PUNCT
ejde-984	343	20	=	=	PUNCT
ejde-984	343	21	tβ(1	tβ(1	NOUN
ejde-984	343	22	)	)	PUNCT
ejde-984	343	23	<	<	X
ejde-984	343	24	0	0	X
ejde-984	343	25	.	.	PUNCT
ejde-984	344	1	indeed	indeed	ADV
ejde-984	344	2	,	,	PUNCT
ejde-984	344	3	jβ(1)(τ	jβ(1)(τ	PROPN
ejde-984	344	4	)	)	PUNCT
ejde-984	344	5	>	>	X
ejde-984	344	6	0	0	PUNCT
ejde-984	344	7	for	for	ADP
ejde-984	344	8	each	each	DET
ejde-984	344	9	τ	τ	X
ejde-984	344	10	∈	∈	PROPN
ejde-984	344	11	(	(	PUNCT
ejde-984	344	12	−∞	−∞	NOUN
ejde-984	344	13	,	,	PUNCT
ejde-984	344	14	tβ(1	tβ(1	NOUN
ejde-984	344	15	)	)	PUNCT
ejde-984	344	16	]	]	PUNCT
ejde-984	344	17	,	,	PUNCT
ejde-984	344	18	and	and	CCONJ
ejde-984	344	19	jβ(1)(0	jβ(1)(0	NUM
ejde-984	344	20	)	)	PUNCT
ejde-984	344	21	=	=	SYM
ejde-984	344	22	eµ(β(1	eµ(β(1	NOUN
ejde-984	344	23	)	)	PUNCT
ejde-984	344	24	)	)	PUNCT
ejde-984	344	25	≤	≤	NUM
ejde-984	344	26	0	0	NUM
ejde-984	344	27	,	,	PUNCT
ejde-984	344	28	it	it	PRON
ejde-984	344	29	is	be	AUX
ejde-984	344	30	necessary	necessary	ADJ
ejde-984	344	31	that	that	SCONJ
ejde-984	344	32	tβ(1	tβ(1	VERB
ejde-984	344	33	)	)	PUNCT
ejde-984	344	34	<	<	X
ejde-984	344	35	0	0	NUM
ejde-984	344	36	.	.	X
ejde-984	344	37	ejde-2025/75	ejde-2025/75	ADJ
ejde-984	344	38	normalized	normalize	VERB
ejde-984	344	39	solutions	solution	NOUN
ejde-984	344	40	of	of	ADP
ejde-984	344	41	fractional	fractional	PROPN
ejde-984	344	42	kirchhoff	kirchhoff	NOUN
ejde-984	344	43	equations	equation	NOUN
ejde-984	344	44	13	13	NUM
ejde-984	344	45	furthermore	furthermore	ADV
ejde-984	344	46	,	,	PUNCT
ejde-984	344	47	lemma	lemma	PROPN
ejde-984	344	48	3.5	3.5	NUM
ejde-984	344	49	implies	imply	VERB
ejde-984	344	50	that	that	SCONJ
ejde-984	344	51	the	the	DET
ejde-984	344	52	function	function	NOUN
ejde-984	344	53	u	u	PROPN
ejde-984	344	54	∈	∈	PROPN
ejde-984	344	55	sc	sc	PROPN
ejde-984	344	56	7→	7→	NUM
ejde-984	344	57	tu	tu	NOUN
ejde-984	344	58	∈	∈	PROPN
ejde-984	344	59	r	r	NOUN
ejde-984	344	60	is	be	AUX
ejde-984	344	61	continuous	continuous	ADJ
ejde-984	344	62	.	.	PUNCT
ejde-984	345	1	then	then	ADV
ejde-984	345	2	there	there	PRON
ejde-984	345	3	exists	exist	VERB
ejde-984	345	4	τγ	τγ	ADP
ejde-984	345	5	∈	∈	PROPN
ejde-984	345	6	(	(	PUNCT
ejde-984	345	7	0	0	NUM
ejde-984	345	8	,	,	PUNCT
ejde-984	345	9	1	1	NUM
ejde-984	345	10	)	)	PUNCT
ejde-984	345	11	such	such	ADJ
ejde-984	345	12	that	that	DET
ejde-984	345	13	tα(τγ)∗β(τγ	tα(τγ)∗β(τγ	NOUN
ejde-984	345	14	)	)	PUNCT
ejde-984	345	15	=	=	SYM
ejde-984	346	1	0	0	X
ejde-984	346	2	.	.	PUNCT
ejde-984	347	1	this	this	PRON
ejde-984	347	2	amounts	amount	VERB
ejde-984	347	3	to	to	ADP
ejde-984	347	4	α(τγ	α(τγ	NUM
ejde-984	347	5	)	)	PUNCT
ejde-984	347	6	∗β(τγ	∗β(τγ	PROPN
ejde-984	347	7	)	)	PUNCT
ejde-984	347	8	=	=	SYM
ejde-984	347	9	tα(τγ)∗β(τγ	tα(τγ)∗β(τγ	NOUN
ejde-984	347	10	)	)	PUNCT
ejde-984	347	11	∗	∗	NOUN
ejde-984	347	12	(	(	PUNCT
ejde-984	347	13	α(τγ	α(τγ	PROPN
ejde-984	347	14	)	)	PUNCT
ejde-984	347	15	∗	∗	NOUN
ejde-984	347	16	β(τγ	β(τγ	NUM
ejde-984	347	17	)	)	PUNCT
ejde-984	347	18	)	)	PUNCT
ejde-984	348	1	∈	∈	NOUN
ejde-984	348	2	pc,µ	pc,µ	NOUN
ejde-984	349	1	=	=	NOUN
ejde-984	349	2	p−	p−	VERB
ejde-984	349	3	c,µ.	c,µ.	NOUN
ejde-984	349	4	by	by	ADP
ejde-984	349	5	(	(	PUNCT
ejde-984	349	6	3.29	3.29	NUM
ejde-984	349	7	)	)	PUNCT
ejde-984	349	8	,	,	PUNCT
ejde-984	349	9	it	it	PRON
ejde-984	349	10	follows	follow	VERB
ejde-984	349	11	that	that	SCONJ
ejde-984	349	12	max	max	PROPN
ejde-984	349	13	γ([0,1	γ([0,1	PROPN
ejde-984	349	14	]	]	PUNCT
ejde-984	349	15	)	)	PUNCT
ejde-984	349	16	ẽµ	ẽµ	PROPN
ejde-984	349	17	≥	≥	PROPN
ejde-984	349	18	ẽµ(γ(τγ	ẽµ(γ(τγ	NOUN
ejde-984	349	19	)	)	PUNCT
ejde-984	349	20	)	)	PUNCT
ejde-984	350	1	=	=	PUNCT
ejde-984	350	2	eµ(α(τγ	eµ(α(τγ	NOUN
ejde-984	350	3	)	)	PUNCT
ejde-984	350	4	∗	∗	NOUN
ejde-984	350	5	β(τγ	β(τγ	NUM
ejde-984	350	6	)	)	PUNCT
ejde-984	350	7	)	)	PUNCT
ejde-984	351	1	≥	≥	PROPN
ejde-984	351	2	inf	inf	PROPN
ejde-984	351	3	p−	p−	NOUN
ejde-984	351	4	c,µ∩sr	c,µ∩sr	NOUN
ejde-984	351	5	c	c	PROPN
ejde-984	352	1	eµ.	eµ.	PROPN
ejde-984	352	2	then	then	ADV
ejde-984	352	3	we	we	PRON
ejde-984	352	4	obtain	obtain	VERB
ejde-984	352	5	that	that	SCONJ
ejde-984	352	6	σ(c	σ(c	PROPN
ejde-984	352	7	,	,	PUNCT
ejde-984	352	8	µ	µ	NOUN
ejde-984	352	9	)	)	PUNCT
ejde-984	352	10	≥	≥	NOUN
ejde-984	352	11	inf	inf	PROPN
ejde-984	352	12	p−	p−	NOUN
ejde-984	352	13	c,µ∩sr	c,µ∩sr	NOUN
ejde-984	352	14	c	c	PROPN
ejde-984	352	15	eµ.	eµ.	PROPN
ejde-984	352	16	(	(	PUNCT
ejde-984	352	17	3.30	3.30	NUM
ejde-984	352	18	)	)	PUNCT
ejde-984	352	19	besides	besides	SCONJ
ejde-984	352	20	,	,	PUNCT
ejde-984	352	21	taking	take	VERB
ejde-984	352	22	u	u	PRON
ejde-984	352	23	∈	∈	NOUN
ejde-984	352	24	p−	p−	NOUN
ejde-984	352	25	c,µ	c,µ	NOUN
ejde-984	352	26	∩	∩	X
ejde-984	352	27	sr	sr	PROPN
ejde-984	352	28	c	c	PROPN
ejde-984	352	29	and	and	CCONJ
ejde-984	352	30	γu	γu	INTJ
ejde-984	352	31	the	the	DET
ejde-984	352	32	corresponding	corresponding	ADJ
ejde-984	352	33	path	path	NOUN
ejde-984	352	34	defined	define	VERB
ejde-984	352	35	in	in	ADP
ejde-984	352	36	(	(	PUNCT
ejde-984	352	37	3.28	3.28	NUM
ejde-984	352	38	)	)	PUNCT
ejde-984	352	39	,	,	PUNCT
ejde-984	352	40	we	we	PRON
ejde-984	352	41	derive	derive	VERB
ejde-984	352	42	that	that	SCONJ
ejde-984	352	43	eµ(u	eµ(u	NOUN
ejde-984	352	44	)	)	PUNCT
ejde-984	352	45	=	=	SYM
ejde-984	352	46	ẽµ(0	ẽµ(0	PROPN
ejde-984	352	47	,	,	PUNCT
ejde-984	352	48	u	u	NOUN
ejde-984	352	49	)	)	PUNCT
ejde-984	352	50	=	=	SYM
ejde-984	352	51	max	max	PROPN
ejde-984	352	52	γu([0,1	γu([0,1	PROPN
ejde-984	352	53	]	]	X
ejde-984	352	54	)	)	PUNCT
ejde-984	352	55	ẽµ	ẽµ	PROPN
ejde-984	352	56	≥	≥	X
ejde-984	352	57	σ(c	σ(c	PROPN
ejde-984	352	58	,	,	PUNCT
ejde-984	352	59	µ	µ	NOUN
ejde-984	352	60	)	)	PUNCT
ejde-984	352	61	,	,	PUNCT
ejde-984	352	62	then	then	ADV
ejde-984	352	63	we	we	PRON
ejde-984	352	64	obtain	obtain	VERB
ejde-984	352	65	that	that	DET
ejde-984	352	66	inf	inf	NOUN
ejde-984	352	67	p−	p−	NOUN
ejde-984	352	68	c,µ∩sr	c,µ∩sr	NOUN
ejde-984	352	69	c	c	PROPN
ejde-984	352	70	eµ	eµ	NOUN
ejde-984	352	71	≥	≥	NOUN
ejde-984	352	72	σ(c	σ(c	PROPN
ejde-984	352	73	,	,	PUNCT
ejde-984	352	74	µ	µ	NOUN
ejde-984	352	75	)	)	PUNCT
ejde-984	352	76	.	.	PUNCT
ejde-984	353	1	(	(	PUNCT
ejde-984	353	2	3.31	3.31	NUM
ejde-984	353	3	)	)	PUNCT
ejde-984	353	4	formulas	formula	NOUN
ejde-984	353	5	(	(	PUNCT
ejde-984	353	6	3.30	3.30	NUM
ejde-984	353	7	)	)	PUNCT
ejde-984	353	8	and	and	CCONJ
ejde-984	353	9	(	(	PUNCT
ejde-984	353	10	3.31	3.31	NUM
ejde-984	353	11	)	)	PUNCT
ejde-984	353	12	imply	imply	VERB
ejde-984	353	13	that	that	SCONJ
ejde-984	353	14	inf	inf	PROPN
ejde-984	353	15	p−	p−	NOUN
ejde-984	353	16	c,µ∩sr	c,µ∩sr	NOUN
ejde-984	353	17	c	c	PROPN
ejde-984	353	18	eµ	eµ	NOUN
ejde-984	354	1	=	=	SYM
ejde-984	354	2	σ(c	σ(c	PROPN
ejde-984	354	3	,	,	PUNCT
ejde-984	354	4	µ	µ	NOUN
ejde-984	354	5	)	)	PUNCT
ejde-984	354	6	.	.	PUNCT
ejde-984	355	1	(	(	PUNCT
ejde-984	355	2	3.32	3.32	NUM
ejde-984	355	3	)	)	PUNCT
ejde-984	355	4	next	next	ADV
ejde-984	355	5	,	,	PUNCT
ejde-984	355	6	we	we	PRON
ejde-984	355	7	prove	prove	VERB
ejde-984	355	8	a	a	DET
ejde-984	355	9	claim	claim	NOUN
ejde-984	355	10	that	that	SCONJ
ejde-984	355	11	:	:	PUNCT
ejde-984	355	12	inf	inf	PROPN
ejde-984	356	1	pc,µ∩sr	pc,µ∩sr	INTJ
ejde-984	356	2	c	c	PROPN
ejde-984	356	3	eµ	eµ	NOUN
ejde-984	356	4	=	=	ADJ
ejde-984	356	5	inf	inf	PROPN
ejde-984	356	6	pc,µ	pc,µ	NOUN
ejde-984	356	7	eµ.	eµ.	PROPN
ejde-984	356	8	(	(	PUNCT
ejde-984	356	9	3.33	3.33	NUM
ejde-984	356	10	)	)	PUNCT
ejde-984	356	11	this	this	PRON
ejde-984	356	12	is	be	AUX
ejde-984	356	13	equivalent	equivalent	ADJ
ejde-984	356	14	to	to	ADP
ejde-984	356	15	verifying	verify	VERB
ejde-984	356	16	that	that	DET
ejde-984	356	17	infpc,µ	infpc,µ	ADJ
ejde-984	356	18	eµ	eµ	X
ejde-984	356	19	≥	≥	NOUN
ejde-984	356	20	infpc,µ∩sr	infpc,µ∩sr	PUNCT
ejde-984	356	21	c	c	AUX
ejde-984	356	22	eµ.	eµ.	PROPN
ejde-984	356	23	suppose	suppose	VERB
ejde-984	356	24	by	by	ADP
ejde-984	356	25	contradiction	contradiction	NOUN
ejde-984	356	26	that	that	SCONJ
ejde-984	356	27	there	there	PRON
ejde-984	356	28	exists	exist	VERB
ejde-984	356	29	u	u	PROPN
ejde-984	356	30	∈	∈	PROPN
ejde-984	356	31	pc,µ	pc,µ	NOUN
ejde-984	356	32	\	\	PROPN
ejde-984	356	33	sr	sr	PROPN
ejde-984	356	34	c	c	PROPN
ejde-984	356	35	with	with	ADP
ejde-984	356	36	eµ(u	eµ(u	NOUN
ejde-984	356	37	)	)	PUNCT
ejde-984	356	38	<	<	X
ejde-984	357	1	infpc,µ∩sr	infpc,µ∩sr	PUNCT
ejde-984	357	2	c	c	X
ejde-984	357	3	eµ.	eµ.	NOUN
ejde-984	357	4	then	then	ADV
ejde-984	357	5	we	we	PRON
ejde-984	357	6	set	set	VERB
ejde-984	357	7	v	v	NOUN
ejde-984	357	8	:	:	PUNCT
ejde-984	357	9	=	=	SYM
ejde-984	357	10	|u|∗	|u|∗	NOUN
ejde-984	357	11	,	,	PUNCT
ejde-984	357	12	the	the	DET
ejde-984	357	13	symmetric	symmetric	ADJ
ejde-984	357	14	decreasing	decrease	VERB
ejde-984	357	15	rearrangement	rearrangement	NOUN
ejde-984	357	16	of	of	ADP
ejde-984	357	17	the	the	DET
ejde-984	357	18	modulus	modulus	NOUN
ejde-984	357	19	of	of	ADP
ejde-984	357	20	u	u	NOUN
ejde-984	357	21	,	,	PUNCT
ejde-984	357	22	which	which	PRON
ejde-984	357	23	belongs	belong	VERB
ejde-984	357	24	to	to	ADP
ejde-984	357	25	sr	sr	PROPN
ejde-984	357	26	c	c	PROPN
ejde-984	357	27	.	.	PUNCT
ejde-984	358	1	by	by	ADP
ejde-984	358	2	lemma	lemma	PROPN
ejde-984	358	3	2.7	2.7	NUM
ejde-984	358	4	,	,	PUNCT
ejde-984	358	5	we	we	PRON
ejde-984	358	6	obtain	obtain	VERB
ejde-984	358	7	eµ(v	eµ(v	NOUN
ejde-984	358	8	)	)	PUNCT
ejde-984	358	9	≤	≤	NOUN
ejde-984	358	10	eµ(u	eµ(u	NOUN
ejde-984	358	11	)	)	PUNCT
ejde-984	358	12	and	and	CCONJ
ejde-984	358	13	pµ(v	pµ(v	VERB
ejde-984	358	14	)	)	PUNCT
ejde-984	358	15	≤	≤	NOUN
ejde-984	358	16	pµ(u	pµ(u	NOUN
ejde-984	358	17	)	)	PUNCT
ejde-984	358	18	.	.	PUNCT
ejde-984	359	1	from	from	ADP
ejde-984	359	2	u	u	PROPN
ejde-984	359	3	∈	∈	PROPN
ejde-984	359	4	pc,µ	pc,µ	NOUN
ejde-984	359	5	\	\	PROPN
ejde-984	359	6	sr	sr	PROPN
ejde-984	359	7	c	c	PROPN
ejde-984	359	8	,	,	PUNCT
ejde-984	359	9	we	we	PRON
ejde-984	359	10	have	have	VERB
ejde-984	359	11	pµ(u	pµ(u	NOUN
ejde-984	359	12	)	)	PUNCT
ejde-984	359	13	=	=	SYM
ejde-984	360	1	0	0	X
ejde-984	360	2	.	.	PUNCT
ejde-984	361	1	if	if	SCONJ
ejde-984	361	2	pµ(v	pµ(v	VERB
ejde-984	361	3	)	)	PUNCT
ejde-984	361	4	=	=	SYM
ejde-984	361	5	0	0	NUM
ejde-984	361	6	,	,	PUNCT
ejde-984	361	7	we	we	PRON
ejde-984	361	8	immediately	immediately	ADV
ejde-984	361	9	derive	derive	VERB
ejde-984	361	10	a	a	DET
ejde-984	361	11	contradiction	contradiction	NOUN
ejde-984	361	12	,	,	PUNCT
ejde-984	361	13	hence	hence	ADV
ejde-984	361	14	we	we	PRON
ejde-984	361	15	assume	assume	VERB
ejde-984	361	16	that	that	PRON
ejde-984	361	17	pµ(v	pµ(v	VERB
ejde-984	361	18	)	)	PUNCT
ejde-984	361	19	<	<	X
ejde-984	361	20	0	0	X
ejde-984	361	21	.	.	PUNCT
ejde-984	362	1	in	in	ADP
ejde-984	362	2	this	this	DET
ejde-984	362	3	case	case	NOUN
