id	sid	tid	token	lemma	pos
ejde-610	1	1	electronic	electronic	ADJ
ejde-610	1	2	journal	journal	NOUN
ejde-610	1	3	of	of	ADP
ejde-610	1	4	differential	differential	ADJ
ejde-610	1	5	equations	equation	NOUN
ejde-610	1	6	,	,	PUNCT
ejde-610	1	7	vol	vol	NOUN
ejde-610	1	8	.	.	NOUN
ejde-610	1	9	2024	2024	NUM
ejde-610	1	10	(	(	PUNCT
ejde-610	1	11	2024	2024	NUM
ejde-610	1	12	)	)	PUNCT
ejde-610	1	13	,	,	PUNCT
ejde-610	1	14	no	no	INTJ
ejde-610	1	15	.	.	NOUN
ejde-610	1	16	18	18	NUM
ejde-610	1	17	,	,	PUNCT
ejde-610	1	18	pp	pp	PROPN
ejde-610	1	19	.	.	PUNCT
ejde-610	2	1	1–11	1–11	PROPN
ejde-610	2	2	.	.	PUNCT
ejde-610	3	1	issn	issn	PROPN
ejde-610	3	2	:	:	PUNCT
ejde-610	3	3	1072	1072	NUM
ejde-610	3	4	-	-	SYM
ejde-610	3	5	6691	6691	NUM
ejde-610	3	6	.	.	PUNCT
ejde-610	4	1	url	url	PROPN
ejde-610	4	2	:	:	PUNCT
ejde-610	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-610	4	4	,	,	PUNCT
ejde-610	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-610	4	6	doi	doi	PROPN
ejde-610	4	7	:	:	PUNCT
ejde-610	4	8	10.58997	10.58997	NUM
ejde-610	4	9	/	/	SYM
ejde-610	4	10	ejde.2024.18	ejde.2024.18	ADJ
ejde-610	4	11	existence	existence	NOUN
ejde-610	4	12	of	of	ADP
ejde-610	4	13	high	high	ADJ
ejde-610	4	14	energy	energy	NOUN
ejde-610	4	15	solutions	solution	NOUN
ejde-610	4	16	for	for	ADP
ejde-610	4	17	superlinear	superlinear	NOUN
ejde-610	4	18	coupled	couple	VERB
ejde-610	4	19	klein	klein	PROPN
ejde-610	4	20	-	-	PUNCT
ejde-610	4	21	gordons	gordon	NOUN
ejde-610	4	22	and	and	CCONJ
ejde-610	4	23	born	bear	VERB
ejde-610	4	24	-	-	PUNCT
ejde-610	4	25	infeld	infeld	NOUN
ejde-610	4	26	equations	equation	NOUN
ejde-610	4	27	lixia	lixia	PROPN
ejde-610	4	28	wang	wang	PROPN
ejde-610	4	29	,	,	PUNCT
ejde-610	4	30	pingping	pingpe	VERB
ejde-610	4	31	zhao	zhao	PROPN
ejde-610	4	32	,	,	PUNCT
ejde-610	4	33	dong	dong	PROPN
ejde-610	4	34	zhang	zhang	PROPN
ejde-610	4	35	communicated	communicate	VERB
ejde-610	4	36	by	by	ADP
ejde-610	4	37	marko	marko	PROPN
ejde-610	4	38	squassina	squassina	PROPN
ejde-610	4	39	abstract	abstract	PROPN
ejde-610	4	40	.	.	PUNCT
ejde-610	5	1	in	in	ADP
ejde-610	5	2	this	this	DET
ejde-610	5	3	article	article	NOUN
ejde-610	5	4	,	,	PUNCT
ejde-610	5	5	we	we	PRON
ejde-610	5	6	study	study	VERB
ejde-610	5	7	the	the	DET
ejde-610	5	8	system	system	NOUN
ejde-610	5	9	of	of	ADP
ejde-610	5	10	klein	klein	PROPN
ejde-610	5	11	-	-	PUNCT
ejde-610	5	12	gordon	gordon	PROPN
ejde-610	5	13	and	and	CCONJ
ejde-610	5	14	borninfeld	borninfeld	NOUN
ejde-610	5	15	equations	equation	NOUN
ejde-610	5	16	−∆u+	−∆u+	VERB
ejde-610	5	17	v	v	NOUN
ejde-610	5	18	(	(	PUNCT
ejde-610	5	19	x)u−	x)u−	PROPN
ejde-610	5	20	(	(	PUNCT
ejde-610	5	21	2ω	2ω	PROPN
ejde-610	5	22	+	+	X
ejde-610	5	23	φ)φu	φ)φu	PROPN
ejde-610	5	24	=	=	SYM
ejde-610	5	25	f(x	f(x	PROPN
ejde-610	5	26	,	,	PUNCT
ejde-610	5	27	u	u	NOUN
ejde-610	5	28	)	)	PUNCT
ejde-610	5	29	,	,	PUNCT
ejde-610	5	30	x	x	PROPN
ejde-610	5	31	∈	∈	PROPN
ejde-610	5	32	r3	r3	PROPN
ejde-610	5	33	,	,	PUNCT
ejde-610	5	34	∆φ+	∆φ+	PROPN
ejde-610	5	35	β∆4φ	β∆4φ	PROPN
ejde-610	5	36	=	=	SYM
ejde-610	5	37	4π(ω	4π(ω	NUM
ejde-610	5	38	+	+	CCONJ
ejde-610	6	1	φ)u2	φ)u2	PROPN
ejde-610	6	2	,	,	PUNCT
ejde-610	6	3	x	x	SYM
ejde-610	6	4	∈	∈	PROPN
ejde-610	6	5	r3	r3	PROPN
ejde-610	6	6	,	,	PUNCT
ejde-610	6	7	where	where	SCONJ
ejde-610	6	8	∆4φ	∆4φ	PROPN
ejde-610	6	9	=	=	SYM
ejde-610	6	10	div(|∇φ|2∇φ	div(|∇φ|2∇φ	PROPN
ejde-610	6	11	)	)	PUNCT
ejde-610	6	12	,	,	PUNCT
ejde-610	6	13	ω	ω	PROPN
ejde-610	6	14	is	be	AUX
ejde-610	6	15	a	a	DET
ejde-610	6	16	positive	positive	ADJ
ejde-610	6	17	constant	constant	NOUN
ejde-610	6	18	.	.	PUNCT
ejde-610	7	1	assuming	assume	VERB
ejde-610	7	2	that	that	SCONJ
ejde-610	7	3	the	the	DET
ejde-610	7	4	primitive	primitive	NOUN
ejde-610	7	5	of	of	ADP
ejde-610	7	6	f(x	f(x	PROPN
ejde-610	7	7	,	,	PUNCT
ejde-610	7	8	u	u	NOUN
ejde-610	7	9	)	)	PUNCT
ejde-610	7	10	is	be	AUX
ejde-610	7	11	of	of	ADP
ejde-610	7	12	2	2	NUM
ejde-610	7	13	-	-	PUNCT
ejde-610	7	14	superlinear	superlinear	NOUN
ejde-610	7	15	growth	growth	NOUN
ejde-610	7	16	in	in	ADP
ejde-610	7	17	u	u	NOUN
ejde-610	7	18	at	at	ADP