ejde-984	362	4	,	,	PUNCT
ejde-984	362	5	from	from	ADP
ejde-984	362	6	lemma	lemma	PROPN
ejde-984	362	7	3.5	3.5	NUM
ejde-984	362	8	,	,	PUNCT
ejde-984	362	9	we	we	PRON
ejde-984	362	10	know	know	VERB
ejde-984	362	11	that	that	SCONJ
ejde-984	362	12	tv	tv	NOUN
ejde-984	362	13	<	<	X
ejde-984	362	14	0	0	NUM
ejde-984	362	15	.	.	PUNCT
ejde-984	363	1	but	but	CCONJ
ejde-984	363	2	then	then	ADV
ejde-984	363	3	we	we	PRON
ejde-984	363	4	obtain	obtain	VERB
ejde-984	363	5	a	a	DET
ejde-984	363	6	contradiction	contradiction	NOUN
ejde-984	363	7	in	in	ADP
ejde-984	363	8	the	the	DET
ejde-984	363	9	following	following	ADJ
ejde-984	363	10	way	way	NOUN
ejde-984	363	11	eµ(u	eµ(u	NUM
ejde-984	363	12	)	)	PUNCT
ejde-984	363	13	<	<	X
ejde-984	363	14	eµ(tv	eµ(tv	PROPN
ejde-984	363	15	∗	∗	PROPN
ejde-984	363	16	v	v	NOUN
ejde-984	363	17	)	)	PUNCT
ejde-984	363	18	=	=	SYM
ejde-984	363	19	ae2stv	ae2stv	NOUN
ejde-984	363	20	2	2	NUM
ejde-984	363	21	|(−∆)s/2v|22	|(−∆)s/2v|22	NUM
ejde-984	363	22	+	+	CCONJ
ejde-984	363	23	be4stv	be4stv	NOUN
ejde-984	363	24	4	4	NUM
ejde-984	363	25	|(−∆)s/2v|42	|(−∆)s/2v|42	NUM
ejde-984	363	26	−	−	NOUN
ejde-984	363	27	µ	µ	PRON
ejde-984	363	28	q	q	NOUN
ejde-984	363	29	eqδs	eqδs	NOUN
ejde-984	363	30	,	,	PUNCT
ejde-984	363	31	qstv	qstv	NOUN
ejde-984	363	32	|v|qq	|v|qq	NOUN
ejde-984	363	33	−	−	ADP
ejde-984	363	34	1	1	NUM
ejde-984	363	35	pδs	pδs	NOUN
ejde-984	363	36	,	,	PUNCT
ejde-984	363	37	p	p	X
ejde-984	363	38	[	[	PUNCT
ejde-984	363	39	ae2stv	ae2stv	NOUN
ejde-984	363	40	|(−∆)s/2v|22	|(−∆)s/2v|22	PART
ejde-984	363	41	+	+	CCONJ
ejde-984	363	42	be4stv	be4stv	NOUN
ejde-984	363	43	|(−∆)s/2v|42	|(−∆)s/2v|42	NUM
ejde-984	363	44	−	−	NOUN
ejde-984	363	45	µδs	µδ	NOUN
ejde-984	363	46	,	,	PUNCT
ejde-984	363	47	qe	qe	PROPN
ejde-984	363	48	qδs	qδs	PROPN
ejde-984	363	49	,	,	PUNCT
ejde-984	363	50	qstv	qstv	NOUN
ejde-984	363	51	|v|qq	|v|qq	NOUN
ejde-984	363	52	]	]	PUNCT
ejde-984	364	1	=	=	SYM
ejde-984	364	2	a	a	PRON
ejde-984	364	3	(	(	PUNCT
ejde-984	364	4	1	1	NUM
ejde-984	364	5	2	2	NUM
ejde-984	364	6	−	−	NOUN
ejde-984	364	7	1	1	NUM
ejde-984	364	8	pδs	pδs	NOUN
ejde-984	364	9	,	,	PUNCT
ejde-984	364	10	p	p	NOUN
ejde-984	364	11	)	)	PUNCT
ejde-984	364	12	e2stv	e2stv	NOUN
ejde-984	364	13	|(−∆)s/2v|22	|(−∆)s/2v|22	PART
ejde-984	365	1	+	+	CCONJ
ejde-984	365	2	b	b	X
ejde-984	365	3	(	(	PUNCT
ejde-984	365	4	1	1	NUM
ejde-984	365	5	4	4	NUM
ejde-984	365	6	−	−	NUM
ejde-984	365	7	1	1	NUM
ejde-984	365	8	pδs	pδs	NOUN
ejde-984	365	9	,	,	PUNCT
ejde-984	365	10	p	p	NOUN
ejde-984	365	11	)	)	PUNCT
ejde-984	365	12	e4stv	e4stv	NOUN
ejde-984	365	13	|(−∆)s/2v|42	|(−∆)s/2v|42	NUM
ejde-984	365	14	−	−	NOUN
ejde-984	365	15	µ	µ	X
ejde-984	365	16	q	q	X
ejde-984	365	17	(	(	PUNCT
ejde-984	365	18	1−	1−	NUM
ejde-984	365	19	qδs	qδs	NOUN
ejde-984	365	20	,	,	PUNCT
ejde-984	365	21	q	q	PROPN
ejde-984	365	22	pδs	pδs	PROPN
ejde-984	365	23	,	,	PUNCT
ejde-984	365	24	p	p	NOUN
ejde-984	365	25	)	)	PUNCT
ejde-984	365	26	eqδs	eqδs	NOUN
ejde-984	365	27	,	,	PUNCT
ejde-984	365	28	qstv	qstv	NOUN
ejde-984	365	29	|v|qq	|v|qq	NOUN
ejde-984	365	30	≤	≤	NUM
ejde-984	365	31	a	a	DET
ejde-984	365	32	(	(	PUNCT
ejde-984	365	33	1	1	NUM
ejde-984	365	34	2	2	NUM
ejde-984	365	35	−	−	NOUN
ejde-984	365	36	1	1	NUM
ejde-984	365	37	pδs	pδs	NOUN
ejde-984	365	38	,	,	PUNCT
ejde-984	365	39	p	p	NOUN
ejde-984	365	40	)	)	PUNCT
ejde-984	365	41	|(−∆)s/2u|22	|(−∆)s/2u|22	X
ejde-984	366	1	+	+	CCONJ
ejde-984	366	2	b	b	X
ejde-984	366	3	(	(	PUNCT
ejde-984	366	4	1	1	NUM
ejde-984	366	5	4	4	NUM
ejde-984	366	6	−	−	NUM
ejde-984	366	7	1	1	NUM
ejde-984	366	8	pδs	pδs	NOUN
ejde-984	366	9	,	,	PUNCT
ejde-984	366	10	p	p	NOUN
ejde-984	366	11	)	)	PUNCT
ejde-984	366	12	|(−∆)s/2u|42	|(−∆)s/2u|42	PROPN
ejde-984	366	13	−	−	PROPN
ejde-984	366	14	µ	µ	PROPN
ejde-984	366	15	q	q	X
ejde-984	366	16	(	(	PUNCT
ejde-984	366	17	1−	1−	NUM
ejde-984	366	18	qδs	qδs	NOUN
ejde-984	366	19	,	,	PUNCT
ejde-984	366	20	q	q	PROPN
ejde-984	366	21	pδs	pδs	PROPN
ejde-984	366	22	,	,	PUNCT
ejde-984	366	23	p	p	NOUN
ejde-984	366	24	)	)	PUNCT
ejde-984	366	25	|u|qq	|u|qq	NOUN
ejde-984	366	26	=	=	PUNCT
ejde-984	366	27	eµ(u	eµ(u	NOUN
ejde-984	366	28	)	)	PUNCT
ejde-984	366	29	,	,	PUNCT
ejde-984	366	30	where	where	SCONJ
ejde-984	366	31	we	we	PRON
ejde-984	366	32	used	use	VERB
ejde-984	366	33	that	that	DET
ejde-984	366	34	tv	tv	NOUN
ejde-984	366	35	∗	∗	NOUN
ejde-984	366	36	v	v	NOUN
ejde-984	366	37	and	and	CCONJ
ejde-984	366	38	u	u	NOUN
ejde-984	366	39	lie	lie	VERB
ejde-984	366	40	in	in	ADP
ejde-984	366	41	pc,µ.	pc,µ.	PROPN
ejde-984	366	42	this	this	PRON
ejde-984	366	43	proves	prove	VERB
ejde-984	366	44	that	that	PRON
ejde-984	366	45	infpc,µ∩sr	infpc,µ∩sr	PROPN
ejde-984	366	46	c	c	AUX
ejde-984	366	47	eµ	eµ	NOUN
ejde-984	366	48	=	=	SYM
ejde-984	366	49	infpc,µ	infpc,µ	ADJ
ejde-984	366	50	eµ.	eµ.	NOUN
ejde-984	366	51	by	by	ADP
ejde-984	366	52	the	the	DET
ejde-984	366	53	above	above	ADJ
ejde-984	366	54	claim	claim	NOUN
ejde-984	366	55	,	,	PUNCT
ejde-984	366	56	(	(	PUNCT
ejde-984	366	57	3.33	3.33	NUM
ejde-984	366	58	)	)	PUNCT
ejde-984	366	59	,	,	PUNCT
ejde-984	366	60	and	and	CCONJ
ejde-984	366	61	lemma	lemma	PROPN
ejde-984	366	62	3.6	3.6	NUM
ejde-984	366	63	,	,	PUNCT
ejde-984	366	64	we	we	PRON
ejde-984	366	65	obtain	obtain	VERB
ejde-984	366	66	that	that	SCONJ
ejde-984	366	67	m(c	m(c	PROPN
ejde-984	366	68	,	,	PUNCT
ejde-984	366	69	µ	µ	NOUN
ejde-984	366	70	)	)	PUNCT
ejde-984	366	71	=	=	SYM
ejde-984	366	72	σ(c	σ(c	PROPN
ejde-984	366	73	,	,	PUNCT
ejde-984	366	74	µ	µ	NOUN
ejde-984	366	75	)	)	PUNCT
ejde-984	366	76	.	.	PUNCT
ejde-984	367	1	(	(	PUNCT
ejde-984	367	2	3.34	3.34	NUM
ejde-984	367	3	)	)	PUNCT
ejde-984	367	4	we	we	PRON
ejde-984	367	5	also	also	ADV
ejde-984	367	6	obtain	obtain	VERB
ejde-984	367	7	that	that	SCONJ
ejde-984	367	8	m(c	m(c	PROPN
ejde-984	367	9	,	,	PUNCT
ejde-984	367	10	µ	µ	NOUN
ejde-984	367	11	)	)	PUNCT
ejde-984	367	12	=	=	SYM
ejde-984	367	13	σ(c	σ(c	PROPN
ejde-984	367	14	,	,	PUNCT
ejde-984	367	15	µ	µ	NOUN
ejde-984	367	16	)	)	PUNCT
ejde-984	367	17	>	>	X
ejde-984	367	18	sup	sup	NOUN
ejde-984	367	19	(	(	PUNCT
ejde-984	367	20	āk∪e0	āk∪e0	PROPN
ejde-984	367	21	µ)∩sr	µ)∩sr	PROPN
ejde-984	367	22	c	c	NOUN
ejde-984	367	23	eµ	eµ	NOUN
ejde-984	367	24	=	=	NOUN
ejde-984	367	25	sup	sup	NOUN
ejde-984	367	26	(	(	PUNCT
ejde-984	367	27	(	(	PUNCT
ejde-984	367	28	0,āk)∪(0,e0	0,āk)∪(0,e0	NOUN
ejde-984	367	29	µ))∩(r×sr	µ))∩(r×sr	PROPN
ejde-984	367	30	c	c	PROPN
ejde-984	367	31	)	)	PUNCT
ejde-984	368	1	ẽµ.	ẽµ.	PROPN
ejde-984	368	2	(	(	PUNCT
ejde-984	368	3	3.35	3.35	NUM
ejde-984	368	4	)	)	PUNCT
ejde-984	368	5	the	the	DET
ejde-984	368	6	rest	rest	NOUN
ejde-984	368	7	of	of	ADP
ejde-984	368	8	the	the	DET
ejde-984	368	9	proof	proof	NOUN
ejde-984	368	10	is	be	AUX
ejde-984	368	11	the	the	DET
ejde-984	368	12	same	same	ADJ
ejde-984	368	13	as	as	ADP
ejde-984	368	14	in	in	ADP
ejde-984	368	15	soave	soave	PROPN
ejde-984	368	16	[	[	X
ejde-984	368	17	24	24	NUM
ejde-984	368	18	]	]	PUNCT
ejde-984	368	19	,	,	PUNCT
ejde-984	368	20	existence	existence	NOUN
ejde-984	368	21	of	of	ADP
ejde-984	368	22	a	a	DET
ejde-984	368	23	second	second	ADJ
ejde-984	368	24	critical	critical	ADJ
ejde-984	368	25	point	point	NOUN
ejde-984	368	26	of	of	ADP
ejde-984	368	27	mountain	mountain	NOUN
ejde-984	368	28	pass	pass	NOUN
ejde-984	368	29	type	type	NOUN
ejde-984	368	30	for	for	ADP
ejde-984	368	31	eµ|sc	eµ|sc	X
ejde-984	368	32	,	,	PUNCT
ejde-984	368	33	thus	thus	ADV
ejde-984	368	34	we	we	PRON
ejde-984	368	35	omit	omit	VERB
ejde-984	368	36	it	it	PRON
ejde-984	368	37	here	here	ADV
ejde-984	368	38	(	(	PUNCT
ejde-984	368	39	namely	namely	ADV
ejde-984	368	40	,	,	PUNCT
ejde-984	368	41	we	we	PRON
ejde-984	368	42	use	use	VERB
ejde-984	368	43	lemma	lemma	PROPN
ejde-984	368	44	2.5	2.5	NUM
ejde-984	368	45	to	to	PART
ejde-984	368	46	obtain	obtain	VERB
ejde-984	368	47	a	a	DET
ejde-984	368	48	ps	ps	NOUN
ejde-984	368	49	sequence	sequence	NOUN
ejde-984	368	50	{	{	PUNCT
ejde-984	368	51	un	un	PROPN
ejde-984	368	52	}	}	PUNCT
ejde-984	368	53	for	for	ADP
ejde-984	368	54	eµ|sr	eµ|sr	ADJ
ejde-984	368	55	c	c	X
ejde-984	368	56	at	at	ADP
ejde-984	368	57	level	level	NOUN
ejde-984	368	58	σ(c	σ(c	PROPN
ejde-984	368	59	,	,	PUNCT
ejde-984	368	60	µ	µ	NOUN
ejde-984	368	61	)	)	PUNCT
ejde-984	368	62	>	>	X
ejde-984	368	63	0	0	PUNCT
ejde-984	369	1	and	and	CCONJ
ejde-984	369	2	dist(un	dist(un	ADJ
ejde-984	369	3	,	,	PUNCT
ejde-984	369	4	pc,µ	pc,µ	NOUN
ejde-984	369	5	)	)	PUNCT
ejde-984	370	1	→	→	SYM
ejde-984	370	2	0	0	NUM
ejde-984	370	3	,	,	PUNCT
ejde-984	370	4	i.e.	i.e.	X
ejde-984	370	5	,	,	PUNCT
ejde-984	370	6	pµ(un	pµ(un	PROPN
ejde-984	370	7	)	)	PUNCT
ejde-984	371	1	→	→	SYM
ejde-984	371	2	0	0	NUM
ejde-984	371	3	)	)	PUNCT
ejde-984	371	4	.	.	PUNCT
ejde-984	372	1	to	to	PART
ejde-984	372	2	verify	verify	VERB
ejde-984	372	3	that	that	SCONJ
ejde-984	372	4	u	u	NOUN
ejde-984	372	5	is	be	AUX
ejde-984	372	6	a	a	DET
ejde-984	372	7	ground	ground	NOUN
ejde-984	372	8	state	state	NOUN
ejde-984	372	9	,	,	PUNCT
ejde-984	372	10	we	we	PRON
ejde-984	372	11	show	show	VERB
ejde-984	372	12	that	that	SCONJ
ejde-984	372	13	u	u	NOUN
ejde-984	372	14	achieves	achieve	VERB
ejde-984	372	15	infpc,µ	infpc,µ	ADJ
ejde-984	372	16	eµ	eµ	PROPN
ejde-984	372	17	=	=	SYM
ejde-984	372	18	m(c	m(c	PROPN
ejde-984	372	19	,	,	PUNCT
ejde-984	372	20	µ	µ	NOUN
ejde-984	372	21	)	)	PUNCT
ejde-984	372	22	.	.	PUNCT
ejde-984	373	1	from	from	ADP
ejde-984	373	2	the	the	DET
ejde-984	373	3	above	above	ADJ
ejde-984	373	4	proof	proof	NOUN
ejde-984	373	5	,	,	PUNCT
ejde-984	373	6	we	we	PRON
ejde-984	373	7	know	know	VERB
ejde-984	373	8	that	that	SCONJ
ejde-984	373	9	σ(c	σ(c	PROPN
ejde-984	373	10	,	,	PUNCT
ejde-984	373	11	µ	µ	NOUN
ejde-984	373	12	)	)	PUNCT
ejde-984	373	13	=	=	SYM
ejde-984	374	1	infγ∈τ̃	infγ∈τ̃	PROPN
ejde-984	374	2	max(τ	max(τ	PROPN
ejde-984	374	3	,	,	PUNCT
ejde-984	374	4	u)∈γ([0,1	u)∈γ([0,1	NOUN
ejde-984	374	5	]	]	PUNCT
ejde-984	374	6	)	)	PUNCT
ejde-984	374	7	ẽµ(τ	ẽµ(τ	PROPN
ejde-984	374	8	,	,	PUNCT
ejde-984	374	9	u	u	NOUN
ejde-984	374	10	)	)	PUNCT
ejde-984	374	11	=	=	SYM
ejde-984	374	12	eµ(u	eµ(u	NOUN
ejde-984	374	13	)	)	PUNCT
ejde-984	374	14	=	=	PUNCT
ejde-984	375	1	infpc,µ∩sr	infpc,µ∩sr	PROPN
ejde-984	375	2	c	c	X
ejde-984	375	3	eµ	eµ	VERB
ejde-984	375	4	,	,	PUNCT
ejde-984	375	5	hence	hence	ADV
ejde-984	375	6	we	we	PRON
ejde-984	375	7	have	have	VERB
ejde-984	375	8	to	to	PART
ejde-984	375	9	show	show	VERB
ejde-984	375	10	that	that	SCONJ
ejde-984	375	11	infpc,µ	infpc,µ	NOUN
ejde-984	375	12	eµ	eµ	NOUN
ejde-984	375	13	=	=	PUNCT
ejde-984	375	14	infpc,µ∩sr	infpc,µ∩sr	ADP
ejde-984	375	15	c	c	X
ejde-984	375	16	eµ.	eµ.	VERB
ejde-984	375	17	by	by	ADP
ejde-984	375	18	claim	claim	NOUN
ejde-984	375	19	(	(	PUNCT
ejde-984	375	20	3.33	3.33	NUM
ejde-984	375	21	)	)	PUNCT
ejde-984	375	22	,	,	PUNCT
ejde-984	375	23	this	this	DET
ejde-984	375	24	equality	equality	NOUN
ejde-984	375	25	holds	hold	VERB
ejde-984	375	26	,	,	PUNCT
ejde-984	375	27	hence	hence	ADV
ejde-984	375	28	u	u	NOUN
ejde-984	375	29	is	be	AUX
ejde-984	375	30	a	a	DET
ejde-984	375	31	ground	ground	NOUN
ejde-984	375	32	state	state	NOUN
ejde-984	375	33	.	.	PUNCT
ejde-984	376	1	therefore	therefore	ADV
ejde-984	376	2	,	,	PUNCT
ejde-984	376	3	theorem	theorem	VERB
ejde-984	376	4	2.9	2.9	NUM
ejde-984	376	5	is	be	AUX
ejde-984	376	6	proved	prove	VERB
ejde-984	376	7	.	.	PUNCT
ejde-984	377	1	□	□	PUNCT
ejde-984	377	2	14	14	NUM
ejde-984	377	3	z.	z.	PROPN
ejde-984	377	4	guo	guo	PROPN
ejde-984	377	5	,	,	PUNCT
ejde-984	377	6	t.	t.	PROPN
ejde-984	377	7	zhang	zhang	PROPN
ejde-984	377	8	ejde-2025/75	ejde-2025/75	ADJ
ejde-984	377	9	proof	proof	NOUN
ejde-984	377	10	of	of	ADP
ejde-984	377	11	theorem	theorem	NOUN
ejde-984	377	12	2.10	2.10	NUM
ejde-984	377	13	.	.	PUNCT
ejde-984	378	1	we	we	PRON
ejde-984	378	2	start	start	VERB
ejde-984	378	3	by	by	ADP
ejde-984	378	4	describing	describe	VERB
ejde-984	378	5	the	the	DET
ejde-984	378	6	structure	structure	NOUN
ejde-984	378	7	of	of	ADP
ejde-984	378	8	zc,µ	zc,µ	PROPN
ejde-984	378	9	of	of	ADP
ejde-984	378	10	ground	ground	NOUN
ejde-984	378	11	states	state	NOUN
ejde-984	378	12	.	.	PUNCT
ejde-984	379	1	if	if	SCONJ
ejde-984	379	2	u	u	PROPN
ejde-984	379	3	∈	∈	PROPN
ejde-984	379	4	zc,µ	zc,µ	NUM
ejde-984	379	5	,	,	PUNCT
ejde-984	379	6	then	then	ADV
ejde-984	379	7	u	u	PROPN
ejde-984	379	8	∈	∈	PROPN
ejde-984	379	9	pc,µ	pc,µ	NOUN
ejde-984	379	10	and	and	CCONJ
ejde-984	379	11	eµ(u	eµ(u	NOUN
ejde-984	379	12	)	)	PUNCT
ejde-984	380	1	=	=	SYM
ejde-984	380	2	m(c	m(c	PROPN
ejde-984	380	3	,	,	PUNCT
ejde-984	380	4	µ	µ	NOUN
ejde-984	380	5	)	)	PUNCT
ejde-984	380	6	=	=	SYM
ejde-984	380	7	infpc,µ	infpc,µ	ADJ
ejde-984	380	8	eµ.	eµ.	VERB
ejde-984	380	9	we	we	PRON
ejde-984	380	10	claim	claim	VERB
ejde-984	380	11	that	that	SCONJ
ejde-984	380	12	u	u	PROPN
ejde-984	380	13	∈	∈	PROPN
ejde-984	380	14	zc,µ	zc,µ	PUNCT
ejde-984	380	15	⇒	⇒	VERB
ejde-984	380	16	|u|	|u|	PROPN
ejde-984	380	17	∈	∈	PROPN
ejde-984	380	18	zc,µ	zc,µ	NOUN
ejde-984	380	19	,	,	PUNCT
ejde-984	380	20	|(−∆)s/2|u||2	|(−∆)s/2|u||2	PROPN
ejde-984	380	21	=	=	SYM
ejde-984	380	22	|(−∆)s/2u|2	|(−∆)s/2u|2	PROPN
ejde-984	380	23	.	.	PUNCT
ejde-984	380	24	(	(	PUNCT
ejde-984	380	25	3.36	3.36	NUM
ejde-984	380	26	)	)	PUNCT
ejde-984	380	27	to	to	PART
ejde-984	380	28	prove	prove	VERB
ejde-984	380	29	the	the	DET
ejde-984	380	30	claim	claim	NOUN
ejde-984	380	31	,	,	PUNCT
ejde-984	380	32	we	we	PRON
ejde-984	380	33	observe	observe	VERB
ejde-984	380	34	that	that	SCONJ
ejde-984	380	35	eµ(|u|	eµ(|u|	NOUN
ejde-984	380	36	)	)	PUNCT
ejde-984	380	37	≤	≤	NOUN
ejde-984	380	38	eµ(u	eµ(u	NOUN
ejde-984	380	39	)	)	PUNCT
ejde-984	380	40	and	and	CCONJ
ejde-984	380	41	pµ(|u|	pµ(|u|	NUM
ejde-984	380	42	)	)	PUNCT
ejde-984	380	43	≤	≤	NOUN
ejde-984	380	44	pµ(u	pµ(u	NOUN
ejde-984	380	45	)	)	PUNCT
ejde-984	380	46	=	=	SYM
ejde-984	381	1	0	0	X
ejde-984	381	2	.	.	PUNCT
ejde-984	381	3	then	then	ADV
ejde-984	381	4	by	by	ADP
ejde-984	381	5	lemma	lemma	PROPN
ejde-984	381	6	3.5	3.5	NUM
ejde-984	381	7	,	,	PUNCT
ejde-984	381	8	there	there	PRON
ejde-984	381	9	exists	exist	VERB
ejde-984	381	10	t|u|	t|u|	PROPN
ejde-984	381	11	≤	≤	NOUN
ejde-984	381	12	0	0	NUM
ejde-984	381	13	with	with	ADP
ejde-984	381	14	t|u|	t|u|	PROPN
ejde-984	381	15	∗	∗	NOUN
ejde-984	381	16	|u|	|u|	PROPN
ejde-984	381	17	∈	∈	PROPN
ejde-984	381	18	pc,µ	pc,µ	NOUN
ejde-984	381	19	and	and	CCONJ
ejde-984	381	20	by	by	ADP
ejde-984	381	21	definition	definition	NOUN
ejde-984	381	22	of	of	ADP
ejde-984	381	23	t|u|	t|u|	PROPN
ejde-984	381	24	,	,	PUNCT
ejde-984	381	25	one	one	NUM
ejde-984	381	26	has	have	VERB
ejde-984	381	27	m(c	m(c	PROPN
ejde-984	381	28	,	,	PUNCT
ejde-984	381	29	µ	µ	NOUN
ejde-984	381	30	)	)	PUNCT
ejde-984	381	31	≤	≤	PUNCT
ejde-984	382	1	eµ(t|u|	eµ(t|u|	PROPN
ejde-984	382	2	∗	∗	PROPN
ejde-984	382	3	|u|	|u|	PROPN
ejde-984	382	4	)	)	PUNCT
ejde-984	383	1	=	=	SYM
ejde-984	383	2	a	a	DET
ejde-984	383	3	(	(	PUNCT
ejde-984	383	4	1	1	NUM
ejde-984	383	5	2	2	NUM
ejde-984	383	6	1	1	NUM
ejde-984	383	7	pδs	pδ	NOUN
ejde-984	383	8	,	,	PUNCT
ejde-984	383	9	p	p	NOUN
ejde-984	383	10	)	)	PUNCT
ejde-984	383	11	e2st|u|	e2st|u|	NOUN
ejde-984	383	12	|(−∆)s/2|u||22	|(−∆)s/2|u||22	PROPN
ejde-984	384	1	+	+	CCONJ
ejde-984	384	2	b	b	X
ejde-984	384	3	(	(	PUNCT
ejde-984	384	4	1	1	NUM
ejde-984	384	5	4	4	NUM
ejde-984	384	6	−	−	NUM
ejde-984	384	7	1	1	NUM
ejde-984	384	8	pδs	pδs	NOUN
ejde-984	384	9	,	,	PUNCT
ejde-984	384	10	p	p	NOUN
ejde-984	384	11	)	)	PUNCT
ejde-984	384	12	e4st|u|	e4st|u|	PROPN
ejde-984	384	13	|(−∆)s/2|u||42	|(−∆)s/2|u||42	PROPN
ejde-984	384	14	−	−	PROPN
ejde-984	384	15	µ	µ	NOUN
ejde-984	384	16	q	q	X
ejde-984	384	17	(	(	PUNCT
ejde-984	384	18	1−	1−	NUM
ejde-984	384	19	qδs	qδs	NOUN
ejde-984	384	20	,	,	PUNCT
ejde-984	384	21	q	q	PROPN
ejde-984	384	22	pδs	pδs	PROPN
ejde-984	384	23	,	,	PUNCT
ejde-984	384	24	p	p	NOUN
ejde-984	384	25	)	)	PUNCT
ejde-984	384	26	eqδs	eqδs	NOUN
ejde-984	384	27	,	,	PUNCT
ejde-984	384	28	qst|u|	qst|u|	PROPN
ejde-984	384	29	|u|qq	|u|qq	VERB
ejde-984	384	30	≤	≤	NUM
ejde-984	384	31	a	a	DET
ejde-984	384	32	(	(	PUNCT
ejde-984	384	33	1	1	NUM
ejde-984	384	34	2	2	NUM
ejde-984	384	35	−	−	NOUN
ejde-984	384	36	1	1	NUM
ejde-984	384	37	pδs	pδs	NOUN
ejde-984	384	38	,	,	PUNCT
ejde-984	384	39	p	p	NOUN
ejde-984	384	40	)	)	PUNCT
ejde-984	384	41	e2st|u|	e2st|u|	NOUN
ejde-984	384	42	|(−∆)s/2u|22	|(−∆)s/2u|22	X
ejde-984	385	1	+	+	CCONJ
ejde-984	385	2	b	b	X
ejde-984	385	3	(	(	PUNCT
ejde-984	385	4	1	1	NUM
ejde-984	385	5	4	4	NUM
ejde-984	385	6	−	−	NUM
ejde-984	385	7	1	1	NUM
ejde-984	385	8	pδs	pδs	NOUN
ejde-984	385	9	,	,	PUNCT
ejde-984	385	10	p	p	NOUN
ejde-984	385	11	)	)	PUNCT
ejde-984	385	12	e4st|u|	e4st|u|	PROPN
ejde-984	385	13	|(−∆)s/2u|42	|(−∆)s/2u|42	NUM
ejde-984	385	14	−	−	PROPN
ejde-984	385	15	µ	µ	NOUN
ejde-984	385	16	q	q	X
ejde-984	385	17	(	(	PUNCT
ejde-984	385	18	1−	1−	NUM
ejde-984	385	19	qδs	qδs	NOUN
ejde-984	385	20	,	,	PUNCT
ejde-984	385	21	q	q	PROPN
ejde-984	385	22	pδs	pδs	PROPN
ejde-984	385	23	,	,	PUNCT
ejde-984	385	24	p	p	NOUN
ejde-984	385	25	)	)	PUNCT
ejde-984	385	26	eqδs	eqδs	NOUN
ejde-984	385	27	,	,	PUNCT
ejde-984	385	28	qst|u|	qst|u|	PROPN
ejde-984	385	29	|u|qq	|u|qq	VERB
ejde-984	385	30	≤	≤	NOUN
ejde-984	385	31	[	[	PUNCT
ejde-984	385	32	a	a	DET
ejde-984	385	33	(	(	PUNCT
ejde-984	385	34	1	1	NUM
ejde-984	385	35	2	2	NUM
ejde-984	385	36	−	−	NOUN
ejde-984	385	37	1	1	NUM
ejde-984	385	38	pδs	pδs	NOUN
ejde-984	385	39	,	,	PUNCT
ejde-984	385	40	p	p	NOUN
ejde-984	385	41	)	)	PUNCT
ejde-984	385	42	|(−∆)s/2u|22	|(−∆)s/2u|22	PROPN
ejde-984	386	1	+	+	CCONJ
ejde-984	386	2	b	b	X
ejde-984	386	3	(	(	PUNCT
ejde-984	386	4	1	1	NUM
ejde-984	386	5	4	4	NUM
ejde-984	386	6	−	−	NUM
ejde-984	386	7	1	1	NUM
ejde-984	386	8	pδs	pδs	NOUN
ejde-984	386	9	,	,	PUNCT
ejde-984	386	10	p	p	NOUN
ejde-984	386	11	)	)	PUNCT
ejde-984	386	12	|(−∆)s/2u|42	|(−∆)s/2u|42	NUM
ejde-984	386	13	−	−	PROPN
ejde-984	386	14	µ	µ	PROPN
ejde-984	386	15	q	q	X
ejde-984	386	16	(	(	PUNCT
ejde-984	386	17	1−	1−	NUM
ejde-984	386	18	qδs	qδs	NOUN
ejde-984	386	19	,	,	PUNCT
ejde-984	386	20	q	q	PROPN
ejde-984	386	21	pδs	pδs	PROPN
ejde-984	386	22	,	,	PUNCT
ejde-984	386	23	p	p	NOUN
ejde-984	386	24	)	)	PUNCT
ejde-984	386	25	|u|qq	|u|qq	NOUN
ejde-984	386	26	]	]	PUNCT
ejde-984	386	27	ekst|u|	ekst|u|	X
ejde-984	386	28	=	=	SYM
ejde-984	386	29	ekst|u|eµ(u	ekst|u|eµ(u	X
ejde-984	386	30	)	)	PUNCT
ejde-984	387	1	=	=	SYM
ejde-984	387	2	ekst|u|m(c	ekst|u|m(c	PROPN
ejde-984	387	3	,	,	PUNCT
ejde-984	387	4	µ	µ	NOUN
ejde-984	387	5	)	)	PUNCT
ejde-984	387	6	,	,	PUNCT
ejde-984	387	7	where	where	SCONJ
ejde-984	387	8	k	k	PROPN
ejde-984	387	9	=	=	SYM
ejde-984	387	10	min	min	PROPN
ejde-984	387	11	{	{	PUNCT
ejde-984	387	12	2	2	NUM
ejde-984	387	13	,	,	PUNCT
ejde-984	387	14	qδs	qδs	NOUN
ejde-984	387	15	,	,	PUNCT
ejde-984	387	16	q	q	NOUN
ejde-984	387	17	}	}	PUNCT
ejde-984	387	18	,	,	PUNCT
ejde-984	387	19	and	and	CCONJ
ejde-984	387	20	we	we	PRON
ejde-984	387	21	used	use	VERB
ejde-984	387	22	the	the	DET
ejde-984	387	23	fact	fact	NOUN
ejde-984	388	1	that	that	SCONJ
ejde-984	388	2	u	u	NOUN
ejde-984	388	3	,	,	PUNCT
ejde-984	388	4	t|u|	t|u|	PROPN
ejde-984	388	5	∗	∗	NOUN
ejde-984	388	6	|u|	|u|	PROPN
ejde-984	388	7	∈	∈	PROPN
ejde-984	388	8	pc,µ	pc,µ	NOUN
ejde-984	388	9	,	,	PUNCT
ejde-984	388	10	and	and	CCONJ
ejde-984	388	11	eµ(u	eµ(u	NOUN
ejde-984	388	12	)	)	PUNCT
ejde-984	388	13	=	=	SYM
ejde-984	388	14	m(c	m(c	PROPN
ejde-984	388	15	,	,	PUNCT
ejde-984	388	16	µ	µ	NOUN
ejde-984	388	17	)	)	PUNCT
ejde-984	388	18	.	.	PUNCT
ejde-984	389	1	by	by	ADP
ejde-984	389	2	t|u|	t|u|	PROPN
ejde-984	389	3	≤	≤	NUM
ejde-984	389	4	0	0	NUM
ejde-984	389	5	,	,	PUNCT
ejde-984	389	6	we	we	PRON
ejde-984	389	7	deduce	deduce	VERB
ejde-984	389	8	that	that	SCONJ
ejde-984	389	9	necessarily	necessarily	ADV
ejde-984	389	10	t|u|	t|u|	ADJ
ejde-984	389	11	=	=	SYM
ejde-984	389	12	0	0	PROPN
ejde-984	389	13	,	,	PUNCT
ejde-984	389	14	that	that	PRON
ejde-984	389	15	is	be	AUX
ejde-984	389	16	pµ(|u|	pµ(|u|	PROPN
ejde-984	389	17	)	)	PUNCT
ejde-984	390	1	=	=	SYM
ejde-984	390	2	0	0	NUM
ejde-984	390	3	,	,	PUNCT
ejde-984	390	4	and	and	CCONJ
ejde-984	390	5	since	since	SCONJ
ejde-984	390	6	also	also	ADV
ejde-984	390	7	pµ(u	pµ(u	X
ejde-984	390	8	)	)	PUNCT
ejde-984	390	9	=	=	SYM
ejde-984	390	10	0	0	NUM
ejde-984	390	11	,	,	PUNCT
ejde-984	390	12	it	it	PRON
ejde-984	390	13	holds	hold	VERB
ejde-984	390	14	that	that	SCONJ
ejde-984	390	15	|u|	|u|	PROPN
ejde-984	390	16	∈	∈	PROPN
ejde-984	390	17	pc,µ	pc,µ	NOUN
ejde-984	390	18	,	,	PUNCT
ejde-984	390	19	|(−∆)s/2|u||2	|(−∆)s/2|u||2	PROPN
ejde-984	390	20	=	=	SYM
ejde-984	390	21	|(−∆)s/2u|2	|(−∆)s/2u|2	PROPN
ejde-984	390	22	and	and	CCONJ
ejde-984	390	23	eµ(|u|	eµ(|u|	PROPN
ejde-984	390	24	)	)	PUNCT
ejde-984	390	25	=	=	SYM
ejde-984	391	1	m(c	m(c	PROPN
ejde-984	391	2	,	,	PUNCT
ejde-984	391	3	µ	µ	NOUN
ejde-984	391	4	)	)	PUNCT
ejde-984	391	5	.	.	PUNCT
ejde-984	392	1	this	this	PRON
ejde-984	392	2	proves	prove	VERB
ejde-984	392	3	claim	claim	NOUN
ejde-984	392	4	(	(	PUNCT
ejde-984	392	5	3.36	3.36	NUM
ejde-984	392	6	)	)	PUNCT
ejde-984	392	7	.	.	PUNCT
ejde-984	393	1	after	after	ADP
ejde-984	393	2	proving	prove	VERB
ejde-984	393	3	that	that	SCONJ
ejde-984	393	4	|u|	|u|	ADV
ejde-984	393	5	minimizes	minimize	VERB
ejde-984	393	6	eµ	eµ	NOUN
ejde-984	393	7	on	on	ADP
ejde-984	393	8	pc,µ	pc,µ	NOUN
ejde-984	393	9	,	,	PUNCT
ejde-984	393	10	we	we	PRON
ejde-984	393	11	obtain	obtain	VERB
ejde-984	393	12	that	that	SCONJ
ejde-984	393	13	|u|	|u|	PROPN
ejde-984	393	14	is	be	AUX
ejde-984	393	15	a	a	DET
ejde-984	393	16	nonnegative	nonnegative	ADJ
ejde-984	393	17	solution	solution	NOUN
ejde-984	393	18	to	to	ADP
ejde-984	393	19	(	(	PUNCT