ejde-610	7	19	infinity	infinity	NOUN
ejde-610	7	20	,	,	PUNCT
ejde-610	7	21	we	we	PRON
ejde-610	7	22	prove	prove	VERB
ejde-610	7	23	the	the	DET
ejde-610	7	24	existence	existence	NOUN
ejde-610	7	25	of	of	ADP
ejde-610	7	26	multiple	multiple	ADJ
ejde-610	7	27	solutions	solution	NOUN
ejde-610	7	28	using	use	VERB
ejde-610	7	29	the	the	DET
ejde-610	7	30	fountain	fountain	NOUN
ejde-610	7	31	theorem	theorem	VERB
ejde-610	7	32	.	.	PUNCT
ejde-610	8	1	here	here	ADV
ejde-610	8	2	the	the	DET
ejde-610	8	3	potential	potential	ADJ
ejde-610	8	4	v	v	NOUN
ejde-610	8	5	are	be	AUX
ejde-610	8	6	allowed	allow	VERB
ejde-610	8	7	to	to	PART
ejde-610	8	8	be	be	AUX
ejde-610	8	9	a	a	DET
ejde-610	8	10	sign	sign	NOUN
ejde-610	8	11	-	-	PUNCT
ejde-610	8	12	changing	change	VERB
ejde-610	8	13	function	function	NOUN
ejde-610	8	14	.	.	PUNCT
ejde-610	9	1	1	1	X
ejde-610	9	2	.	.	X
ejde-610	9	3	introduction	introduction	NOUN
ejde-610	9	4	and	and	CCONJ
ejde-610	9	5	main	main	ADJ
ejde-610	9	6	results	result	NOUN
ejde-610	9	7	in	in	ADP
ejde-610	9	8	this	this	DET
ejde-610	9	9	article	article	NOUN
ejde-610	9	10	,	,	PUNCT
ejde-610	9	11	we	we	PRON
ejde-610	9	12	study	study	VERB
ejde-610	9	13	klein	klein	PROPN
ejde-610	9	14	-	-	PUNCT
ejde-610	9	15	gordon	gordon	PROPN
ejde-610	9	16	equation	equation	NOUN
ejde-610	9	17	using	use	VERB
ejde-610	9	18	born	bear	VERB
ejde-610	9	19	-	-	PUNCT
ejde-610	9	20	infeld	infeld	NOUN
ejde-610	9	21	theory	theory	NOUN
ejde-610	9	22	−∆u+	−∆u+	NOUN
ejde-610	9	23	v	v	NOUN
ejde-610	9	24	(	(	PUNCT
ejde-610	9	25	x)u−	x)u−	PROPN
ejde-610	9	26	(	(	PUNCT
ejde-610	9	27	2ω	2ω	PROPN
ejde-610	9	28	+	+	X
ejde-610	10	1	φ)φu	φ)φu	PROPN
ejde-610	10	2	=	=	SYM
ejde-610	10	3	f(x	f(x	PROPN
ejde-610	10	4	,	,	PUNCT
ejde-610	10	5	u	u	NOUN
ejde-610	10	6	)	)	PUNCT
ejde-610	10	7	,	,	PUNCT
ejde-610	10	8	x	x	PROPN
ejde-610	10	9	∈	∈	PROPN
ejde-610	10	10	r3	r3	PROPN
ejde-610	10	11	,	,	PUNCT
ejde-610	10	12	∆φ+	∆φ+	PROPN
ejde-610	10	13	β∆4φ	β∆4φ	PROPN
ejde-610	10	14	=	=	SYM
ejde-610	10	15	4π(ω	4π(ω	NUM
ejde-610	10	16	+	+	CCONJ
ejde-610	10	17	φ)u2	φ)u2	PROPN
ejde-610	10	18	,	,	PUNCT
ejde-610	10	19	x	x	SYM
ejde-610	10	20	∈	∈	PROPN
ejde-610	10	21	r3	r3	PROPN
ejde-610	10	22	,	,	PUNCT
ejde-610	10	23	(	(	PUNCT
ejde-610	10	24	1.1	1.1	NUM
ejde-610	10	25	)	)	PUNCT
ejde-610	10	26	where	where	SCONJ
ejde-610	10	27	ω	ω	NOUN
ejde-610	10	28	is	be	AUX
ejde-610	10	29	a	a	DET
ejde-610	10	30	positive	positive	ADJ
ejde-610	10	31	constant	constant	ADJ
ejde-610	10	32	,	,	PUNCT
ejde-610	10	33	v	v	NOUN
ejde-610	10	34	∈	∈	PROPN
ejde-610	10	35	c(r3,r	c(r3,r	PROPN
ejde-610	10	36	)	)	PUNCT
ejde-610	10	37	,	,	PUNCT
ejde-610	10	38	and	and	CCONJ
ejde-610	10	39	f	f	PROPN
ejde-610	10	40	∈	∈	PROPN
ejde-610	10	41	c(r3	c(r3	NOUN
ejde-610	10	42	×r	×r	NOUN
ejde-610	10	43	,	,	PUNCT
ejde-610	10	44	r	r	NOUN
ejde-610	10	45	)	)	PUNCT
ejde-610	10	46	.	.	PUNCT
ejde-610	11	1	by	by	ADP
ejde-610	11	2	using	use	VERB
ejde-610	11	3	the	the	DET
ejde-610	11	4	local	local	ADJ
ejde-610	11	5	linking	linking	NOUN
ejde-610	11	6	theorem	theorem	NOUN
ejde-610	11	7	and	and	CCONJ
ejde-610	11	8	the	the	DET
ejde-610	11	9	fountain	fountain	NOUN
ejde-610	11	10	theorem	theorem	VERB
ejde-610	11	11	,	,	PUNCT
ejde-610	11	12	we	we	PRON
ejde-610	11	13	obtain	obtain	VERB
ejde-610	11	14	multiple	multiple	ADJ
ejde-610	11	15	solutions	solution	NOUN
ejde-610	11	16	for	for	ADP
ejde-610	11	17	(	(	PUNCT
ejde-610	11	18	1.1	1.1	NUM
ejde-610	11	19	)	)	PUNCT
ejde-610	11	20	.	.	PUNCT
ejde-610	12	1	it	it	PRON
ejde-610	12	2	is	be	AUX
ejde-610	12	3	well	well	ADV
ejde-610	12	4	known	know	VERB
ejde-610	12	5	that	that	SCONJ
ejde-610	12	6	klein	klein	PROPN
ejde-610	12	7	-	-	PUNCT
ejde-610	12	8	gordon	gordon	PROPN
ejde-610	12	9	equation	equation	NOUN
ejde-610	12	10	can	can	AUX
ejde-610	12	11	be	be	AUX
ejde-610	12	12	used	use	VERB
ejde-610	12	13	in	in	ADP
ejde-610	12	14	theory	theory	NOUN
ejde-610	12	15	of	of	ADP
ejde-610	12	16	electrically	electrically	ADV
ejde-610	12	17	charged	charge	VERB
ejde-610	12	18	fields	field	NOUN
ejde-610	12	19	[	[	X
ejde-610	12	20	16	16	NUM
ejde-610	12	21	]	]	PUNCT