ejde-984	393	20	1.1	1.1	NUM
ejde-984	393	21	)	)	PUNCT
ejde-984	393	22	for	for	ADP
ejde-984	393	23	some	some	DET
ejde-984	393	24	λ	λ	PROPN
ejde-984	393	25	∈	∈	PROPN
ejde-984	393	26	r	r	NOUN
ejde-984	393	27	,	,	PUNCT
ejde-984	393	28	by	by	ADP
ejde-984	393	29	lemma	lemma	PROPN
ejde-984	393	30	3.4	3.4	NUM
ejde-984	393	31	.	.	PUNCT
ejde-984	394	1	by	by	ADP
ejde-984	394	2	regularity	regularity	NOUN
ejde-984	394	3	and	and	CCONJ
ejde-984	394	4	the	the	DET
ejde-984	394	5	strong	strong	ADJ
ejde-984	394	6	maximum	maximum	ADJ
ejde-984	394	7	principle	principle	NOUN
ejde-984	394	8	,	,	PUNCT
ejde-984	394	9	it	it	PRON
ejde-984	394	10	is	be	AUX
ejde-984	394	11	a	a	DET
ejde-984	394	12	c2	c2	PROPN
ejde-984	394	13	positive	positive	ADJ
ejde-984	394	14	solution	solution	NOUN
ejde-984	394	15	.	.	PUNCT
ejde-984	395	1	using	use	VERB
ejde-984	395	2	also	also	ADV
ejde-984	395	3	that	that	DET
ejde-984	395	4	|(−∆)s/2|u||2	|(−∆)s/2|u||2	PROPN
ejde-984	395	5	=	=	SYM
ejde-984	395	6	|(−∆)s/2u|2	|(−∆)s/2u|2	PROPN
ejde-984	395	7	,	,	PUNCT
ejde-984	395	8	and	and	CCONJ
ejde-984	395	9	u	u	PROPN
ejde-984	395	10	∈	∈	PROPN
ejde-984	395	11	zc,µ	zc,µ	PROPN
ejde-984	395	12	(	(	PUNCT
ejde-984	395	13	then	then	ADV
ejde-984	395	14	we	we	PRON
ejde-984	395	15	obtain	obtain	VERB
ejde-984	395	16	|u|	|u|	PROPN
ejde-984	395	17	∈	∈	PROPN
ejde-984	395	18	zc,µ	zc,µ	NUM
ejde-984	395	19	,	,	PUNCT
ejde-984	395	20	namely	namely	ADV
ejde-984	395	21	,	,	PUNCT
ejde-984	395	22	eµ(|u|	eµ(|u|	PROPN
ejde-984	395	23	)	)	PUNCT
ejde-984	396	1	=	=	SYM
ejde-984	396	2	m(c	m(c	PROPN
ejde-984	396	3	,	,	PUNCT
ejde-984	396	4	µ	µ	NOUN
ejde-984	396	5	)	)	PUNCT
ejde-984	396	6	)	)	PUNCT
ejde-984	396	7	,	,	PUNCT
ejde-984	396	8	we	we	PRON
ejde-984	396	9	will	will	AUX
ejde-984	396	10	prove	prove	VERB
ejde-984	396	11	that	that	SCONJ
ejde-984	396	12	eiθ|u|	eiθ|u|	PROPN
ejde-984	396	13	∈	∈	PROPN
ejde-984	396	14	zc,µ	zc,µ	PROPN
ejde-984	396	15	for	for	ADP
ejde-984	396	16	any	any	DET
ejde-984	396	17	θ	θ	PROPN
ejde-984	396	18	∈	∈	PROPN
ejde-984	396	19	r.	r.	NOUN
ejde-984	396	20	by	by	ADP
ejde-984	396	21	the	the	DET
ejde-984	396	22	definition	definition	NOUN
ejde-984	396	23	of	of	ADP
ejde-984	396	24	eµ(u	eµ(u	NOUN
ejde-984	396	25	)	)	PUNCT
ejde-984	396	26	and	and	CCONJ
ejde-984	396	27	that	that	SCONJ
ejde-984	396	28	the	the	DET
ejde-984	396	29	modulus	modulus	NOUN
ejde-984	396	30	of	of	ADP
ejde-984	396	31	eiθ	eiθ	NOUN
ejde-984	396	32	=	=	NOUN
ejde-984	396	33	1	1	NUM
ejde-984	396	34	for	for	ADP
ejde-984	396	35	any	any	DET
ejde-984	396	36	θ	θ	PROPN
ejde-984	396	37	∈	∈	PROPN
ejde-984	396	38	r	r	NOUN
ejde-984	396	39	,	,	PUNCT
ejde-984	396	40	we	we	PRON
ejde-984	396	41	obtain	obtain	VERB
ejde-984	396	42	that	that	PRON
ejde-984	396	43	eµ(e	eµ(e	PUNCT
ejde-984	396	44	iθ|u|	iθ|u|	ADJ
ejde-984	396	45	)	)	PUNCT
ejde-984	396	46	=	=	PUNCT
ejde-984	396	47	a	a	DET
ejde-984	396	48	2	2	NUM
ejde-984	396	49	|(−∆)s/2eiθ|u||22	|(−∆)s/2eiθ|u||22	NOUN
ejde-984	396	50	+	+	CCONJ
ejde-984	396	51	b	b	SYM
ejde-984	396	52	4	4	NUM
ejde-984	396	53	|(−∆)s/2eiθ|u||42	|(−∆)s/2eiθ|u||42	PROPN
ejde-984	396	54	−	−	PROPN
ejde-984	396	55	µ	µ	PRON
ejde-984	396	56	q	q	ADJ
ejde-984	396	57	|eiθ|u||qq	|eiθ|u||qq	NOUN
ejde-984	396	58	−	−	NOUN
ejde-984	396	59	1	1	NUM
ejde-984	396	60	p	p	NOUN
ejde-984	396	61	|eiθ|u||pp	|eiθ|u||pp	PROPN
ejde-984	396	62	=	=	PUNCT
ejde-984	396	63	a	a	DET
ejde-984	396	64	2	2	NUM
ejde-984	396	65	|(−∆)s/2|u||22	|(−∆)s/2|u||22	NOUN
ejde-984	396	66	+	+	CCONJ
ejde-984	396	67	b	b	SYM
ejde-984	396	68	4	4	NUM
ejde-984	396	69	|(−∆)s/2|u||42	|(−∆)s/2|u||42	PROPN
ejde-984	396	70	−	−	PROPN
ejde-984	396	71	µ	µ	PRON
ejde-984	396	72	q	q	NOUN
ejde-984	396	73	|u|qq	|u|qq	NOUN
ejde-984	396	74	−	−	PROPN
ejde-984	396	75	1	1	NUM
ejde-984	396	76	p	p	PROPN
ejde-984	396	77	|u|pp	|u|pp	NOUN
ejde-984	396	78	=	=	SYM
ejde-984	396	79	eµ(|u|	eµ(|u|	PROPN
ejde-984	396	80	)	)	PUNCT
ejde-984	397	1	=	=	SYM
ejde-984	397	2	m(c	m(c	PROPN
ejde-984	397	3	,	,	PUNCT
ejde-984	397	4	µ	µ	NOUN
ejde-984	397	5	)	)	PUNCT
ejde-984	397	6	.	.	PUNCT
ejde-984	398	1	in	in	ADP
ejde-984	398	2	the	the	DET
ejde-984	398	3	remaining	remain	VERB
ejde-984	398	4	of	of	ADP
ejde-984	398	5	this	this	DET
ejde-984	398	6	proof	proof	NOUN
ejde-984	398	7	,	,	PUNCT
ejde-984	398	8	we	we	PRON
ejde-984	398	9	prove	prove	VERB
ejde-984	398	10	that	that	SCONJ
ejde-984	398	11	if	if	SCONJ
ejde-984	398	12	u	u	PROPN
ejde-984	398	13	∈	∈	PROPN
ejde-984	398	14	zc,µ	zc,µ	NUM
ejde-984	398	15	,	,	PUNCT
ejde-984	398	16	then	then	ADV
ejde-984	398	17	the	the	DET
ejde-984	398	18	associated	associated	ADJ
ejde-984	398	19	lagrange	lagrange	PROPN
ejde-984	398	20	multiplier	multipli	ADJ
ejde-984	398	21	λ	λ	PROPN
ejde-984	398	22	is	be	AUX
ejde-984	398	23	negative	negative	ADJ
ejde-984	398	24	.	.	PUNCT
ejde-984	399	1	recalling	recall	VERB
ejde-984	399	2	that	that	SCONJ
ejde-984	399	3	u	u	PROPN
ejde-984	399	4	∈	∈	PROPN
ejde-984	399	5	pc,µ	pc,µ	NOUN
ejde-984	399	6	,	,	PUNCT
ejde-984	399	7	then	then	ADV
ejde-984	399	8	pµ(u	pµ(u	NOUN
ejde-984	399	9	)	)	PUNCT
ejde-984	399	10	=	=	SYM
ejde-984	399	11	0	0	NUM
ejde-984	399	12	,	,	PUNCT
ejde-984	399	13	and	and	CCONJ
ejde-984	399	14	we	we	PRON
ejde-984	399	15	have	have	VERB
ejde-984	399	16	that	that	PRON
ejde-984	399	17	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	399	18	+	+	NUM
ejde-984	399	19	b|(−∆)s/2u|42	b|(−∆)s/2u|42	NOUN
ejde-984	399	20	=	=	SYM
ejde-984	399	21	µδs	µδ	NOUN
ejde-984	399	22	,	,	PUNCT
ejde-984	399	23	q|u|qq	q|u|qq	PROPN
ejde-984	399	24	+	+	CCONJ
ejde-984	399	25	δs	δs	NOUN
ejde-984	399	26	,	,	PUNCT
ejde-984	399	27	p|u|pp	p|u|pp	PROPN
ejde-984	399	28	≤	≤	NUM
ejde-984	399	29	δs	δs	NOUN
ejde-984	399	30	,	,	PUNCT
ejde-984	399	31	p|u|pp	p|u|pp	PROPN
ejde-984	399	32	.	.	PUNCT
ejde-984	400	1	then	then	ADV
ejde-984	400	2	,	,	PUNCT
ejde-984	400	3	by	by	ADP
ejde-984	400	4	the	the	DET
ejde-984	400	5	fractional	fractional	ADJ
ejde-984	400	6	gagliardo	gagliardo	NOUN
ejde-984	400	7	-	-	PUNCT
ejde-984	400	8	nirenberg	nirenberg	NOUN
ejde-984	400	9	inequality	inequality	NOUN
ejde-984	400	10	,	,	PUNCT
ejde-984	400	11	we	we	PRON
ejde-984	400	12	derive	derive	VERB
ejde-984	400	13	that	that	SCONJ
ejde-984	400	14	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	400	15	≤	≤	ADJ
ejde-984	400	16	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	401	1	+	+	CCONJ
ejde-984	401	2	b|(−∆)s/2u|42	b|(−∆)s/2u|42	PROPN
ejde-984	401	3	≤	≤	NUM
ejde-984	401	4	δs	δs	NOUN
ejde-984	401	5	,	,	PUNCT
ejde-984	401	6	p|u|pp	p|u|pp	PROPN
ejde-984	401	7	≤	≤	NUM
ejde-984	401	8	δs	δs	NOUN
ejde-984	401	9	,	,	PUNCT
ejde-984	401	10	pc(s	pc(s	X
ejde-984	401	11	,	,	PUNCT
ejde-984	401	12	p	p	ADJ
ejde-984	401	13	)	)	PUNCT
ejde-984	401	14	p|(−∆)s/2u|pδs	p|(−∆)s/2u|pδs	PROPN
ejde-984	401	15	,	,	PUNCT
ejde-984	401	16	p2	p2	X
ejde-984	401	17	|u|p(1−δs	|u|p(1−δs	NOUN
ejde-984	401	18	,	,	PUNCT
ejde-984	401	19	p	p	NOUN
ejde-984	401	20	)	)	PUNCT
ejde-984	401	21	2	2	NUM
ejde-984	401	22	.	.	PUNCT
ejde-984	402	1	next	next	ADV
ejde-984	402	2	,	,	PUNCT
ejde-984	402	3	we	we	PRON
ejde-984	402	4	prove	prove	VERB
ejde-984	402	5	the	the	DET
ejde-984	402	6	claim	claim	NOUN
ejde-984	402	7	that	that	SCONJ
ejde-984	402	8	u	u	PRON
ejde-984	402	9	̸≡	̸≡	VERB
ejde-984	402	10	0	0	NUM
ejde-984	402	11	.	.	PUNCT
ejde-984	403	1	since	since	SCONJ
ejde-984	403	2	otherwise	otherwise	ADV
ejde-984	403	3	eµ(u	eµ(u	NOUN
ejde-984	403	4	)	)	PUNCT
ejde-984	403	5	=	=	SYM
ejde-984	403	6	0	0	X
ejde-984	403	7	=	=	SYM
ejde-984	403	8	m(c	m(c	PROPN
ejde-984	403	9	,	,	PUNCT
ejde-984	403	10	µ	µ	NOUN
ejde-984	403	11	)	)	PUNCT
ejde-984	403	12	,	,	PUNCT
ejde-984	403	13	in	in	ADP
ejde-984	403	14	contradiction	contradiction	NOUN
ejde-984	403	15	with	with	ADP
ejde-984	403	16	lemma	lemma	PROPN
ejde-984	403	17	3.6	3.6	NUM
ejde-984	403	18	.	.	PUNCT
ejde-984	403	19	by	by	ADP
ejde-984	403	20	the	the	DET
ejde-984	403	21	definition	definition	NOUN
ejde-984	403	22	of	of	ADP
ejde-984	403	23	pc,µ	pc,µ	NOUN
ejde-984	403	24	,	,	PUNCT
ejde-984	403	25	we	we	PRON
ejde-984	403	26	obtain	obtain	VERB
ejde-984	403	27	|u|2	|u|2	PROPN
ejde-984	403	28	=	=	SYM
ejde-984	403	29	c	c	NOUN
ejde-984	403	30	,	,	PUNCT
ejde-984	403	31	then	then	ADV
ejde-984	403	32	we	we	PRON
ejde-984	403	33	can	can	AUX
ejde-984	403	34	deduce	deduce	VERB
ejde-984	403	35	that	that	SCONJ
ejde-984	403	36	|(−∆)s/2u|2	|(−∆)s/2u|2	PROPN
ejde-984	403	37	≥	≥	PRON
ejde-984	403	38	(	(	PUNCT
ejde-984	403	39	a	a	DET
ejde-984	403	40	δs	δs	NOUN
ejde-984	403	41	,	,	PUNCT
ejde-984	403	42	pc(s	pc(s	NUM
ejde-984	403	43	,	,	PUNCT
ejde-984	403	44	p)pcp(1−δs	p)pcp(1−δs	ADJ
ejde-984	403	45	,	,	PUNCT
ejde-984	403	46	p	p	NOUN
ejde-984	403	47	)	)	PUNCT
ejde-984	403	48	)	)	PUNCT
ejde-984	403	49	1	1	NUM
ejde-984	403	50	pδs	pδs	NOUN
ejde-984	403	51	,	,	PUNCT
ejde-984	403	52	p−2	p−2	PROPN
ejde-984	403	53	.	.	PUNCT
ejde-984	404	1	(	(	PUNCT
ejde-984	404	2	3.37	3.37	NUM
ejde-984	404	3	)	)	PUNCT
ejde-984	404	4	ejde-2025/75	ejde-2025/75	NOUN
ejde-984	404	5	normalized	normalize	VERB
ejde-984	404	6	solutions	solution	NOUN
ejde-984	404	7	of	of	ADP
ejde-984	404	8	fractional	fractional	PROPN
ejde-984	404	9	kirchhoff	kirchhoff	NOUN
ejde-984	404	10	equations	equation	NOUN
ejde-984	404	11	15	15	NUM
ejde-984	404	12	now	now	ADV
ejde-984	404	13	,	,	PUNCT
ejde-984	404	14	since	since	SCONJ
ejde-984	404	15	u	u	NOUN
ejde-984	404	16	is	be	AUX
ejde-984	404	17	a	a	DET
ejde-984	404	18	weak	weak	ADJ
ejde-984	404	19	radial	radial	ADJ
ejde-984	404	20	and	and	CCONJ
ejde-984	404	21	positive	positive	ADJ
ejde-984	404	22	solution	solution	NOUN
ejde-984	404	23	to	to	ADP
ejde-984	404	24	(	(	PUNCT
ejde-984	404	25	a+	a+	PRON
ejde-984	404	26	b	b	NUM
ejde-984	404	27	∫	∫	PROPN
ejde-984	404	28	r3	r3	PROPN
ejde-984	404	29	|(−∆)s/2u|2dx)(−∆)su	|(−∆)s/2u|2dx)(−∆)su	NOUN
ejde-984	404	30	=	=	PUNCT
ejde-984	404	31	λu+	λu+	NOUN
ejde-984	404	32	µ|u|q−2u+	µ|u|q−2u+	PROPN
ejde-984	404	33	|u|p−2u	|u|p−2u	PROPN
ejde-984	404	34	in	in	ADP
ejde-984	404	35	r3	r3	PROPN
ejde-984	404	36	.	.	PUNCT
ejde-984	405	1	(	(	PUNCT
ejde-984	405	2	3.38	3.38	NUM
ejde-984	405	3	)	)	PUNCT
ejde-984	405	4	by	by	ADP
ejde-984	405	5	the	the	DET
ejde-984	405	6	pohožaev	pohožaev	NOUN
ejde-984	405	7	identity	identity	NOUN
ejde-984	405	8	,	,	PUNCT
ejde-984	405	9	we	we	PRON
ejde-984	405	10	infer	infer	VERB
ejde-984	405	11	that	that	SCONJ
ejde-984	405	12	pµ(u	pµ(u	NOUN
ejde-984	405	13	)	)	PUNCT
ejde-984	405	14	=	=	SYM
ejde-984	405	15	0	0	NUM
ejde-984	405	16	,	,	PUNCT
ejde-984	405	17	i.e.	i.e.	X
ejde-984	405	18	,	,	PUNCT
ejde-984	405	19	δs	δs	NOUN
ejde-984	405	20	,	,	PUNCT
ejde-984	405	21	p|u|pp	p|u|pp	PROPN
ejde-984	405	22	=	=	SYM
ejde-984	405	23	a|(−∆)s/2u|22	a|(−∆)s/2u|22	NOUN
ejde-984	405	24	+	+	NUM
ejde-984	405	25	b|(−∆)s/2u|42	b|(−∆)s/2u|42	PROPN
ejde-984	405	26	−	−	NOUN
ejde-984	405	27	µδs	µδ	NOUN
ejde-984	405	28	,	,	PUNCT
ejde-984	405	29	q|u|qq	q|u|qq	PROPN
ejde-984	405	30	.	.	PUNCT
ejde-984	406	1	(	(	PUNCT
ejde-984	406	2	3.39	3.39	NUM
ejde-984	406	3	)	)	PUNCT
ejde-984	406	4	testing	testing	NOUN
ejde-984	406	5	(	(	PUNCT
ejde-984	406	6	3.38	3.38	NUM
ejde-984	406	7	)	)	PUNCT
ejde-984	406	8	with	with	ADP
ejde-984	406	9	u	u	NOUN
ejde-984	406	10	and	and	CCONJ
ejde-984	406	11	using	use	VERB
ejde-984	406	12	(	(	PUNCT
ejde-984	406	13	3.39	3.39	NUM
ejde-984	406	14	)	)	PUNCT
ejde-984	406	15	,	,	PUNCT
ejde-984	406	16	we	we	PRON
ejde-984	406	17	obtain	obtain	VERB
ejde-984	406	18	that	that	DET
ejde-984	406	19	λ|u|22	λ|u|22	PRON
ejde-984	406	20	=	=	PUNCT
ejde-984	406	21	a	a	PRON
ejde-984	406	22	(	(	PUNCT
ejde-984	406	23	1−	1−	NUM
ejde-984	406	24	1	1	NUM
ejde-984	406	25	δs	δs	NOUN
ejde-984	406	26	,	,	PUNCT
ejde-984	406	27	p	p	NOUN
ejde-984	406	28	)	)	PUNCT
ejde-984	406	29	|(−∆)s/2u|22	|(−∆)s/2u|22	PROPN
ejde-984	407	1	+	+	CCONJ
ejde-984	407	2	b	b	X
ejde-984	407	3	(	(	PUNCT
ejde-984	407	4	1−	1−	NUM
ejde-984	407	5	1	1	NUM
ejde-984	407	6	δs	δs	NOUN
ejde-984	407	7	,	,	PUNCT
ejde-984	407	8	p	p	NOUN
ejde-984	407	9	)	)	PUNCT
ejde-984	407	10	|(−∆)s/2u|42	|(−∆)s/2u|42	NUM
ejde-984	407	11	+	+	CCONJ
ejde-984	407	12	µ	µ	X
ejde-984	407	13	(	(	PUNCT
ejde-984	407	14	δs	δs	NOUN
ejde-984	407	15	,	,	PUNCT
ejde-984	407	16	q	q	NOUN
ejde-984	407	17	δs	δs	NOUN
ejde-984	407	18	,	,	PUNCT
ejde-984	407	19	p	p	NOUN
ejde-984	407	20	−	−	PROPN
ejde-984	407	21	1	1	NUM
ejde-984	407	22	)	)	PUNCT
ejde-984	407	23	|u|qq	|u|qq	NOUN
ejde-984	407	24	,	,	PUNCT
ejde-984	407	25	where	where	SCONJ
ejde-984	407	26	1−	1−	NUM
ejde-984	407	27	1	1	NUM
ejde-984	407	28	δs	δs	NOUN
ejde-984	407	29	,	,	PUNCT
ejde-984	407	30	p	p	X
ejde-984	407	31	<	<	X
ejde-984	407	32	0	0	PUNCT
ejde-984	407	33	since	since	SCONJ
ejde-984	407	34	0	0	NUM
ejde-984	407	35	<	<	X
ejde-984	407	36	δs	δs	NOUN
ejde-984	407	37	,	,	PUNCT
ejde-984	407	38	p	p	X
ejde-984	407	39	<	<	X
ejde-984	407	40	1	1	NUM
ejde-984	407	41	,	,	PUNCT
ejde-984	407	42	while	while	SCONJ
ejde-984	407	43	µ	µ	X
ejde-984	407	44	(	(	PUNCT
ejde-984	407	45	δs	δs	NOUN
ejde-984	407	46	,	,	PUNCT
ejde-984	407	47	q	q	NOUN
ejde-984	407	48	δs	δs	NOUN
ejde-984	407	49	,	,	PUNCT
ejde-984	407	50	p	p	NOUN
ejde-984	407	51	−1	−1	NOUN
ejde-984	407	52	)	)	PUNCT
ejde-984	407	53	>	>	X
ejde-984	407	54	0	0	PUNCT
ejde-984	407	55	since	since	SCONJ
ejde-984	407	56	µ	µ	X
ejde-984	407	57	<	<	X
ejde-984	407	58	0	0	NUM
ejde-984	407	59	.	.	PUNCT
ejde-984	407	60	using	use	VERB
ejde-984	407	61	again	again	ADV
ejde-984	407	62	the	the	DET
ejde-984	407	63	fractional	fractional	ADJ
ejde-984	407	64	gagliardo	gagliardo	NOUN
ejde-984	407	65	-	-	PUNCT
ejde-984	407	66	nirenberg	nirenberg	NOUN
ejde-984	407	67	inequality	inequality	NOUN
ejde-984	407	68	and	and	CCONJ
ejde-984	407	69	estimate	estimate	NOUN
ejde-984	407	70	(	(	PUNCT
ejde-984	407	71	3.37	3.37	NUM
ejde-984	407	72	)	)	PUNCT
ejde-984	407	73	,	,	PUNCT
ejde-984	407	74	we	we	PRON
ejde-984	407	75	infer	infer	VERB
ejde-984	407	76	that	that	SCONJ
ejde-984	407	77	λ|u|22	λ|u|22	PRON
ejde-984	407	78	≤	≤	ADV
ejde-984	407	79	a	a	DET
ejde-984	407	80	(	(	PUNCT
ejde-984	407	81	1−	1−	NUM
ejde-984	407	82	1	1	NUM
ejde-984	407	83	δs	δs	NOUN
ejde-984	407	84	,	,	PUNCT
ejde-984	407	85	p	p	NOUN
ejde-984	407	86	)	)	PUNCT
ejde-984	407	87	|(−∆)s/2u|22	|(−∆)s/2u|22	PROPN
ejde-984	408	1	+	+	CCONJ
ejde-984	408	2	b	b	X
ejde-984	408	3	(	(	PUNCT
ejde-984	408	4	1−	1−	NUM
ejde-984	408	5	1	1	NUM
ejde-984	408	6	δs	δs	NOUN
ejde-984	408	7	,	,	PUNCT
ejde-984	408	8	p	p	NOUN
ejde-984	408	9	)	)	PUNCT
ejde-984	408	10	|(−∆)s/2u|42	|(−∆)s/2u|42	NUM
ejde-984	408	11	+	+	CCONJ
ejde-984	408	12	µ	µ	X
ejde-984	408	13	(	(	PUNCT
ejde-984	408	14	δs	δs	NOUN
ejde-984	408	15	,	,	PUNCT
ejde-984	408	16	q	q	NOUN
ejde-984	408	17	δs	δs	NOUN
ejde-984	408	18	,	,	PUNCT
ejde-984	408	19	p	p	NOUN
ejde-984	408	20	−	−	PROPN
ejde-984	408	21	1	1	NUM
ejde-984	408	22	)	)	PUNCT
ejde-984	408	23	c(s	c(	NOUN
ejde-984	408	24	,	,	PUNCT
ejde-984	408	25	q)q|(−∆)s/2u|qδs	q)q|(−∆)s/2u|qδs	NOUN
ejde-984	408	26	,	,	PUNCT
ejde-984	408	27	q2	q2	PROPN
ejde-984	408	28	|u|q(1−δs	|u|q(1−δs	PROPN
ejde-984	408	29	,	,	PUNCT
ejde-984	408	30	q	q	NOUN
ejde-984	408	31	)	)	PUNCT
ejde-984	408	32	2	2	NUM
ejde-984	408	33	≤	≤	PROPN
ejde-984	408	34	|(−∆)s/2u|qδs	|(−∆)s/2u|qδs	NOUN
ejde-984	408	35	,	,	PUNCT
ejde-984	408	36	q2	q2	NOUN
ejde-984	408	37	[	[	PUNCT
ejde-984	408	38	a	a	DET
ejde-984	408	39	(	(	PUNCT
ejde-984	408	40	1−	1−	NUM
ejde-984	408	41	1	1	NUM
ejde-984	408	42	δs	δs	NOUN
ejde-984	408	43	,	,	PUNCT
ejde-984	408	44	p	p	NOUN
ejde-984	408	45	)	)	PUNCT
ejde-984	408	46	|(−∆)s/2u|2−qδs	|(−∆)s/2u|2−qδs	NOUN
ejde-984	408	47	,	,	PUNCT
ejde-984	408	48	q	q	NOUN
ejde-984	408	49	2	2	NUM
ejde-984	408	50	+	+	SYM
ejde-984	408	51	b	b	X
ejde-984	408	52	(	(	PUNCT
ejde-984	408	53	1−	1−	NUM
ejde-984	408	54	1	1	NUM
ejde-984	408	55	δs	δs	NOUN
ejde-984	408	56	,	,	PUNCT
ejde-984	408	57	p	p	NOUN
ejde-984	408	58	)	)	PUNCT
ejde-984	408	59	|(−∆)s/2u|4−qδs	|(−∆)s/2u|4−qδs	PROPN
ejde-984	408	60	,	,	PUNCT
ejde-984	408	61	q	q	NOUN
ejde-984	408	62	2	2	NUM
ejde-984	408	63	+	+	SYM
ejde-984	408	64	µ	µ	X
ejde-984	408	65	(	(	PUNCT
ejde-984	408	66	δs	δs	NOUN
ejde-984	408	67	,	,	PUNCT
ejde-984	408	68	q	q	NOUN
ejde-984	408	69	δs	δs	NOUN
ejde-984	408	70	,	,	PUNCT
ejde-984	408	71	p	p	NOUN
ejde-984	408	72	−	−	PROPN
ejde-984	408	73	1	1	NUM
ejde-984	408	74	)	)	PUNCT
ejde-984	408	75	c(s	c(	NOUN
ejde-984	408	76	,	,	PUNCT
ejde-984	408	77	q)qcq(1−δs	q)qcq(1−δs	PROPN
ejde-984	408	78	,	,	PUNCT
ejde-984	408	79	q	q	NOUN
ejde-984	408	80	)	)	PUNCT
ejde-984	408	81	]	]	PUNCT
ejde-984	408	82	≤	≤	PROPN
ejde-984	408	83	|(−∆)s/2u|qδs	|(−∆)s/2u|qδs	PROPN
ejde-984	408	84	,	,	PUNCT
ejde-984	408	85	q2	q2	NOUN
ejde-984	408	86	[	[	PUNCT
ejde-984	408	87	a	a	DET
ejde-984	408	88	(	(	PUNCT
ejde-984	408	89	1−	1−	NUM
ejde-984	408	90	1	1	NUM
ejde-984	408	91	δs	δs	NOUN
ejde-984	408	92	,	,	PUNCT
ejde-984	408	93	p	p	NOUN
ejde-984	408	94	)	)	PUNCT
ejde-984	408	95	(	(	PUNCT
ejde-984	408	96	a	a	DET
ejde-984	408	97	δs	δs	NOUN
ejde-984	408	98	,	,	PUNCT
ejde-984	408	99	pc(s	pc(s	NUM
ejde-984	408	100	,	,	PUNCT
ejde-984	408	101	p)pcp(1−δs	p)pcp(1−δs	ADJ
ejde-984	408	102	,	,	PUNCT
ejde-984	408	103	p	p	NOUN
ejde-984	408	104	)	)	PUNCT
ejde-984	408	105	)	)	PUNCT
ejde-984	408	106	2−qδs	2−qδs	NOUN
ejde-984	408	107	,	,	PUNCT
ejde-984	408	108	q	q	X
ejde-984	408	109	pδs	pδs	NOUN
ejde-984	408	110	,	,	PUNCT
ejde-984	408	111	p−2	p−2	PROPN
ejde-984	408	112	+	+	CCONJ
ejde-984	408	113	b	b	PROPN
ejde-984	408	114	(	(	PUNCT
ejde-984	408	115	1−	1−	NUM
ejde-984	408	116	a	a	DET
ejde-984	408	117	δs	δs	NOUN
ejde-984	408	118	,	,	PUNCT
ejde-984	408	119	p	p	NOUN
ejde-984	408	120	)	)	PUNCT
ejde-984	408	121	(	(	PUNCT
ejde-984	408	122	1	1	NUM
ejde-984	408	123	δs	δs	NOUN
ejde-984	408	124	,	,	PUNCT
ejde-984	408	125	pc(s	pc(s	NUM
ejde-984	408	126	,	,	PUNCT
ejde-984	408	127	p)pcp(1−δs	p)pcp(1−δs	ADJ
ejde-984	408	128	,	,	PUNCT
ejde-984	408	129	p	p	NOUN
ejde-984	408	130	)	)	PUNCT
ejde-984	408	131	)	)	PUNCT
ejde-984	408	132	4−qδs	4−qδs	PROPN
ejde-984	408	133	,	,	PUNCT
ejde-984	408	134	q	q	PROPN
ejde-984	408	135	pδs	pδs	PROPN
ejde-984	408	136	,	,	PUNCT
ejde-984	408	137	p−2	p−2	PROPN
ejde-984	408	138	+	+	CCONJ
ejde-984	408	139	|µ|	|µ|	PROPN
ejde-984	408	140	(	(	PUNCT
ejde-984	408	141	1−	1−	NUM
ejde-984	408	142	δs	δs	NOUN
ejde-984	408	143	,	,	PUNCT
ejde-984	408	144	q	q	NOUN
ejde-984	408	145	δs	δs	NOUN
ejde-984	408	146	,	,	PUNCT
ejde-984	408	147	p	p	NOUN
ejde-984	408	148	)	)	PUNCT
ejde-984	408	149	c(s	c(	NOUN
ejde-984	408	150	,	,	PUNCT
ejde-984	408	151	q)qcq(1−δs	q)qcq(1−δs	PROPN
ejde-984	408	152	,	,	PUNCT
ejde-984	408	153	q	q	NOUN
ejde-984	408	154	)	)	PUNCT
ejde-984	408	155	]	]	PUNCT
ejde-984	408	156	.	.	PUNCT
ejde-984	409	1	it	it	PRON
ejde-984	409	2	is	be	AUX
ejde-984	409	3	not	not	PART
ejde-984	409	4	difficult	difficult	ADJ
ejde-984	409	5	to	to	PART
ejde-984	409	6	check	check	VERB
ejde-984	409	7	that	that	SCONJ
ejde-984	409	8	the	the	DET
ejde-984	409	9	right	right	ADJ
ejde-984	409	10	hand	hand	NOUN
ejde-984	409	11	side	side	NOUN
ejde-984	409	12	is	be	AUX
ejde-984	409	13	strictly	strictly	ADV
ejde-984	409	14	negative	negative	ADJ
ejde-984	409	15	when	when	SCONJ
ejde-984	409	16	(	(	PUNCT
ejde-984	409	17	2.9	2.9	NUM
ejde-984	409	18	)	)	PUNCT
ejde-984	409	19	holds	hold	VERB
ejde-984	409	20	,	,	PUNCT
ejde-984	409	21	finally	finally	ADV
ejde-984	409	22	implying	imply	VERB
ejde-984	409	23	that	that	SCONJ
ejde-984	409	24	λ	λ	PROPN
ejde-984	409	25	<	<	X
ejde-984	409	26	0	0	NUM
ejde-984	409	27	,	,	PUNCT
ejde-984	409	28	as	as	SCONJ
ejde-984	409	29	desired	desire	VERB
ejde-984	409	30	,	,	PUNCT
ejde-984	409	31	hence	hence	ADV
ejde-984	409	32	the	the	DET
ejde-984	409	33	proof	proof	NOUN
ejde-984	409	34	is	be	AUX
ejde-984	409	35	complete	complete	ADJ
ejde-984	409	36	.	.	PUNCT
ejde-984	410	1	□	□	PUNCT
ejde-984	410	2	acknowledgents	acknowledgent	NOUN
ejde-984	410	3	.	.	PUNCT
ejde-984	411	1	this	this	DET
ejde-984	411	2	work	work	NOUN
ejde-984	411	3	was	be	AUX
ejde-984	411	4	supported	support	VERB
ejde-984	411	5	by	by	ADP
ejde-984	411	6	a	a	DET
ejde-984	411	7	special	special	ADJ
ejde-984	411	8	fund	fund	NOUN
ejde-984	411	9	for	for	ADP
ejde-984	411	10	basic	basic	ADJ
ejde-984	411	11	scientific	scientific	ADJ
ejde-984	411	12	research	research	NOUN
ejde-984	411	13	expenses	expense	NOUN
ejde-984	411	14	of	of	ADP
ejde-984	411	15	universities	university	NOUN
ejde-984	411	16	in	in	ADP
ejde-984	411	17	the	the	DET
ejde-984	411	18	liaoning	liaoning	NOUN
ejde-984	411	19	province	province	NOUN
ejde-984	411	20	(	(	PUNCT
ejde-984	411	21	no	no	INTJ
ejde-984	411	22	.	.	PUNCT
ejde-984	412	1	lj212410165018	lj212410165018	PROPN
ejde-984	412	2	)	)	PUNCT
ejde-984	412	3	.	.	PUNCT
ejde-984	413	1	references	reference	NOUN
ejde-984	413	2	[	[	X
ejde-984	413	3	1	1	NUM
ejde-984	413	4	]	]	PUNCT
ejde-984	413	5	c.	c.	PROPN
ejde-984	413	6	o.	o.	PROPN
ejde-984	413	7	alves	alves	PROPN
ejde-984	413	8	,	,	PUNCT
ejde-984	413	9	f.	f.	PROPN
ejde-984	413	10	j.	j.	PROPN
ejde-984	413	11	s.	s.	PROPN
ejde-984	413	12	a.	a.	PROPN
ejde-984	413	13	corrêa	corrêa	PROPN
ejde-984	413	14	,	,	PUNCT
ejde-984	413	15	t.	t.	PROPN
ejde-984	413	16	f.	f.	PROPN
ejde-984	413	17	ma	ma	PROPN
ejde-984	413	18	;	;	PUNCT
ejde-984	413	19	positive	positive	ADJ
ejde-984	413	20	solutions	solution	NOUN
ejde-984	413	21	for	for	ADP
ejde-984	413	22	a	a	DET
ejde-984	413	23	quasilinear	quasilinear	NOUN
ejde-984	413	24	elliptic	elliptic	ADJ
ejde-984	413	25	equation	equation	NOUN
ejde-984	413	26	of	of	ADP
ejde-984	413	27	kirchhoff	kirchhoff	NOUN
ejde-984	413	28	type	type	NOUN
ejde-984	413	29	,	,	PUNCT
ejde-984	413	30	comput	comput	NOUN
ejde-984	413	31	.	.	PUNCT
ejde-984	414	1	math	math	NOUN