ejde-610	12	22	.	.	PUNCT
ejde-610	13	1	the	the	DET
ejde-610	13	2	born	bear	VERB
ejde-610	13	3	-	-	PUNCT
ejde-610	13	4	infeld	infeld	NOUN
ejde-610	13	5	theory	theory	NOUN
ejde-610	13	6	is	be	AUX
ejde-610	13	7	proposed	propose	VERB
ejde-610	13	8	by	by	ADP
ejde-610	13	9	born	bear	VERB
ejde-610	13	10	[	[	X
ejde-610	13	11	7	7	NUM
ejde-610	13	12	,	,	PUNCT
ejde-610	13	13	8	8	NUM
ejde-610	13	14	,	,	PUNCT
ejde-610	13	15	9	9	NUM
ejde-610	13	16	]	]	PUNCT
ejde-610	13	17	to	to	PART
ejde-610	13	18	overcome	overcome	VERB
ejde-610	13	19	the	the	DET
ejde-610	13	20	infinite	infinite	ADJ
ejde-610	13	21	energy	energy	NOUN
ejde-610	13	22	problem	problem	NOUN
ejde-610	13	23	associated	associate	VERB
ejde-610	13	24	with	with	ADP
ejde-610	13	25	a	a	DET
ejde-610	13	26	point	point	NOUN
ejde-610	13	27	-	-	PUNCT
ejde-610	13	28	charge	charge	NOUN
ejde-610	13	29	source	source	NOUN
ejde-610	13	30	in	in	ADP
ejde-610	13	31	the	the	DET
ejde-610	13	32	original	original	ADJ
ejde-610	13	33	maxwell	maxwell	PROPN
ejde-610	13	34	theory	theory	NOUN
ejde-610	13	35	.	.	PUNCT
ejde-610	14	1	the	the	DET
ejde-610	14	2	presence	presence	NOUN
ejde-610	14	3	of	of	ADP
ejde-610	14	4	the	the	DET
ejde-610	14	5	nonlinear	nonlinear	ADJ
ejde-610	14	6	term	term	NOUN
ejde-610	14	7	f	f	PROPN
ejde-610	14	8	simulates	simulate	VERB
ejde-610	14	9	the	the	DET
ejde-610	14	10	interaction	interaction	NOUN
ejde-610	14	11	between	between	ADP
ejde-610	14	12	many	many	ADJ
ejde-610	14	13	particles	particle	NOUN
ejde-610	14	14	or	or	CCONJ
ejde-610	14	15	external	external	ADJ
ejde-610	14	16	nonlinear	nonlinear	ADJ
ejde-610	14	17	perturbations	perturbation	NOUN
ejde-610	14	18	.	.	PUNCT
ejde-610	15	1	for	for	ADP
ejde-610	15	2	more	more	ADJ
ejde-610	15	3	details	detail	NOUN
ejde-610	15	4	in	in	ADP
ejde-610	15	5	the	the	DET
ejde-610	15	6	physical	physical	ADJ
ejde-610	15	7	aspects	aspect	NOUN
ejde-610	15	8	,	,	PUNCT
ejde-610	15	9	we	we	PRON
ejde-610	15	10	refer	refer	VERB
ejde-610	15	11	the	the	DET
ejde-610	15	12	readers	reader	NOUN
ejde-610	15	13	to	to	ADP
ejde-610	15	14	[	[	X
ejde-610	15	15	5	5	NUM
ejde-610	15	16	,	,	PUNCT
ejde-610	15	17	10	10	NUM
ejde-610	15	18	,	,	PUNCT
ejde-610	15	19	17	17	NUM
ejde-610	15	20	,	,	PUNCT
ejde-610	15	21	21	21	NUM
ejde-610	15	22	,	,	PUNCT
ejde-610	15	23	30	30	NUM
ejde-610	15	24	]	]	PUNCT
ejde-610	15	25	.	.	PUNCT
ejde-610	16	1	in	in	ADP
ejde-610	16	2	recent	recent	ADJ
ejde-610	16	3	years	year	NOUN
ejde-610	16	4	,	,	PUNCT
ejde-610	16	5	the	the	DET
ejde-610	16	6	born	bear	VERB
ejde-610	16	7	-	-	PUNCT
ejde-610	16	8	infeld	infeld	NOUN
ejde-610	16	9	nonlinear	nonlinear	ADJ
ejde-610	16	10	electromagnetism	electromagnetism	NOUN
ejde-610	16	11	has	have	AUX
ejde-610	16	12	become	become	VERB
ejde-610	16	13	more	more	ADV
ejde-610	16	14	important	important	ADJ
ejde-610	16	15	since	since	SCONJ
ejde-610	16	16	its	its	PRON
ejde-610	16	17	relevance	relevance	NOUN
ejde-610	16	18	in	in	ADP
ejde-610	16	19	the	the	DET
ejde-610	16	20	theory	theory	NOUN
ejde-610	16	21	of	of	ADP
ejde-610	16	22	superstring	superstring	NOUN
ejde-610	16	23	and	and	CCONJ
ejde-610	16	24	membranes	membrane	NOUN
ejde-610	16	25	.	.	PUNCT
ejde-610	17	1	by	by	ADP
ejde-610	17	2	using	use	VERB
ejde-610	17	3	variational	variational	ADJ
ejde-610	17	4	methods	method	NOUN
ejde-610	17	5	,	,	PUNCT
ejde-610	17	6	several	several	ADJ
ejde-610	17	7	existence	existence	NOUN
ejde-610	17	8	results	result	VERB
ejde-610	17	9	for	for	ADP
ejde-610	17	10	problem	problem	NOUN
ejde-610	17	11	(	(	PUNCT
ejde-610	17	12	1.1	1.1	NUM
ejde-610	17	13	)	)	PUNCT
ejde-610	17	14	have	have	AUX
ejde-610	17	15	been	be	AUX
ejde-610	17	16	found	find	VERB
ejde-610	17	17	with	with	ADP
ejde-610	17	18	constant	constant	ADJ
ejde-610	17	19	potential	potential	ADJ
ejde-610	17	20	v	v	NOUN
ejde-610	17	21	(	(	PUNCT
ejde-610	17	22	x	x	NOUN
ejde-610	17	23	)	)	PUNCT
ejde-610	17	24	=	=	SYM
ejde-610	17	25	m2	m2	PROPN
ejde-610	17	26	−	−	PROPN
ejde-610	17	27	ω2	ω2	PROPN
ejde-610	17	28	.	.	PUNCT
ejde-610	18	1	next	next	ADV
ejde-610	18	2	we	we	PRON
ejde-610	18	3	recall	recall	VERB
ejde-610	18	4	some	some	PRON