ejde-984	414	2	.	.	PUNCT
ejde-984	415	1	appl	appl	PROPN
ejde-984	415	2	.	.	PROPN
ejde-984	415	3	,	,	PUNCT
ejde-984	415	4	49	49	NUM
ejde-984	415	5	(	(	PUNCT
ejde-984	415	6	2005	2005	NUM
ejde-984	415	7	)	)	PUNCT
ejde-984	415	8	(	(	PUNCT
ejde-984	415	9	1	1	NUM
ejde-984	415	10	):	):	PUNCT
ejde-984	415	11	85–93	85–93	ADJ
ejde-984	415	12	.	.	PUNCT
ejde-984	416	1	[	[	X
ejde-984	416	2	2	2	NUM
ejde-984	416	3	]	]	X
ejde-984	416	4	l.	l.	PROPN
ejde-984	416	5	caffarelli	caffarelli	PROPN
ejde-984	416	6	,	,	PUNCT
ejde-984	416	7	l.	l.	PROPN
ejde-984	416	8	silvestre	silvestre	PROPN
ejde-984	416	9	;	;	PUNCT
ejde-984	416	10	an	an	DET
ejde-984	416	11	extension	extension	NOUN
ejde-984	416	12	problem	problem	NOUN
ejde-984	416	13	related	relate	VERB
ejde-984	416	14	to	to	ADP
ejde-984	416	15	the	the	DET
ejde-984	416	16	fractional	fractional	PROPN
ejde-984	416	17	laplacian	laplacian	PROPN
ejde-984	416	18	,	,	PUNCT
ejde-984	416	19	commun	commun	PROPN
ejde-984	416	20	.	.	PUNCT
ejde-984	417	1	partial	partial	ADJ
ejde-984	417	2	differ	differ	VERB
ejde-984	417	3	.	.	PUNCT
ejde-984	418	1	equations	equation	NOUN
ejde-984	418	2	,	,	PUNCT
ejde-984	418	3	32	32	NUM
ejde-984	418	4	(	(	PUNCT
ejde-984	418	5	2007	2007	NUM
ejde-984	418	6	)	)	PUNCT
ejde-984	418	7	(	(	PUNCT
ejde-984	418	8	8)	8)	NUM
ejde-984	418	9	:	:	PUNCT
ejde-984	418	10	1245–1260	1245–1260	NUM
ejde-984	418	11	.	.	PUNCT
ejde-984	419	1	[	[	X
ejde-984	419	2	3	3	X
ejde-984	419	3	]	]	X
ejde-984	419	4	p.	p.	PROPN
ejde-984	419	5	c.	c.	PROPN
ejde-984	419	6	carrião	carrião	PROPN
ejde-984	419	7	,	,	PUNCT
ejde-984	419	8	o.	o.	PROPN
ejde-984	419	9	h.	h.	PROPN
ejde-984	419	10	miyagaki	miyagaki	PROPN
ejde-984	419	11	,	,	PUNCT
ejde-984	419	12	a.	a.	NOUN
ejde-984	419	13	vicente	vicente	PROPN
ejde-984	419	14	;	;	PUNCT
ejde-984	419	15	normalized	normalize	VERB
ejde-984	419	16	solutions	solution	NOUN
ejde-984	419	17	of	of	ADP
ejde-984	419	18	kirchhoff	kirchhoff	NOUN
ejde-984	419	19	equations	equation	NOUN
ejde-984	419	20	with	with	ADP
ejde-984	419	21	critical	critical	ADJ
ejde-984	419	22	and	and	CCONJ
ejde-984	419	23	subcritical	subcritical	ADJ
ejde-984	419	24	nonlinearities	nonlinearitie	NOUN
ejde-984	419	25	:	:	PUNCT
ejde-984	419	26	the	the	DET
ejde-984	419	27	defocusing	defocuse	VERB
ejde-984	419	28	case	case	NOUN
ejde-984	419	29	,	,	PUNCT
ejde-984	419	30	partial	partial	ADJ
ejde-984	419	31	differ	differ	NOUN
ejde-984	419	32	.	.	PUNCT
ejde-984	420	1	equ	equ	PROPN
ejde-984	420	2	.	.	PUNCT
ejde-984	420	3	appl	appl	PROPN
ejde-984	420	4	.	.	PROPN
ejde-984	420	5	,	,	PUNCT
ejde-984	420	6	3	3	NUM
ejde-984	420	7	(	(	PUNCT
ejde-984	420	8	2022	2022	NUM
ejde-984	420	9	)	)	PUNCT
ejde-984	420	10	(	(	PUNCT
ejde-984	420	11	5	5	NUM
ejde-984	420	12	):	):	PUNCT
ejde-984	420	13	16	16	NUM
ejde-984	420	14	.	.	PUNCT
ejde-984	421	1	[	[	X
ejde-984	421	2	4	4	X
ejde-984	421	3	]	]	PUNCT
ejde-984	421	4	t.	t.	PROPN
ejde-984	421	5	cazenave	cazenave	PROPN
ejde-984	421	6	,	,	PUNCT
ejde-984	421	7	p.-l	p.-l	PROPN
ejde-984	421	8	.	.	PUNCT
ejde-984	422	1	lions	lion	NOUN
ejde-984	422	2	;	;	PUNCT
ejde-984	422	3	orbital	orbital	ADJ
ejde-984	422	4	stability	stability	NOUN
ejde-984	422	5	of	of	ADP
ejde-984	422	6	standing	stand	VERB
ejde-984	422	7	waves	wave	NOUN
ejde-984	422	8	for	for	ADP
ejde-984	422	9	some	some	DET
ejde-984	422	10	nonlinear	nonlinear	ADJ
ejde-984	422	11	schrödinger	schrödinger	NOUN
ejde-984	422	12	equations	equation	NOUN
ejde-984	422	13	,	,	PUNCT
ejde-984	422	14	commun	commun	PROPN
ejde-984	422	15	.	.	PUNCT
ejde-984	422	16	math	math	NOUN
ejde-984	422	17	.	.	PUNCT
ejde-984	423	1	phys	phy	NOUN
ejde-984	423	2	.	.	PUNCT
ejde-984	423	3	85	85	NUM
ejde-984	423	4	(	(	PUNCT
ejde-984	423	5	1982	1982	NUM
ejde-984	423	6	):	):	PUNCT
ejde-984	423	7	549–561	549–561	NUM
ejde-984	423	8	.	.	PUNCT
ejde-984	424	1	[	[	X
ejde-984	424	2	5	5	NUM
ejde-984	424	3	]	]	PUNCT
ejde-984	424	4	s.-y	s.-y	NOUN
ejde-984	424	5	.	.	PUNCT
ejde-984	425	1	a.	a.	PROPN
ejde-984	425	2	chang	chang	PROPN
ejde-984	425	3	,	,	PUNCT
ejde-984	425	4	m.	m.	PROPN
ejde-984	425	5	del	del	PROPN
ejde-984	425	6	mar	mar	PROPN
ejde-984	425	7	gonzález	gonzález	PROPN
ejde-984	425	8	;	;	PUNCT
ejde-984	425	9	fractional	fractional	ADJ
ejde-984	425	10	laplacian	laplacian	NOUN
ejde-984	425	11	in	in	ADP
ejde-984	425	12	conformal	conformal	ADJ
ejde-984	425	13	geometry	geometry	NOUN
ejde-984	425	14	,	,	PUNCT
ejde-984	425	15	adv	adv	PROPN
ejde-984	425	16	.	.	PUNCT
ejde-984	425	17	math	math	PROPN
ejde-984	425	18	.	.	PUNCT
ejde-984	425	19	,	,	PUNCT
ejde-984	425	20	226	226	NUM
ejde-984	425	21	(	(	PUNCT
ejde-984	425	22	2011	2011	NUM
ejde-984	425	23	)	)	PUNCT
ejde-984	425	24	(	(	PUNCT
ejde-984	425	25	2	2	NUM
ejde-984	425	26	):	):	PUNCT
ejde-984	425	27	1410–1432	1410–1432	NUM
ejde-984	425	28	.	.	PUNCT
ejde-984	426	1	[	[	X
ejde-984	426	2	6	6	NUM
ejde-984	426	3	]	]	PUNCT
ejde-984	426	4	w.	w.	PROPN
ejde-984	426	5	chen	chen	PROPN
ejde-984	426	6	,	,	PUNCT
ejde-984	426	7	x.	x.	PROPN
ejde-984	426	8	huang	huang	PROPN
ejde-984	426	9	;	;	PUNCT
ejde-984	426	10	the	the	DET
ejde-984	426	11	existence	existence	NOUN
ejde-984	426	12	of	of	ADP
ejde-984	426	13	normalized	normalize	VERB
ejde-984	426	14	solutions	solution	NOUN
ejde-984	426	15	for	for	ADP
ejde-984	426	16	a	a	DET
ejde-984	426	17	fractional	fractional	ADJ
ejde-984	426	18	kirchhoff	kirchhoff	NOUN
ejde-984	426	19	-	-	PUNCT
ejde-984	426	20	type	type	NOUN
ejde-984	426	21	equation	equation	NOUN
ejde-984	426	22	with	with	ADP
ejde-984	426	23	doubly	doubly	ADV
ejde-984	426	24	critical	critical	ADJ
ejde-984	426	25	exponents	exponent	NOUN
ejde-984	426	26	,	,	PUNCT
ejde-984	426	27	z.	z.	PROPN
ejde-984	426	28	angew	angew	PROPN
ejde-984	426	29	.	.	PUNCT
ejde-984	427	1	math	math	NOUN
ejde-984	427	2	.	.	PUNCT
ejde-984	428	1	phys	phy	NOUN
ejde-984	428	2	.	.	PUNCT
ejde-984	428	3	,	,	PUNCT
ejde-984	428	4	73	73	NUM
ejde-984	428	5	(	(	PUNCT
ejde-984	428	6	2022	2022	NUM
ejde-984	428	7	)	)	PUNCT
ejde-984	428	8	(	(	PUNCT
ejde-984	428	9	6	6	NUM
ejde-984	428	10	):	):	PUNCT
ejde-984	428	11	18	18	NUM
ejde-984	428	12	.	.	PUNCT
ejde-984	429	1	[	[	X
ejde-984	429	2	7	7	X
ejde-984	429	3	]	]	X
ejde-984	429	4	e.	e.	PROPN
ejde-984	429	5	di	di	PROPN
ejde-984	429	6	nezza	nezza	PROPN
ejde-984	429	7	,	,	PUNCT
ejde-984	429	8	g.	g.	PROPN
ejde-984	429	9	palatucci	palatucci	PROPN
ejde-984	429	10	,	,	PUNCT
ejde-984	429	11	and	and	CCONJ
ejde-984	429	12	e.	e.	PROPN
ejde-984	429	13	valdinoci	valdinoci	PROPN
ejde-984	429	14	,	,	PUNCT
ejde-984	429	15	hitchhiker	hitchhiker	NOUN
ejde-984	429	16	’s	’s	PART
ejde-984	429	17	guide	guide	NOUN
ejde-984	429	18	to	to	ADP
ejde-984	429	19	the	the	DET
ejde-984	429	20	fractional	fractional	ADJ
ejde-984	429	21	sobolev	sobolev	NOUN
ejde-984	429	22	spaces	space	NOUN
ejde-984	429	23	,	,	PUNCT
ejde-984	429	24	bull	bull	NOUN
ejde-984	429	25	.	.	PUNCT
ejde-984	430	1	sci	sci	PROPN
ejde-984	430	2	.	.	PUNCT
ejde-984	430	3	math	math	PROPN
ejde-984	430	4	.	.	PUNCT
ejde-984	430	5	,	,	PUNCT
ejde-984	430	6	136	136	NUM
ejde-984	430	7	(	(	PUNCT
ejde-984	430	8	2012	2012	NUM
ejde-984	430	9	)	)	PUNCT
ejde-984	430	10	(	(	PUNCT
ejde-984	430	11	5	5	NUM
ejde-984	430	12	):	):	PUNCT
ejde-984	430	13	521–573	521–573	NUM
ejde-984	430	14	.	.	PUNCT
ejde-984	431	1	[	[	X
ejde-984	431	2	8	8	NUM
ejde-984	431	3	]	]	PUNCT
ejde-984	431	4	s.	s.	PROPN
ejde-984	431	5	ding	ding	PROPN
ejde-984	431	6	;	;	PUNCT
ejde-984	431	7	research	research	NOUN
ejde-984	431	8	of	of	ADP
ejde-984	431	9	the	the	DET
ejde-984	431	10	normalized	normalize	VERB
ejde-984	431	11	solutions	solution	NOUN
ejde-984	431	12	for	for	ADP
ejde-984	431	13	fractional	fractional	ADJ
ejde-984	431	14	kirchhoff	kirchhoff	NOUN
ejde-984	431	15	equations	equation	NOUN
ejde-984	431	16	(	(	PUNCT
ejde-984	431	17	in	in	ADP
ejde-984	431	18	chinese	chinese	PROPN
ejde-984	431	19	)	)	PUNCT
ejde-984	431	20	,	,	PUNCT
ejde-984	431	21	master	master	NOUN
ejde-984	431	22	’s	’s	PART
ejde-984	431	23	thesis	thesis	NOUN
ejde-984	431	24	,	,	PUNCT
ejde-984	431	25	hubei	hubei	PROPN
ejde-984	431	26	normal	normal	ADJ
ejde-984	431	27	university	university	NOUN
ejde-984	431	28	,	,	PUNCT
ejde-984	431	29	2024	2024	NUM
ejde-984	431	30	.	.	PUNCT
ejde-984	432	1	[	[	X
ejde-984	432	2	9	9	NUM
ejde-984	432	3	]	]	PUNCT
ejde-984	432	4	s.	s.	PROPN
ejde-984	432	5	dipierro	dipierro	PROPN
ejde-984	432	6	,	,	PUNCT
ejde-984	432	7	g.	g.	PROPN
ejde-984	432	8	palatucci	palatucci	PROPN
ejde-984	432	9	,	,	PUNCT
ejde-984	432	10	e.	e.	PROPN
ejde-984	432	11	valdinoci	valdinoci	PROPN
ejde-984	432	12	;	;	PUNCT
ejde-984	432	13	existence	existence	NOUN
ejde-984	432	14	and	and	CCONJ
ejde-984	432	15	symmetry	symmetry	NOUN
ejde-984	432	16	results	result	NOUN
ejde-984	432	17	for	for	ADP
ejde-984	432	18	a	a	DET
ejde-984	432	19	schrödinger	schrödinger	NOUN
ejde-984	432	20	type	type	NOUN
ejde-984	432	21	problem	problem	NOUN
ejde-984	432	22	involving	involve	VERB
ejde-984	432	23	the	the	DET
ejde-984	432	24	fractional	fractional	PROPN
ejde-984	432	25	laplacian	laplacian	PROPN
ejde-984	432	26	,	,	PUNCT
ejde-984	432	27	matematiche	matematiche	NOUN
ejde-984	432	28	.	.	PUNCT
ejde-984	433	1	68	68	NUM
ejde-984	433	2	(	(	PUNCT
ejde-984	433	3	2013	2013	NUM
ejde-984	433	4	)	)	PUNCT
ejde-984	433	5	(	(	PUNCT
ejde-984	433	6	1	1	NUM
ejde-984	433	7	):	):	PUNCT
ejde-984	433	8	201–216	201–216	NUM
ejde-984	433	9	.	.	PUNCT
ejde-984	434	1	[	[	X
ejde-984	434	2	10	10	NUM
ejde-984	434	3	]	]	X
ejde-984	434	4	n.	n.	NOUN
ejde-984	434	5	ghoussoub	ghoussoub	NOUN
ejde-984	434	6	;	;	PUNCT
ejde-984	434	7	duality	duality	NOUN
ejde-984	434	8	and	and	CCONJ
ejde-984	434	9	perturbation	perturbation	NOUN
ejde-984	434	10	methods	method	NOUN
ejde-984	434	11	in	in	ADP
ejde-984	434	12	critical	critical	ADJ
ejde-984	434	13	point	point	NOUN
ejde-984	434	14	theory	theory	NOUN
ejde-984	434	15	,	,	PUNCT
ejde-984	434	16	ser	ser	PROPN
ejde-984	434	17	.	.	PUNCT
ejde-984	435	1	camb	camb	PROPN
ejde-984	435	2	.	.	PUNCT
ejde-984	436	1	tracts	tract	NOUN
ejde-984	436	2	math	math	PROPN
ejde-984	436	3	.	.	PUNCT
ejde-984	436	4	,	,	PUNCT
ejde-984	436	5	cambridge	cambridge	PROPN
ejde-984	436	6	university	university	PROPN
ejde-984	436	7	press	press	NOUN
ejde-984	436	8	.	.	PUNCT
ejde-984	437	1	107	107	NUM
ejde-984	437	2	,	,	PUNCT
ejde-984	437	3	1993	1993	NUM
ejde-984	437	4	.	.	PUNCT
ejde-984	438	1	16	16	NUM
ejde-984	438	2	z.	z.	PROPN
ejde-984	438	3	guo	guo	PROPN
ejde-984	438	4	,	,	PUNCT
ejde-984	438	5	t.	t.	PROPN
ejde-984	438	6	zhang	zhang	PROPN
ejde-984	438	7	ejde-2025/75	ejde-2025/75	ADJ
ejde-984	439	1	[	[	X
ejde-984	439	2	11	11	NUM
ejde-984	439	3	]	]	PUNCT
ejde-984	439	4	t.	t.	PROPN
ejde-984	439	5	gou	gou	PROPN
ejde-984	439	6	,	,	PUNCT
ejde-984	439	7	l.	l.	PROPN
ejde-984	439	8	jeanjean	jeanjean	PROPN
ejde-984	439	9	;	;	PUNCT
ejde-984	439	10	multiple	multiple	ADJ
ejde-984	439	11	positive	positive	ADJ
ejde-984	439	12	normalized	normalize	VERB
ejde-984	439	13	solutions	solution	NOUN
ejde-984	439	14	for	for	ADP
ejde-984	439	15	nonlinear	nonlinear	ADJ
ejde-984	439	16	schrödinger	schrödinger	NOUN
ejde-984	439	17	systems	system	NOUN
ejde-984	439	18	,	,	PUNCT
ejde-984	439	19	nonlinearity	nonlinearity	NOUN
ejde-984	439	20	,	,	PUNCT
ejde-984	439	21	31	31	NUM
ejde-984	439	22	(	(	PUNCT
ejde-984	439	23	2018	2018	NUM
ejde-984	439	24	)	)	PUNCT
ejde-984	439	25	(	(	PUNCT
ejde-984	439	26	5	5	NUM
ejde-984	439	27	):	):	PUNCT
ejde-984	439	28	2319–2345	2319–2345	NUM
ejde-984	439	29	.	.	PUNCT
ejde-984	440	1	[	[	X
ejde-984	440	2	12	12	NUM
ejde-984	440	3	]	]	PUNCT
ejde-984	440	4	x.	x.	NOUN
ejde-984	441	1	he	he	PRON
ejde-984	441	2	,	,	PUNCT
ejde-984	441	3	w.	w.	PROPN
ejde-984	441	4	zou	zou	PROPN
ejde-984	441	5	;	;	PUNCT
ejde-984	441	6	multiplicity	multiplicity	NOUN
ejde-984	441	7	of	of	ADP
ejde-984	441	8	concentrating	concentrate	VERB
ejde-984	441	9	solutions	solution	NOUN
ejde-984	441	10	for	for	ADP
ejde-984	441	11	a	a	DET
ejde-984	441	12	class	class	NOUN
ejde-984	441	13	of	of	ADP
ejde-984	441	14	fractional	fractional	PROPN
ejde-984	441	15	kirchhoff	kirchhoff	NOUN
ejde-984	441	16	equation	equation	NOUN
ejde-984	441	17	,	,	PUNCT
ejde-984	441	18	manuscr	manuscr	PROPN
ejde-984	441	19	.	.	PUNCT
ejde-984	442	1	math	math	NOUN
ejde-984	442	2	.	.	PUNCT
ejde-984	443	1	,	,	PUNCT
ejde-984	443	2	158	158	NUM
ejde-984	443	3	(	(	PUNCT
ejde-984	443	4	2019	2019	NUM
ejde-984	443	5	)	)	PUNCT
ejde-984	443	6	(	(	PUNCT
ejde-984	443	7	1	1	NUM
ejde-984	443	8	-	-	SYM
ejde-984	443	9	2	2	NUM
ejde-984	443	10	):	):	PUNCT
ejde-984	443	11	159–203	159–203	NUM
ejde-984	443	12	.	.	PUNCT
ejde-984	444	1	[	[	X
ejde-984	444	2	13	13	NUM
ejde-984	444	3	]	]	PUNCT
ejde-984	445	1	l.	l.	PROPN
ejde-984	445	2	jeanjean	jeanjean	PROPN
ejde-984	445	3	;	;	PUNCT
ejde-984	445	4	existence	existence	NOUN
ejde-984	445	5	of	of	ADP
ejde-984	445	6	solutions	solution	NOUN
ejde-984	445	7	with	with	ADP
ejde-984	445	8	prescribed	prescribed	ADJ
ejde-984	445	9	norm	norm	NOUN
ejde-984	445	10	for	for	ADP
ejde-984	445	11	semilinear	semilinear	PROPN
ejde-984	445	12	elliptic	elliptic	ADJ
ejde-984	445	13	equations	equation	NOUN
ejde-984	445	14	,	,	PUNCT
ejde-984	445	15	nonlinear	nonlinear	ADJ
ejde-984	445	16	anal	anal	NOUN
ejde-984	445	17	.	.	PUNCT
ejde-984	445	18	,	,	PUNCT
ejde-984	445	19	theory	theory	NOUN
ejde-984	445	20	methods	method	NOUN
ejde-984	445	21	appl	appl	PROPN
ejde-984	445	22	.	.	PROPN
ejde-984	445	23	,	,	PUNCT
ejde-984	445	24	28	28	NUM
ejde-984	445	25	(	(	PUNCT
ejde-984	445	26	1997	1997	NUM
ejde-984	445	27	)	)	PUNCT
ejde-984	445	28	(	(	PUNCT
ejde-984	445	29	10	10	NUM
ejde-984	445	30	):	):	PUNCT
ejde-984	445	31	1633–1659	1633–1659	NUM
ejde-984	445	32	.	.	PUNCT
ejde-984	446	1	[	[	X
ejde-984	446	2	14	14	NUM
ejde-984	446	3	]	]	X
ejde-984	446	4	g.	g.	PROPN
ejde-984	446	5	kirchhoff	kirchhoff	PROPN
ejde-984	446	6	;	;	PUNCT
ejde-984	446	7	vorlesungen	vorlesungen	PROPN
ejde-984	446	8	über	über	PROPN
ejde-984	446	9	mathematische	mathematische	PROPN
ejde-984	446	10	physik	physik	PROPN
ejde-984	446	11	.	.	PUNCT
ejde-984	446	12	i.	i.	PROPN
ejde-984	446	13	mechanik	mechanik	PROPN
ejde-984	446	14	.	.	PUNCT
ejde-984	447	1	leipzig	leipzig	PROPN
ejde-984	447	2	.	.	PUNCT
ejde-984	448	1	teubner	teubner	NOUN
ejde-984	448	2	.	.	PUNCT
ejde-984	449	1	1876	1876	NUM
ejde-984	449	2	.	.	PUNCT
ejde-984	450	1	[	[	X
ejde-984	450	2	15	15	NUM
ejde-984	450	3	]	]	X
ejde-984	450	4	n.	n.	NOUN
ejde-984	450	5	laskin	laskin	PROPN
ejde-984	450	6	;	;	PUNCT
ejde-984	450	7	fractional	fractional	ADJ
ejde-984	450	8	quantum	quantum	ADJ
ejde-984	450	9	mechanics	mechanic	NOUN
ejde-984	450	10	and	and	CCONJ
ejde-984	450	11	lévy	lévy	NUM
ejde-984	450	12	path	path	NOUN
ejde-984	450	13	integrals	integral	NOUN
ejde-984	450	14	,	,	PUNCT
ejde-984	450	15	phys	phy	NOUN
ejde-984	450	16	.	.	PUNCT
ejde-984	451	1	lett	lett	PROPN
ejde-984	451	2	.	.	PROPN
ejde-984	451	3	,	,	PUNCT
ejde-984	451	4	a.	a.	PROPN
ejde-984	451	5	,	,	PUNCT
ejde-984	451	6	268	268	NUM
ejde-984	451	7	(	(	PUNCT
ejde-984	451	8	2000	2000	NUM
ejde-984	451	9	)	)	PUNCT
ejde-984	451	10	(	(	PUNCT
ejde-984	451	11	4	4	NUM
ejde-984	451	12	-	-	SYM
ejde-984	451	13	6	6	NUM
ejde-984	451	14	):	):	PUNCT
ejde-984	451	15	298–305	298–305	NUM
ejde-984	451	16	.	.	PUNCT
ejde-984	452	1	[	[	X
ejde-984	452	2	16	16	NUM
ejde-984	452	3	]	]	X
ejde-984	452	4	g.	g.	PROPN
ejde-984	452	5	li	li	PROPN
ejde-984	452	6	,	,	PUNCT
ejde-984	452	7	x.	x.	PROPN
ejde-984	452	8	luo	luo	PROPN
ejde-984	452	9	,	,	PUNCT
ejde-984	452	10	t.	t.	PROPN
ejde-984	452	11	yang	yang	PROPN
ejde-984	452	12	;	;	PUNCT
ejde-984	452	13	normalized	normalize	VERB
ejde-984	452	14	solutions	solution	NOUN
ejde-984	452	15	to	to	ADP
ejde-984	452	16	a	a	DET
ejde-984	452	17	class	class	NOUN
ejde-984	452	18	of	of	ADP
ejde-984	452	19	kirchhoff	kirchhoff	NOUN
ejde-984	452	20	equations	equation	NOUN
ejde-984	452	21	with	with	ADP
ejde-984	452	22	sobolev	sobolev	PROPN
ejde-984	452	23	critical	critical	ADJ
ejde-984	452	24	exponent	exponent	NOUN
ejde-984	452	25	,	,	PUNCT
ejde-984	452	26	ann	ann	PROPN
ejde-984	453	1	.	.	PUNCT
ejde-984	453	2	fenn	fenn	PROPN
ejde-984	453	3	.	.	PUNCT
ejde-984	453	4	math	math	PROPN
ejde-984	453	5	.	.	PUNCT
ejde-984	454	1	,	,	PUNCT
ejde-984	454	2	47	47	NUM
ejde-984	454	3	(	(	PUNCT
ejde-984	454	4	2022	2022	NUM
ejde-984	454	5	)	)	PUNCT
ejde-984	454	6	(	(	PUNCT
ejde-984	454	7	2	2	NUM
ejde-984	454	8	):	):	PUNCT
ejde-984	454	9	895–925	895–925	NUM
ejde-984	454	10	.	.	PUNCT
ejde-984	455	1	[	[	X
ejde-984	455	2	17	17	NUM
ejde-984	455	3	]	]	PUNCT
ejde-984	455	4	z.	z.	PROPN
ejde-984	455	5	liu	liu	PROPN
ejde-984	455	6	,	,	PUNCT
ejde-984	455	7	m.	m.	PROPN
ejde-984	455	8	squassina	squassina	PROPN
ejde-984	455	9	,	,	PUNCT
ejde-984	455	10	j.	j.	PROPN
ejde-984	455	11	zhang	zhang	PROPN
ejde-984	455	12	;	;	PUNCT
ejde-984	455	13	ground	ground	NOUN
ejde-984	455	14	states	state	NOUN
ejde-984	455	15	for	for	ADP
ejde-984	455	16	fractional	fractional	ADJ
ejde-984	455	17	kirchhoff	kirchhoff	NOUN
ejde-984	455	18	equations	equation	NOUN
ejde-984	455	19	with	with	ADP
ejde-984	455	20	critical	critical	ADJ
ejde-984	455	21	nonlinearity	nonlinearity	NOUN
ejde-984	455	22	in	in	ADP
ejde-984	455	23	low	low	ADJ
ejde-984	455	24	dimension	dimension	NOUN
ejde-984	455	25	,	,	PUNCT
ejde-984	455	26	nonlinear	nonlinear	NOUN
ejde-984	455	27	differ	differ	VERB
ejde-984	455	28	.	.	PUNCT
ejde-984	456	1	equ	equ	PROPN
ejde-984	456	2	.	.	PUNCT
ejde-984	456	3	appl	appl	PROPN
ejde-984	456	4	.	.	PROPN
ejde-984	456	5	,	,	PUNCT
ejde-984	456	6	24	24	NUM
ejde-984	456	7	(	(	PUNCT
ejde-984	456	8	2017	2017	NUM
ejde-984	456	9	)	)	PUNCT
ejde-984	456	10	(	(	PUNCT
ejde-984	456	11	4):32	4):32	NUM
ejde-984	456	12	.	.	PUNCT
ejde-984	457	1	[	[	X
ejde-984	457	2	18	18	NUM
ejde-984	457	3	]	]	X
ejde-984	457	4	h.	h.	PROPN
ejde-984	457	5	luo	luo	PROPN
ejde-984	457	6	,	,	PUNCT
ejde-984	457	7	z.	z.	PROPN
ejde-984	457	8	zhang	zhang	PROPN
ejde-984	457	9	;	;	PUNCT
ejde-984	457	10	normalized	normalize	VERB
ejde-984	457	11	solutions	solution	NOUN
ejde-984	457	12	to	to	ADP
ejde-984	457	13	the	the	DET
ejde-984	457	14	fractional	fractional	PROPN
ejde-984	457	15	schrödinger	schrödinger	NOUN
ejde-984	457	16	equations	equation	NOUN
ejde-984	457	17	with	with	ADP
ejde-984	457	18	combined	combined	ADJ
ejde-984	457	19	nonlinearities	nonlinearitie	NOUN
ejde-984	457	20	,	,	PUNCT
ejde-984	457	21	calc	calc	NOUN
ejde-984	457	22	.	.	PUNCT
ejde-984	458	1	var	var	PROPN
ejde-984	458	2	.	.	PUNCT
ejde-984	459	1	partial	partial	ADJ
ejde-984	459	2	differ	differ	VERB
ejde-984	459	3	.	.	PUNCT
ejde-984	460	1	equ	equ	PROPN
ejde-984	460	2	.	.	PROPN
ejde-984	460	3	,	,	PUNCT
ejde-984	460	4	59	59	NUM
ejde-984	460	5	(	(	PUNCT
ejde-984	460	6	2020	2020	NUM
ejde-984	460	7	)	)	PUNCT
ejde-984	460	8	(	(	PUNCT
ejde-984	460	9	4	4	NUM
ejde-984	460	10	):	):	PUNCT
ejde-984	460	11	35	35	NUM
ejde-984	460	12	.	.	PUNCT
ejde-984	461	1	[	[	X
ejde-984	461	2	19	19	NUM
ejde-984	461	3	]	]	X
ejde-984	461	4	j.	j.	PROPN
ejde-984	461	5	r.	r.	PROPN
ejde-984	461	6	metzler	metzler	PROPN
ejde-984	461	7	;	;	PUNCT
ejde-984	461	8	the	the	DET
ejde-984	461	9	random	random	ADJ
ejde-984	461	10	walks	walk	VERB
ejde-984	461	11	guide	guide	NOUN
ejde-984	461	12	to	to	ADP
ejde-984	461	13	anomalous	anomalous	ADJ
ejde-984	461	14	diffusion	diffusion	NOUN
ejde-984	461	15	:	:	PUNCT
ejde-984	461	16	a	a	DET
ejde-984	461	17	fractional	fractional	ADJ
ejde-984	461	18	dynamics	dynamic	NOUN
ejde-984	461	19	approach	approach	NOUN
ejde-984	461	20	,	,	PUNCT
ejde-984	461	21	phys	phy	NOUN
ejde-984	461	22	.	.	PUNCT
ejde-984	462	1	rep	rep	PROPN
ejde-984	462	2	.	.	PROPN
ejde-984	462	3	,	,	PUNCT
ejde-984	462	4	2000	2000	NUM
ejde-984	462	5	.	.	PUNCT
ejde-984	463	1	[	[	X
ejde-984	463	2	20	20	NUM
ejde-984	463	3	]	]	PUNCT
ejde-984	463	4	x.	x.	NOUN
ejde-984	463	5	mingqi	mingqi	PROPN
ejde-984	463	6	,	,	PUNCT
ejde-984	463	7	v.	v.	PROPN
ejde-984	463	8	d.	d.	PROPN
ejde-984	463	9	rădulescu	rădulescu	PROPN
ejde-984	463	10	,	,	PUNCT
ejde-984	463	11	b.	b.	PROPN
ejde-984	463	12	zhang	zhang	PROPN
ejde-984	463	13	;	;	PUNCT
ejde-984	463	14	fractional	fractional	PROPN
ejde-984	463	15	kirchhoff	kirchhoff	NOUN
ejde-984	463	16	problems	problem	NOUN
ejde-984	463	17	with	with	ADP
ejde-984	463	18	critical	critical	ADJ
ejde-984	463	19	trudinger	trudinger	NOUN
ejde-984	463	20	-	-	PUNCT
ejde-984	463	21	moser	moser	PROPN
ejde-984	463	22	nonlinearity	nonlinearity	PROPN
ejde-984	463	23	,	,	PUNCT
ejde-984	463	24	calc	calc	NOUN
ejde-984	463	25	.	.	PUNCT
ejde-984	464	1	var	var	PROPN
ejde-984	464	2	.	.	PUNCT
ejde-984	465	1	partial	partial	ADJ
ejde-984	465	2	differ	differ	VERB
ejde-984	465	3	.	.	PUNCT
ejde-984	466	1	equ	equ	PROPN
ejde-984	466	2	.	.	PROPN
ejde-984	466	3	,	,	PUNCT
ejde-984	466	4	58	58	NUM
ejde-984	466	5	(	(	PUNCT
ejde-984	466	6	2019	2019	NUM
ejde-984	466	7	)	)	PUNCT
ejde-984	466	8	(	(	PUNCT
ejde-984	466	9	2	2	NUM
ejde-984	466	10	):	):	PUNCT
ejde-984	466	11	27	27	NUM
ejde-984	466	12	.	.	PUNCT
ejde-984	467	1	[	[	X
ejde-984	467	2	21	21	NUM
ejde-984	467	3	]	]	X
ejde-984	467	4	s.	s.	PROPN
ejde-984	467	5	pohožaev	pohožaev	PROPN
ejde-984	467	6	;	;	PUNCT
ejde-984	467	7	a	a	DET
ejde-984	467	8	certain	certain	ADJ
ejde-984	467	9	class	class	NOUN
ejde-984	467	10	of	of	ADP
ejde-984	467	11	quasilinear	quasilinear	PROPN
ejde-984	467	12	hyperbolic	hyperbolic	ADJ
ejde-984	467	13	equations	equation	NOUN
ejde-984	467	14	,	,	PUNCT
ejde-984	467	15	mat	mat	PROPN
ejde-984	467	16	.	.	PUNCT
ejde-984	467	17	sb	sb	PROPN
ejde-984	467	18	.	.	PROPN
ejde-984	467	19	96	96	NUM
ejde-984	467	20	(	(	PUNCT
ejde-984	467	21	1975	1975	NUM
ejde-984	467	22	)	)	PUNCT
ejde-984	467	23	(	(	PUNCT
ejde-984	467	24	138	138	NUM