ejde-610	18	5	of	of	ADP
ejde-610	18	6	them	they	PRON
ejde-610	18	7	.	.	PUNCT
ejde-610	19	1	2020	2020	NUM
ejde-610	19	2	mathematics	mathematic	NOUN
ejde-610	19	3	subject	subject	ADJ
ejde-610	19	4	classification	classification	NOUN
ejde-610	19	5	.	.	PUNCT
ejde-610	20	1	35b33	35b33	NUM
ejde-610	20	2	,	,	PUNCT
ejde-610	20	3	35j65	35j65	NUM
ejde-610	20	4	,	,	PUNCT
ejde-610	20	5	35q55	35q55	NUM
ejde-610	20	6	.	.	PUNCT
ejde-610	21	1	key	key	ADJ
ejde-610	21	2	words	word	NOUN
ejde-610	21	3	and	and	CCONJ
ejde-610	21	4	phrases	phrase	NOUN
ejde-610	21	5	.	.	PUNCT
ejde-610	22	1	klein	klein	PROPN
ejde-610	22	2	-	-	PUNCT
ejde-610	22	3	gordon	gordon	PROPN
ejde-610	22	4	equation	equation	NOUN
ejde-610	22	5	;	;	PUNCT
ejde-610	22	6	born	bear	VERB
ejde-610	22	7	-	-	PUNCT
ejde-610	22	8	infeld	infeld	NOUN
ejde-610	22	9	theory	theory	NOUN
ejde-610	22	10	;	;	PUNCT
ejde-610	22	11	superlinear	superlinear	NOUN
ejde-610	22	12	;	;	PUNCT
ejde-610	22	13	fountain	fountain	NOUN
ejde-610	22	14	theorem	theorem	NOUN
ejde-610	22	15	.	.	PUNCT
ejde-610	23	1	©	©	NOUN
ejde-610	23	2	2024	2024	NUM
ejde-610	23	3	.	.	PUNCT
ejde-610	24	1	this	this	DET
ejde-610	24	2	work	work	NOUN
ejde-610	24	3	is	be	AUX
ejde-610	24	4	licensed	license	VERB
ejde-610	24	5	under	under	ADP
ejde-610	24	6	a	a	DET
ejde-610	24	7	cc	cc	NOUN
ejde-610	24	8	by	by	ADP
ejde-610	24	9	4.0	4.0	NUM
ejde-610	24	10	license	license	NOUN
ejde-610	24	11	.	.	PUNCT
ejde-610	25	1	submitted	submit	VERB
ejde-610	25	2	october	october	PROPN
ejde-610	25	3	30	30	NUM
ejde-610	25	4	,	,	PUNCT
ejde-610	25	5	2023	2023	NUM
ejde-610	25	6	.	.	PUNCT
ejde-610	26	1	published	publish	VERB
ejde-610	26	2	february	february	PROPN
ejde-610	26	3	16	16	NUM
ejde-610	26	4	,	,	PUNCT
ejde-610	26	5	2024	2024	NUM
ejde-610	26	6	.	.	PUNCT
ejde-610	26	7	1	1	NUM
ejde-610	26	8	2	2	NUM
ejde-610	26	9	l.	l.	PROPN
ejde-610	26	10	wang	wang	PROPN
ejde-610	26	11	,	,	PUNCT
ejde-610	26	12	p.	p.	PROPN
ejde-610	26	13	zhao	zhao	PROPN
ejde-610	26	14	,	,	PUNCT
ejde-610	26	15	d.	d.	PROPN
ejde-610	26	16	zhang	zhang	PROPN
ejde-610	26	17	ejde-2024/18	ejde-2024/18	PROPN
ejde-610	26	18	in	in	ADP
ejde-610	26	19	2002	2002	NUM
ejde-610	26	20	,	,	PUNCT
ejde-610	26	21	d’avenia	d’avenia	PROPN
ejde-610	26	22	et	et	PROPN
ejde-610	26	23	al	al	PROPN
ejde-610	27	1	[	[	X
ejde-610	27	2	15	15	NUM
ejde-610	27	3	]	]	PUNCT
ejde-610	27	4	considered	consider	VERB
ejde-610	27	5	for	for	ADP
ejde-610	27	6	the	the	DET
ejde-610	27	7	klein	klein	PROPN
ejde-610	27	8	-	-	PUNCT
ejde-610	27	9	gordon	gordon	PROPN
ejde-610	27	10	equation	equation	NOUN
ejde-610	27	11	on	on	ADP
ejde-610	27	12	r3	r3	PROPN
ejde-610	27	13	−∆u+	−∆u+	PROPN
ejde-610	28	1	[	[	X
ejde-610	28	2	m2	m2	PROPN
ejde-610	28	3	−	−	PROPN
ejde-610	28	4	(	(	PUNCT
ejde-610	28	5	ω	ω	PROPN
ejde-610	28	6	+	+	NUM
ejde-610	28	7	φ)2]φu	φ)2]φu	NOUN
ejde-610	28	8	=	=	SYM
ejde-610	28	9	f(x	f(x	PROPN
ejde-610	28	10	,	,	PUNCT
ejde-610	28	11	u	u	NOUN
ejde-610	28	12	)	)	PUNCT
ejde-610	28	13	,	,	PUNCT
ejde-610	28	14	x	x	PROPN
ejde-610	28	15	∈	∈	PROPN
ejde-610	28	16	r3	r3	PROPN
ejde-610	28	17	,	,	PUNCT
ejde-610	28	18	∆φ+	∆φ+	PROPN
ejde-610	28	19	β∆4φ	β∆4φ	PROPN
ejde-610	28	20	=	=	SYM
ejde-610	28	21	4π(ω	4π(ω	NUM
ejde-610	28	22	+	+	CCONJ
ejde-610	28	23	φ)u2	φ)u2	PROPN
ejde-610	28	24	,	,	PUNCT
ejde-610	28	25	x	x	SYM
ejde-610	28	26	∈	∈	PROPN
ejde-610	28	27	r3	r3	PROPN
ejde-610	28	28	,	,	PUNCT
ejde-610	28	29	(	(	PUNCT
ejde-610	28	30	1.2	1.2	NUM
ejde-610	28	31	)	)	PUNCT
ejde-610	28	32	with	with	ADP
ejde-610	28	33	pure	pure	ADJ
ejde-610	28	34	power	power	NOUN
ejde-610	28	35	nonlinearity	nonlinearity	NOUN
ejde-610	28	36	,	,	PUNCT
ejde-610	28	37	i.e.	i.e.	X
ejde-610	28	38	,	,	PUNCT
ejde-610	28	39	f(x	f(x	PROPN
ejde-610	28	40	,	,	PUNCT
ejde-610	28	41	u	u	NOUN
ejde-610	28	42	)	)	PUNCT
ejde-610	28	43	=	=	SYM
ejde-610	28	44	|u|p−2u	|u|p−2u	PROPN
ejde-610	28	45	,	,	PUNCT
ejde-610	28	46	where	where	SCONJ
ejde-610	28	47	ω	ω	PROPN
ejde-610	28	48	and	and	CCONJ
ejde-610	28	49	m	m	PROPN
ejde-610	28	50	are	be	AUX
ejde-610	28	51	constants	constant	NOUN
ejde-610	28	52	.	.	PUNCT
ejde-610	29	1	by	by	ADP
ejde-610	29	2	using	use	VERB
ejde-610	29	3	the	the	DET
ejde-610	29	4	mountain	mountain	NOUN
ejde-610	29	5	pass	pass	NOUN
ejde-610	29	6	theorem	theorem	NOUN
ejde-610	29	7	,	,	PUNCT
ejde-610	29	8	they	they	PRON
ejde-610	29	9	proved	prove	VERB
ejde-610	29	10	that	that	SCONJ
ejde-610	29	11	(	(	PUNCT
ejde-610	29	12	1.2	1.2	NUM
ejde-610	29	13	)	)	PUNCT
ejde-610	29	14	has	have	VERB
ejde-610	29	15	infinitely	infinitely	ADV
ejde-610	29	16	many	many	ADJ
ejde-610	29	17	radially	radially	ADV
ejde-610	29	18	symmetric	symmetric	ADJ
ejde-610	29	19	solutions	solution	NOUN
ejde-610	29	20	under	under	ADP
ejde-610	29	21	the	the	DET
ejde-610	29	22	assumptions	assumption	NOUN
ejde-610	29	23	that	that	PRON
ejde-610	29	24	|m|	|m|	VERB
ejde-610	29	25	>	>	X
ejde-610	29	26	ω	ω	PROPN
ejde-610	29	27	and	and	CCONJ
ejde-610	29	28	4	4	NUM
ejde-610	29	29	<	<	X
ejde-610	29	30	p	p	X
ejde-610	29	31	<	<	X
ejde-610	29	32	6	6	NUM
ejde-610	29	33	.	.	PUNCT
ejde-610	29	34	mugnai	mugnai	ADJ
ejde-610	29	35	[	[	X
ejde-610	29	36	21	21	NUM
ejde-610	29	37	]	]	PUNCT
ejde-610	29	38	covered	cover	VERB
ejde-610	29	39	the	the	DET
ejde-610	29	40	case	case	NOUN
ejde-610	29	41	2	2	NUM
ejde-610	29	42	<	<	X
ejde-610	29	43	p	p	X
ejde-610	29	44	≤	≤	NOUN
ejde-610	29	45	4	4	NUM
ejde-610	29	46	assuming	assume	VERB
ejde-610	29	47	√	√	PROPN
ejde-610	29	48	p−2	p−2	PROPN
ejde-610	29	49	2	2	NUM
ejde-610	29	50	|m|	|m|	VERB
ejde-610	29	51	>	>	X
ejde-610	29	52	ω	ω	PROPN
ejde-610	29	53	>	>	X
ejde-610	29	54	0	0	X
ejde-610	29	55	.	.	PUNCT
ejde-610	30	1	later	later	ADV
ejde-610	30	2	,	,	PUNCT
ejde-610	30	3	for	for	ADP
ejde-610	30	4	f(x	f(x	PROPN
ejde-610	30	5	,	,	PUNCT
ejde-610	30	6	u	u	NOUN
ejde-610	30	7	)	)	PUNCT
ejde-610	30	8	=	=	SYM
ejde-610	30	9	|u|p−2u+	|u|p−2u+	PROPN
ejde-610	30	10	|u|2∗−2u	|u|2∗−2u	PROPN
ejde-610	30	11	,	,	PUNCT
ejde-610	30	12	i.e.	i.e.	X
ejde-610	30	13	the	the	DET
ejde-610	30	14	critical	critical	ADJ
ejde-610	30	15	sobolev	sobolev	NOUN
ejde-610	30	16	case	case	NOUN
ejde-610	30	17	was	be	AUX
ejde-610	30	18	studied	study	VERB
ejde-610	30	19	in	in	ADP
ejde-610	30	20	[	[	X
ejde-610	30	21	23	23	NUM
ejde-610	30	22	]	]	PUNCT
ejde-610	30	23	.	.	PUNCT
ejde-610	31	1	the	the	DET
ejde-610	31	2	authors	author	NOUN
ejde-610	31	3	obtained	obtain	VERB
ejde-610	31	4	a	a	DET
ejde-610	31	5	nontrivial	nontrivial	ADJ
ejde-610	31	6	solution	solution	NOUN
ejde-610	31	7	under	under	ADP
ejde-610	31	8	the	the	DET
ejde-610	31	9	conditions	condition	NOUN
ejde-610	31	10	4	4	NUM
ejde-610	31	11	<	<	X
ejde-610	31	12	p	p	X
ejde-610	31	13	<	<	X
ejde-610	31	14	6	6	NUM
ejde-610	31	15	and	and	CCONJ
ejde-610	31	16	m	m	PROPN
ejde-610	31	17	>	>	X
ejde-610	32	1	ω	ω	X
ejde-610	32	2	.	.	PUNCT
ejde-610	33	1	the	the	DET
ejde-610	33	2	authors	author	NOUN
ejde-610	33	3	in	in	ADP
ejde-610	33	4	[	[	X
ejde-610	33	5	19	19	NUM
ejde-610	33	6	]	]	PUNCT
ejde-610	33	7	improved	improve	VERB
ejde-610	33	8	the	the	DET
ejde-610	33	9	result	result	NOUN
ejde-610	33	10	of	of	ADP
ejde-610	33	11	[	[	X
ejde-610	33	12	23	23	NUM
ejde-610	33	13	]	]	PUNCT
ejde-610	33	14	and	and	CCONJ
ejde-610	33	15	studied	study	VERB
ejde-610	33	16	the	the	DET
ejde-610	33	17	existence	existence	NOUN
ejde-610	33	18	of	of	ADP
ejde-610	33	19	ground	ground	NOUN
ejde-610	33	20	state	state	NOUN
ejde-610	33	21	solution	solution	NOUN
ejde-610	33	22	.	.	PUNCT
ejde-610	34	1	zhang	zhang	PROPN
ejde-610	34	2	and	and	CCONJ
ejde-610	34	3	liu	liu	PROPN
ejde-610	35	1	[	[	X
ejde-610	35	2	32	32	NUM
ejde-610	35	3	]	]	PUNCT
ejde-610	35	4	considered	consider	VERB
ejde-610	35	5	the	the	DET
ejde-610	35	6	existence	existence	NOUN
ejde-610	35	7	and	and	CCONJ
ejde-610	35	8	multiplicity	multiplicity	NOUN
ejde-610	35	9	of	of	ADP
ejde-610	35	10	sign	sign	NOUN
ejde-610	35	11	-	-	PUNCT
ejde-610	35	12	changing	change	VERB
ejde-610	35	13	solutions	solution	NOUN
ejde-610	35	14	by	by	ADP
ejde-610	35	15	the	the	DET
ejde-610	35	16	method	method	NOUN
ejde-610	35	17	of	of	ADP
ejde-610	35	18	invariant	invariant	ADJ
ejde-610	35	19	sets	set	NOUN
ejde-610	35	20	of	of	ADP
ejde-610	35	21	descending	descend	VERB