ejde-984	467	25	):	):	PUNCT
ejde-984	467	26	152	152	NUM
ejde-984	467	27	-	-	SYM
ejde-984	467	28	168	168	NUM
ejde-984	467	29	.	.	PUNCT
ejde-984	468	1	[	[	X
ejde-984	468	2	22	22	NUM
ejde-984	468	3	]	]	PUNCT
ejde-984	468	4	m.	m.	NOUN
ejde-984	468	5	h.	h.	PROPN
ejde-984	468	6	protter	protter	PROPN
ejde-984	468	7	,	,	PUNCT
ejde-984	468	8	h.	h.	PROPN
ejde-984	468	9	f.	f.	PROPN
ejde-984	468	10	weinberger	weinberger	PROPN
ejde-984	468	11	;	;	PUNCT
ejde-984	468	12	maximum	maximum	ADJ
ejde-984	468	13	-	-	PUNCT
ejde-984	468	14	principles	principle	NOUN
ejde-984	468	15	in	in	ADP
ejde-984	468	16	differential	differential	ADJ
ejde-984	468	17	equations	equation	NOUN
ejde-984	468	18	,	,	PUNCT
ejde-984	468	19	prentice	prentice	NOUN
ejde-984	468	20	-	-	PUNCT
ejde-984	468	21	hall	hall	NOUN
ejde-984	468	22	partial	partial	ADJ
ejde-984	468	23	differential	differential	NOUN
ejde-984	468	24	equations	equation	NOUN
ejde-984	468	25	series	series	PROPN
ejde-984	468	26	.	.	PUNCT
ejde-984	469	1	englewood	englewood	PROPN
ejde-984	469	2	cliffs	cliffs	PROPN
ejde-984	469	3	,	,	PUNCT
ejde-984	469	4	nj	nj	PROPN
ejde-984	469	5	1967	1967	NUM
ejde-984	469	6	.	.	PUNCT
ejde-984	470	1	[	[	X
ejde-984	470	2	23	23	NUM
ejde-984	470	3	]	]	PUNCT
ejde-984	470	4	l.	l.	PROPN
ejde-984	470	5	silvestre	silvestre	PROPN
ejde-984	470	6	;	;	PUNCT
ejde-984	470	7	regularity	regularity	NOUN
ejde-984	470	8	of	of	ADP
ejde-984	470	9	the	the	DET
ejde-984	470	10	obstacle	obstacle	NOUN
ejde-984	470	11	problem	problem	NOUN
ejde-984	470	12	for	for	ADP
ejde-984	470	13	a	a	DET
ejde-984	470	14	fractional	fractional	ADJ
ejde-984	470	15	power	power	NOUN
ejde-984	470	16	of	of	ADP
ejde-984	470	17	the	the	DET
ejde-984	470	18	laplace	laplace	NOUN
ejde-984	470	19	operator	operator	NOUN
ejde-984	470	20	,	,	PUNCT
ejde-984	470	21	commun	commun	PROPN
ejde-984	470	22	.	.	PUNCT
ejde-984	471	1	pure	pure	ADJ
ejde-984	471	2	appl	appl	PROPN
ejde-984	471	3	.	.	PUNCT
ejde-984	471	4	math	math	PROPN
ejde-984	471	5	.	.	PUNCT
ejde-984	472	1	,	,	PUNCT
ejde-984	472	2	60	60	NUM
ejde-984	472	3	(	(	PUNCT
ejde-984	472	4	2007	2007	NUM
ejde-984	472	5	)	)	PUNCT
ejde-984	472	6	(	(	PUNCT
ejde-984	472	7	1	1	NUM
ejde-984	472	8	):	):	PUNCT
ejde-984	472	9	67–112	67–112	NUM
ejde-984	472	10	.	.	PUNCT
ejde-984	473	1	[	[	X
ejde-984	473	2	24	24	NUM
ejde-984	473	3	]	]	X
ejde-984	473	4	n.	n.	PROPN
ejde-984	473	5	soave	soave	PROPN
ejde-984	473	6	;	;	PUNCT
ejde-984	473	7	normalized	normalize	VERB
ejde-984	473	8	ground	ground	NOUN
ejde-984	473	9	states	state	NOUN
ejde-984	473	10	for	for	ADP
ejde-984	473	11	the	the	DET
ejde-984	473	12	nls	nls	NOUN
ejde-984	473	13	equation	equation	NOUN
ejde-984	473	14	with	with	ADP
ejde-984	473	15	combined	combined	ADJ
ejde-984	473	16	nonlinearities	nonlinearitie	NOUN
ejde-984	473	17	,	,	PUNCT
ejde-984	473	18	j.	j.	PROPN
ejde-984	473	19	differ	differ	VERB
ejde-984	473	20	.	.	PUNCT
ejde-984	474	1	equations	equation	NOUN
ejde-984	474	2	,	,	PUNCT
ejde-984	474	3	269	269	NUM
ejde-984	474	4	(	(	PUNCT
ejde-984	474	5	2020	2020	NUM
ejde-984	474	6	)	)	PUNCT
ejde-984	474	7	(	(	PUNCT
ejde-984	474	8	9	9	NUM
ejde-984	474	9	):	):	PUNCT
ejde-984	474	10	6941–6987	6941–6987	NUM
ejde-984	474	11	.	.	PUNCT
ejde-984	475	1	[	[	X
ejde-984	475	2	25	25	NUM
ejde-984	475	3	]	]	X
ejde-984	475	4	n.	n.	PROPN
ejde-984	475	5	soave	soave	PROPN
ejde-984	475	6	;	;	PUNCT
ejde-984	475	7	normalized	normalize	VERB
ejde-984	475	8	ground	ground	NOUN
ejde-984	475	9	states	state	NOUN
ejde-984	475	10	for	for	ADP
ejde-984	475	11	the	the	DET
ejde-984	475	12	nls	nls	NOUN
ejde-984	475	13	equation	equation	NOUN
ejde-984	475	14	with	with	ADP
ejde-984	475	15	combined	combined	ADJ
ejde-984	475	16	nonlinearities	nonlinearitie	NOUN
ejde-984	475	17	:	:	PUNCT
ejde-984	475	18	the	the	DET
ejde-984	475	19	sobolev	sobolev	ADJ
ejde-984	475	20	critical	critical	ADJ
ejde-984	475	21	case	case	NOUN
ejde-984	475	22	,	,	PUNCT
ejde-984	475	23	j.	j.	PROPN
ejde-984	475	24	funct	funct	PROPN
ejde-984	475	25	.	.	PUNCT
ejde-984	476	1	anal	anal	PROPN
ejde-984	476	2	.	.	PROPN
ejde-984	476	3	,	,	PUNCT
ejde-984	476	4	279	279	NUM
ejde-984	476	5	(	(	PUNCT
ejde-984	476	6	2020	2020	NUM
ejde-984	476	7	)	)	PUNCT
ejde-984	476	8	(	(	PUNCT
ejde-984	476	9	6	6	NUM
ejde-984	476	10	):	):	PUNCT
ejde-984	476	11	42	42	NUM
ejde-984	476	12	.	.	PUNCT
ejde-984	477	1	[	[	X
ejde-984	477	2	26	26	NUM
ejde-984	477	3	]	]	PUNCT
ejde-984	477	4	m.	m.	NOUN
ejde-984	477	5	struwe	struwe	NOUN
ejde-984	477	6	;	;	PUNCT
ejde-984	477	7	variational	variational	ADJ
ejde-984	477	8	methods	method	NOUN
ejde-984	477	9	.	.	PUNCT
ejde-984	478	1	applications	application	NOUN
ejde-984	478	2	to	to	PART
ejde-984	478	3	nonlinear	nonlinear	VERB
ejde-984	478	4	partial	partial	ADJ
ejde-984	478	5	differential	differential	NOUN
ejde-984	478	6	equations	equation	NOUN
ejde-984	478	7	and	and	CCONJ
ejde-984	478	8	hamiltonian	hamiltonian	ADJ
ejde-984	478	9	systems	system	NOUN
ejde-984	478	10	,	,	PUNCT
ejde-984	478	11	2nd	2nd	ADJ
ejde-984	478	12	ed	ed	NOUN
ejde-984	478	13	.	.	PROPN
ejde-984	478	14	,	,	PUNCT
ejde-984	478	15	ser	ser	PROPN
ejde-984	478	16	.	.	PROPN
ejde-984	478	17	ergeb	ergeb	PROPN
ejde-984	478	18	.	.	PUNCT
ejde-984	479	1	math	math	NOUN
ejde-984	479	2	.	.	PUNCT
ejde-984	480	1	grenzgeb	grenzgeb	NOUN
ejde-984	480	2	.	.	PUNCT
ejde-984	481	1	3	3	X
ejde-984	481	2	.	.	NUM
ejde-984	481	3	folge	folge	PROPN
ejde-984	481	4	,	,	PUNCT
ejde-984	481	5	berlin	berlin	PROPN
ejde-984	481	6	:	:	PUNCT
ejde-984	481	7	springer	springer	NOUN
ejde-984	481	8	.	.	PUNCT
ejde-984	482	1	34	34	NUM
ejde-984	482	2	,	,	PUNCT
ejde-984	482	3	1996	1996	NUM
ejde-984	482	4	.	.	PUNCT
ejde-984	483	1	[	[	X
ejde-984	483	2	27	27	NUM
ejde-984	483	3	]	]	PUNCT
ejde-984	483	4	j.	j.	PROPN
ejde-984	483	5	wei	wei	PROPN
ejde-984	483	6	,	,	PUNCT
ejde-984	483	7	y.	y.	PROPN
ejde-984	483	8	wu	wu	PROPN
ejde-984	483	9	;	;	PUNCT
ejde-984	483	10	normalized	normalize	VERB
ejde-984	483	11	solutions	solution	NOUN
ejde-984	483	12	for	for	ADP
ejde-984	483	13	schrödinger	schrödinger	NOUN
ejde-984	483	14	equations	equation	NOUN
ejde-984	483	15	with	with	ADP
ejde-984	483	16	critical	critical	ADJ
ejde-984	483	17	sobolev	sobolev	NOUN
ejde-984	483	18	exponent	exponent	NOUN
ejde-984	483	19	and	and	CCONJ
ejde-984	483	20	mixed	mixed	ADJ
ejde-984	483	21	nonlinearities	nonlinearitie	NOUN
ejde-984	483	22	,	,	PUNCT
ejde-984	483	23	j.	j.	PROPN
ejde-984	483	24	funct	funct	PROPN
ejde-984	483	25	.	.	PUNCT
ejde-984	484	1	anal	anal	PROPN
ejde-984	484	2	.	.	PROPN
ejde-984	484	3	,	,	PUNCT
ejde-984	484	4	283	283	NUM
ejde-984	484	5	(	(	PUNCT
ejde-984	484	6	2022	2022	NUM
ejde-984	484	7	)	)	PUNCT
ejde-984	484	8	(	(	PUNCT
ejde-984	484	9	6	6	NUM
ejde-984	484	10	):	):	PUNCT
ejde-984	484	11	46	46	NUM
ejde-984	484	12	.	.	PUNCT
ejde-984	485	1	[	[	X
ejde-984	485	2	28	28	NUM
ejde-984	485	3	]	]	X
ejde-984	485	4	h.	h.	PROPN
ejde-984	485	5	ye	ye	PROPN
ejde-984	485	6	,	,	PUNCT
ejde-984	485	7	the	the	DET
ejde-984	485	8	sharp	sharp	ADJ
ejde-984	485	9	existence	existence	NOUN
ejde-984	485	10	of	of	ADP
ejde-984	485	11	constrained	constrain	VERB
ejde-984	485	12	minimizers	minimizer	NOUN
ejde-984	485	13	for	for	ADP
ejde-984	485	14	a	a	DET
ejde-984	485	15	class	class	NOUN
ejde-984	485	16	of	of	ADP
ejde-984	485	17	nonlinear	nonlinear	PROPN
ejde-984	485	18	kirchhoff	kirchhoff	PROPN
ejde-984	485	19	equations	equations	PROPN
ejde-984	485	20	,	,	PUNCT
ejde-984	485	21	math	math	NOUN
ejde-984	485	22	.	.	PUNCT
ejde-984	486	1	methods	method	NOUN
ejde-984	486	2	appl	appl	PROPN
ejde-984	486	3	.	.	PUNCT
ejde-984	487	1	sci	sci	PROPN
ejde-984	487	2	.	.	PROPN
ejde-984	487	3	,	,	PUNCT
ejde-984	487	4	38	38	NUM
ejde-984	487	5	(	(	PUNCT
ejde-984	487	6	2015	2015	NUM
ejde-984	487	7	)	)	PUNCT
ejde-984	487	8	(	(	PUNCT
ejde-984	487	9	13	13	NUM
ejde-984	487	10	):	):	PUNCT
ejde-984	487	11	2663–2679	2663–2679	NUM
ejde-984	487	12	.	.	PUNCT
ejde-984	488	1	[	[	X
ejde-984	488	2	29	29	NUM
ejde-984	488	3	]	]	PUNCT
ejde-984	488	4	m.	m.	NOUN
ejde-984	488	5	zhen	zhen	PROPN
ejde-984	488	6	,	,	PUNCT
ejde-984	488	7	b.	b.	PROPN
ejde-984	488	8	zhang	zhang	PROPN
ejde-984	488	9	;	;	PUNCT
ejde-984	488	10	normalized	normalize	VERB
ejde-984	488	11	ground	ground	NOUN
ejde-984	488	12	states	state	NOUN
ejde-984	488	13	for	for	ADP
ejde-984	488	14	the	the	DET
ejde-984	488	15	critical	critical	ADJ
ejde-984	488	16	fractional	fractional	ADJ
ejde-984	488	17	nls	nls	NOUN
ejde-984	488	18	equation	equation	NOUN
ejde-984	488	19	with	with	ADP
ejde-984	488	20	a	a	DET
ejde-984	488	21	perturbation	perturbation	NOUN
ejde-984	488	22	,	,	PUNCT
ejde-984	488	23	rev	rev	PROPN
ejde-984	488	24	.	.	PROPN
ejde-984	488	25	mat	mat	PROPN
ejde-984	488	26	.	.	PROPN
ejde-984	488	27	complut	complut	PROPN
ejde-984	488	28	.	.	PUNCT
ejde-984	488	29	,	,	PUNCT
ejde-984	488	30	35	35	NUM
ejde-984	488	31	(	(	PUNCT
ejde-984	488	32	2022	2022	NUM
ejde-984	488	33	)	)	PUNCT
ejde-984	488	34	(	(	PUNCT
ejde-984	488	35	1	1	NUM
ejde-984	488	36	):	):	PUNCT
ejde-984	488	37	89–132	89–132	PROPN
ejde-984	488	38	.	.	PUNCT
ejde-984	489	1	zhenyu	zhenyu	PROPN
ejde-984	489	2	guo	guo	PROPN
ejde-984	489	3	school	school	PROPN
ejde-984	489	4	of	of	ADP
ejde-984	489	5	mathematics	mathematic	NOUN
ejde-984	489	6	,	,	PUNCT
ejde-984	489	7	liaoning	liaoning	NOUN
ejde-984	489	8	normal	normal	ADJ
ejde-984	489	9	university	university	NOUN
ejde-984	489	10	,	,	PUNCT
ejde-984	489	11	dalian	dalian	PROPN
ejde-984	489	12	116029	116029	NUM
ejde-984	489	13	,	,	PUNCT
ejde-984	489	14	china	china	PROPN
ejde-984	489	15	email	email	NOUN
ejde-984	489	16	address	address	NOUN
ejde-984	489	17	:	:	PUNCT
ejde-984	490	1	guozy@163.com	guozy@163.com	PROPN
ejde-984	490	2	tianqing	tianqe	VERB
ejde-984	490	3	zhang	zhang	PROPN
ejde-984	490	4	shenyang	shenyang	PROPN
ejde-984	490	5	railway	railway	PROPN
ejde-984	490	6	no	no	PROPN
ejde-984	490	7	.	.	PROPN
ejde-984	490	8	5	5	NUM
ejde-984	490	9	primary	primary	ADJ
ejde-984	490	10	school	school	NOUN
ejde-984	490	11	,	,	PUNCT
ejde-984	490	12	shenyang	shenyang	PROPN
ejde-984	490	13	110001	110001	NUM
ejde-984	490	14	,	,	PUNCT
ejde-984	490	15	china	china	PROPN
ejde-984	490	16	email	email	NOUN
ejde-984	490	17	address	address	NOUN
ejde-984	490	18	:	:	PUNCT
ejde-984	490	19	tqingzhang@163.com	tqingzhang@163.com	X
ejde-984	490	20	1	1	X
ejde-984	490	21	.	.	PUNCT
ejde-984	490	22	introduction	introduction	NOUN
ejde-984	490	23	2	2	NUM
ejde-984	490	24	.	.	PUNCT
ejde-984	490	25	preliminaries	preliminary	NOUN
ejde-984	490	26	and	and	CCONJ
ejde-984	490	27	main	main	ADJ
ejde-984	490	28	results	result	NOUN
ejde-984	490	29	3	3	NUM
ejde-984	490	30	.	.	PUNCT
ejde-984	490	31	subcritical	subcritical	ADJ
ejde-984	490	32	case	case	NOUN
ejde-984	490	33	acknowledgents	acknowledgent	NOUN
ejde-984	490	34	references	reference	NOUN