ejde-610	35	22	flow	flow	NOUN
ejde-610	35	23	.	.	PUNCT
ejde-610	36	1	recently	recently	ADV
ejde-610	36	2	,	,	PUNCT
ejde-610	36	3	for	for	ADP
ejde-610	36	4	general	general	ADJ
ejde-610	36	5	potential	potential	ADJ
ejde-610	36	6	v	v	X
ejde-610	36	7	(	(	PUNCT
ejde-610	36	8	x	x	NOUN
ejde-610	36	9	)	)	PUNCT
ejde-610	36	10	,	,	PUNCT
ejde-610	36	11	chen	chen	PROPN
ejde-610	36	12	and	and	CCONJ
ejde-610	36	13	song	song	NOUN
ejde-610	36	14	[	[	X
ejde-610	36	15	14	14	NUM
ejde-610	36	16	]	]	PUNCT
ejde-610	36	17	obtained	obtain	VERB
ejde-610	36	18	the	the	DET
ejde-610	36	19	existence	existence	NOUN
ejde-610	36	20	of	of	ADP
ejde-610	36	21	multiple	multiple	ADJ
ejde-610	36	22	nontrivial	nontrivial	ADJ
ejde-610	36	23	solutions	solution	NOUN
ejde-610	36	24	for	for	ADP
ejde-610	36	25	(	(	PUNCT
ejde-610	36	26	1.1	1.1	NUM
ejde-610	36	27	)	)	PUNCT
ejde-610	36	28	with	with	ADP
ejde-610	36	29	the	the	DET
ejde-610	36	30	nonlinearity	nonlinearity	NOUN
ejde-610	36	31	f(x	f(x	PROPN
ejde-610	36	32	,	,	PUNCT
ejde-610	36	33	u	u	NOUN
ejde-610	36	34	)	)	PUNCT
ejde-610	36	35	=	=	SYM
ejde-610	36	36	λk(x)|u|q−2u	λk(x)|u|q−2u	PUNCT
ejde-610	37	1	+	+	X
ejde-610	37	2	g(x)|u|p−2u	g(x)|u|p−2u	PROPN
ejde-610	37	3	;	;	PUNCT
ejde-610	37	4	that	that	PRON
ejde-610	37	5	is	is	ADV
ejde-610	37	6	,	,	PUNCT
ejde-610	37	7	the	the	DET
ejde-610	37	8	klein	klein	PROPN
ejde-610	37	9	-	-	PUNCT
ejde-610	37	10	gordon	gordon	PROPN
ejde-610	37	11	equation	equation	NOUN
ejde-610	37	12	with	with	ADP
ejde-610	37	13	concave	concave	ADJ
ejde-610	37	14	and	and	CCONJ
ejde-610	37	15	convex	convex	NOUN
ejde-610	37	16	nonlinearities	nonlinearitie	NOUN
ejde-610	37	17	coupled	couple	VERB
ejde-610	37	18	with	with	ADP
ejde-610	37	19	born	bear	VERB
ejde-610	37	20	-	-	PUNCT
ejde-610	37	21	infeld	infeld	NOUN
ejde-610	37	22	equations	equation	NOUN
ejde-610	37	23	on	on	ADP
ejde-610	37	24	r3	r3	PROPN
ejde-610	37	25	.	.	PUNCT
ejde-610	38	1	other	other	ADJ
ejde-610	38	2	related	relate	VERB
ejde-610	38	3	results	result	NOUN
ejde-610	38	4	about	about	ADP
ejde-610	38	5	homogeneous	homogeneous	ADJ
ejde-610	38	6	klein	klein	PROPN
ejde-610	38	7	-	-	PUNCT
ejde-610	38	8	gordon	gordon	PROPN
ejde-610	38	9	equation	equation	NOUN
ejde-610	38	10	with	with	ADP
ejde-610	38	11	born	bear	VERB
ejde-610	38	12	-	-	PUNCT
ejde-610	38	13	infeld	infeld	NOUN
ejde-610	38	14	equations	equation	NOUN
ejde-610	38	15	can	can	AUX
ejde-610	38	16	be	be	AUX
ejde-610	38	17	found	find	VERB
ejde-610	38	18	in	in	ADP
ejde-610	38	19	[	[	X
ejde-610	38	20	1	1	NUM
ejde-610	38	21	,	,	PUNCT
ejde-610	38	22	11	11	NUM
ejde-610	38	23	,	,	PUNCT
ejde-610	38	24	24	24	NUM
ejde-610	38	25	,	,	PUNCT
ejde-610	38	26	25	25	NUM
ejde-610	38	27	,	,	PUNCT
ejde-610	38	28	28	28	NUM
ejde-610	38	29	,	,	PUNCT
ejde-610	38	30	31	31	NUM
ejde-610	38	31	]	]	PUNCT
ejde-610	38	32	.	.	PUNCT
ejde-610	39	1	next	next	ADV
ejde-610	39	2	,	,	PUNCT
ejde-610	39	3	we	we	PRON
ejde-610	39	4	consider	consider	VERB
ejde-610	39	5	the	the	DET
ejde-610	39	6	non	non	ADJ
ejde-610	39	7	-	-	ADJ
ejde-610	39	8	homogeneous	homogeneous	ADJ
ejde-610	39	9	case	case	NOUN
ejde-610	39	10	,	,	PUNCT
ejde-610	39	11	that	that	PRON
ejde-610	39	12	is	be	AUX
ejde-610	39	13	f(x	f(x	PROPN
ejde-610	39	14	,	,	PUNCT
ejde-610	39	15	u	u	NOUN
ejde-610	39	16	)	)	PUNCT
ejde-610	39	17	is	be	AUX
ejde-610	39	18	instead	instead	ADV
ejde-610	39	19	of	of	ADP
ejde-610	39	20	f(x	f(x	PROPN
ejde-610	39	21	,	,	PUNCT
ejde-610	39	22	u)+	u)+	PROPN
ejde-610	39	23	h(x	h(x	PROPN
ejde-610	39	24	)	)	PUNCT
ejde-610	39	25	.	.	PUNCT
ejde-610	40	1	chen	chen	PROPN
ejde-610	40	2	and	and	CCONJ
ejde-610	40	3	li	li	PROPN
ejde-610	41	1	[	[	X
ejde-610	41	2	12	12	NUM
ejde-610	41	3	]	]	PUNCT
ejde-610	41	4	proved	prove	VERB
ejde-610	41	5	that	that	SCONJ
ejde-610	41	6	(	(	PUNCT
ejde-610	41	7	1.1	1.1	NUM
ejde-610	41	8	)	)	PUNCT
ejde-610	41	9	has	have	VERB
ejde-610	41	10	two	two	NUM
ejde-610	41	11	nontrivial	nontrivial	ADJ
ejde-610	41	12	radially	radially	ADV
ejde-610	41	13	symmetric	symmetric	ADJ
ejde-610	41	14	solutions	solution	NOUN
ejde-610	41	15	if	if	SCONJ
ejde-610	41	16	f(x	f(x	PROPN
ejde-610	41	17	,	,	PUNCT
ejde-610	41	18	u	u	NOUN
ejde-610	41	19	)	)	PUNCT
ejde-610	41	20	=	=	SYM
ejde-610	41	21	|u|p−2u	|u|p−2u	PROPN
ejde-610	41	22	and	and	CCONJ
ejde-610	41	23	h(x	h(x	PROPN
ejde-610	41	24	)	)	PUNCT
ejde-610	41	25	is	be	AUX
ejde-610	41	26	radially	radially	ADV
ejde-610	41	27	symmetric	symmetric	ADJ
ejde-610	41	28	.	.	PUNCT
ejde-610	42	1	in	in	ADP
ejde-610	42	2	[	[	X
ejde-610	42	3	26	26	NUM
ejde-610	42	4	]	]	PUNCT
ejde-610	42	5	,	,	PUNCT
ejde-610	42	6	the	the	DET
ejde-610	42	7	authors	author	NOUN
ejde-610	42	8	obtain	obtain	VERB
ejde-610	42	9	the	the	DET
ejde-610	42	10	existence	existence	NOUN
ejde-610	42	11	of	of	ADP
ejde-610	42	12	two	two	NUM
ejde-610	42	13	solutions	solution	NOUN
ejde-610	42	14	by	by	ADP
ejde-610	42	15	the	the	DET
ejde-610	42	16	mountain	mountain	NOUN
ejde-610	42	17	pass	pass	NOUN
ejde-610	42	18	theorem	theorem	NOUN
ejde-610	42	19	and	and	CCONJ
ejde-610	42	20	the	the	DET
ejde-610	42	21	ekeland	ekeland	NOUN
ejde-610	42	22	’s	’s	PART
ejde-610	42	23	variational	variational	ADJ
ejde-610	42	24	principle	principle	NOUN
ejde-610	42	25	in	in	ADP
ejde-610	42	26	critical	critical	ADJ
ejde-610	42	27	point	point	NOUN
ejde-610	42	28	theory	theory	NOUN
ejde-610	42	29	for	for	ADP
ejde-610	42	30	general	general	ADJ
ejde-610	42	31	f(x	f(x	PROPN
ejde-610	42	32	,	,	PUNCT
ejde-610	42	33	u	u	NOUN
ejde-610	42	34	)	)	PUNCT
ejde-610	42	35	.	.	PUNCT
ejde-610	43	1	in	in	ADP
ejde-610	43	2	[	[	X
ejde-610	43	3	27	27	NUM
ejde-610	43	4	]	]	PUNCT
ejde-610	43	5	,	,	PUNCT
ejde-610	43	6	the	the	DET
ejde-610	43	7	authors	author	NOUN
ejde-610	43	8	consider	consider	VERB
ejde-610	43	9	the	the	DET
ejde-610	43	10	existence	existence	NOUN
ejde-610	43	11	of	of	ADP
ejde-610	43	12	multiple	multiple	ADJ
ejde-610	43	13	solutions	solution	NOUN
ejde-610	43	14	for	for	ADP
ejde-610	43	15	nonhomogeneous	nonhomogeneous	ADJ
ejde-610	43	16	kleingordon	kleingordon	NOUN
ejde-610	43	17	equation	equation	NOUN
ejde-610	43	18	with	with	ADP
ejde-610	43	19	sign	sign	NOUN
ejde-610	43	20	-	-	PUNCT
ejde-610	43	21	changing	change	VERB
ejde-610	43	22	potential	potential	NOUN
ejde-610	43	23	coupled	couple	VERB
ejde-610	43	24	with	with	ADP
ejde-610	43	25	born	bear	VERB
ejde-610	43	26	-	-	PUNCT
ejde-610	43	27	infeld	infeld	NOUN
ejde-610	43	28	theory	theory	NOUN
ejde-610	43	29	.	.	PUNCT
ejde-610	44	1	motivated	motivate	VERB
ejde-610	44	2	by	by	ADP
ejde-610	44	3	the	the	DET
ejde-610	44	4	above	above	ADJ
ejde-610	44	5	works	work	NOUN
ejde-610	44	6	,	,	PUNCT
ejde-610	44	7	we	we	PRON
ejde-610	44	8	consider	consider	VERB
ejde-610	44	9	system	system	NOUN
ejde-610	44	10	(	(	PUNCT
ejde-610	44	11	1.1	1.1	NUM
ejde-610	44	12	)	)	PUNCT
ejde-610	44	13	with	with	ADP
ejde-610	44	14	more	more	ADV
ejde-610	44	15	general	general	ADJ
ejde-610	44	16	potential	potential	ADJ
ejde-610	44	17	v	v	NOUN
ejde-610	44	18	(	(	PUNCT
ejde-610	44	19	x	x	NOUN
ejde-610	44	20	)	)	PUNCT
ejde-610	44	21	and	and	CCONJ
ejde-610	44	22	the	the	DET
ejde-610	44	23	primitive	primitive	NOUN
ejde-610	44	24	of	of	ADP
ejde-610	44	25	f(x	f(x	PROPN
ejde-610	44	26	,	,	PUNCT
ejde-610	44	27	u	u	NOUN
ejde-610	44	28	)	)	PUNCT
ejde-610	44	29	is	be	AUX
ejde-610	44	30	of	of	ADP
ejde-610	44	31	2	2	NUM
ejde-610	44	32	-	-	PUNCT
ejde-610	44	33	superlinear	superlinear	NOUN
ejde-610	44	34	growth	growth	NOUN
ejde-610	44	35	in	in	ADP
ejde-610	44	36	u	u	NOUN
ejde-610	44	37	at	at	ADP
ejde-610	44	38	infinity	infinity	NOUN
ejde-610	44	39	.	.	PUNCT
ejde-610	45	1	precisely	precisely	ADV
ejde-610	45	2	,	,	PUNCT
ejde-610	45	3	we	we	PRON
ejde-610	45	4	make	make	VERB
ejde-610	45	5	the	the	DET
ejde-610	45	6	following	follow	VERB
ejde-610	45	7	assumptions	assumption	NOUN
ejde-610	45	8	.	.	PUNCT
ejde-610	46	1	(	(	PUNCT
ejde-610	46	2	a1	a1	PROPN
ejde-610	46	3	)	)	PUNCT
ejde-610	46	4	v	v	NOUN
ejde-610	46	5	∈	∈	PROPN
ejde-610	46	6	c(r3,r	c(r3,r	PROPN
ejde-610	46	7	)	)	PUNCT
ejde-610	46	8	is	be	AUX
ejde-610	46	9	bounded	bound	VERB
ejde-610	46	10	below	below	ADP
ejde-610	46	11	and	and	CCONJ
ejde-610	46	12	,	,	PUNCT
ejde-610	46	13	for	for	ADP
ejde-610	46	14	every	every	DET
ejde-610	46	15	c	c	PROPN
ejde-610	46	16	>	>	X
ejde-610	46	17	0	0	PROPN
ejde-610	46	18	,	,	PUNCT
ejde-610	46	19	meas{x	meas{x	PROPN
ejde-610	46	20	∈	∈	PROPN
ejde-610	46	21	r3	r3	PROPN
ejde-610	46	22	:	:	PUNCT
ejde-610	46	23	v	v	X
ejde-610	46	24	(	(	PUNCT
ejde-610	46	25	x	x	NOUN
ejde-610	46	26	)	)	PUNCT
ejde-610	46	27	≤	≤	NOUN
ejde-610	47	1	c	c	X
ejde-610	47	2	}	}	PUNCT
ejde-610	47	3	<	<	X
ejde-610	47	4	+	+	PROPN
ejde-610	47	5	∞	∞	PROPN
ejde-610	47	6	,	,	PUNCT
ejde-610	47	7	where	where	SCONJ
ejde-610	47	8	meas	meas	PROPN
ejde-610	47	9	denotes	denote	VERB
ejde-610	47	10	the	the	DET
ejde-610	47	11	lebesgue	lebesgue	NOUN
ejde-610	47	12	measures	measure	NOUN
ejde-610	47	13	;	;	PUNCT
ejde-610	47	14	(	(	PUNCT
ejde-610	47	15	a2	a2	PROPN
ejde-610	47	16	)	)	PUNCT
ejde-610	47	17	f	f	PROPN
ejde-610	47	18	∈	∈	PROPN
ejde-610	47	19	c(r3	c(r3	NOUN
ejde-610	47	20	×	×	NOUN
ejde-610	47	21	r	r	NOUN
ejde-610	47	22	,	,	PUNCT
ejde-610	47	23	r	r	NOUN
ejde-610	47	24	)	)	PUNCT
ejde-610	47	25	and	and	CCONJ
ejde-610	47	26	there	there	PRON
ejde-610	47	27	exist	exist	VERB
ejde-610	47	28	c	c	PROPN
ejde-610	47	29	>	>	PUNCT
ejde-610	47	30	0	0	PUNCT
ejde-610	48	1	and	and	CCONJ
ejde-610	48	2	p	p	NOUN
ejde-610	48	3	∈	∈	PROPN
ejde-610	48	4	(	(	PUNCT
ejde-610	48	5	2	2	NUM
ejde-610	48	6	,	,	PUNCT
ejde-610	48	7	6	6	NUM
ejde-610	48	8	)	)	PUNCT
ejde-610	48	9	such	such	ADJ
ejde-610	48	10	that	that	SCONJ
ejde-610	48	11	|f(x	|f(x	PROPN
ejde-610	48	12	,	,	PUNCT
ejde-610	48	13	t)|	t)|	ADJ
ejde-610	48	14	≤	≤	ADJ
ejde-610	48	15	c(1	c(1	NOUN
ejde-610	48	16	+	+	CCONJ
ejde-610	48	17	|t|p−1	|t|p−1	NUM
ejde-610	48	18	)	)	PUNCT
ejde-610	48	19	;	;	PUNCT
ejde-610	48	20	(	(	PUNCT
ejde-610	48	21	a3	a3	NOUN
ejde-610	48	22	)	)	PUNCT
ejde-610	48	23	f(x	f(x	PROPN
ejde-610	48	24	,	,	PUNCT
ejde-610	48	25	t	t	PROPN
ejde-610	48	26	)	)	PUNCT
ejde-610	48	27	=	=	PUNCT
ejde-610	48	28	o(t	o(t	NOUN
ejde-610	48	29	)	)	PUNCT
ejde-610	48	30	uniformly	uniformly	ADV
ejde-610	48	31	in	in	ADP
ejde-610	48	32	x	x	PUNCT
ejde-610	48	33	as	as	ADP
ejde-610	48	34	t→	t→	X
ejde-610	48	35	0	0	NUM
ejde-610	48	36	;	;	PUNCT
ejde-610	48	37	(	(	PUNCT
ejde-610	48	38	a4	a4	NOUN
ejde-610	48	39	)	)	PUNCT
ejde-610	48	40	f(x	f(x	PROPN
ejde-610	48	41	,	,	PUNCT
ejde-610	48	42	t)/t→	t)/t→	X
ejde-610	49	1	+	+	NOUN
ejde-610	49	2	∞	∞	NOUN
ejde-610	49	3	uniformly	uniformly	ADV
ejde-610	49	4	in	in	ADP
ejde-610	49	5	x	x	PUNCT
ejde-610	49	6	as	as	ADP
ejde-610	49	7	|t|	|t|	PROPN
ejde-610	49	8	→	→	SYM
ejde-610	49	9	+	+	NOUN
ejde-610	49	10	∞	∞	NUM
ejde-610	49	11	;	;	PUNCT
ejde-610	49	12	(	(	PUNCT
ejde-610	49	13	a5	a5	PROPN
ejde-610	49	14	)	)	PUNCT
ejde-610	49	15	there	there	PRON
ejde-610	49	16	exists	exist	VERB
ejde-610	49	17	θ	θ	PROPN
ejde-610	49	18	>	>	X
ejde-610	49	19	2	2	NUM
ejde-610	49	20	and	and	CCONJ
ejde-610	49	21	b	b	NOUN
ejde-610	49	22	>	>	X
ejde-610	49	23	0	0	NUM
ejde-610	49	24	such	such	ADJ
ejde-610	49	25	that	that	SCONJ
ejde-610	49	26	f(x	f(x	PROPN
ejde-610	49	27	,	,	PUNCT
ejde-610	49	28	t	t	PROPN
ejde-610	49	29	)	)	PUNCT
ejde-610	49	30	:	:	PUNCT
ejde-610	50	1	=	=	SYM
ejde-610	50	2	1	1	NUM
ejde-610	50	3	θf(x	θf(x	NOUN
ejde-610	50	4	,	,	PUNCT
ejde-610	50	5	t)t−f	t)t−f	NOUN
ejde-610	50	6	(	(	PUNCT
ejde-610	50	7	x	x	NOUN
ejde-610	50	8	,	,	PUNCT
ejde-610	50	9	t	t	PROPN
ejde-610	50	10	)	)	PUNCT
ejde-610	50	11	≥	≥	NOUN
ejde-610	50	12	−bt2	−bt2	NOUN
ejde-610	50	13	,	,	PUNCT
ejde-610	50	14	where	where	SCONJ
ejde-610	50	15	f	f	PROPN
ejde-610	50	16	(	(	PUNCT
ejde-610	50	17	x	x	PROPN
ejde-610	50	18	,	,	PUNCT
ejde-610	50	19	t	t	PROPN
ejde-610	50	20	)	)	PUNCT
ejde-610	50	21	:	:	PUNCT
ejde-610	50	22	=	=	PUNCT
ejde-610	50	23	∫	∫	PROPN
ejde-610	50	24	t	t	PROPN
ejde-610	50	25	0	0	NUM
ejde-610	50	26	(	(	PUNCT
ejde-610	50	27	x	x	NOUN
ejde-610	50	28	,	,	PUNCT
ejde-610	50	29	s)ds	s)ds	PROPN
ejde-610	50	30	.	.	PUNCT
ejde-610	51	1	the	the	DET
ejde-610	51	2	condition	condition	NOUN
ejde-610	51	3	(	(	PUNCT
ejde-610	51	4	ar	ar	NOUN
ejde-610	51	5	)	)	PUNCT
ejde-610	51	6	there	there	PRON
ejde-610	51	7	exists	exist	VERB
ejde-610	51	8	µ	µ	X
ejde-610	51	9	>	>	X
ejde-610	51	10	4	4	NUM
ejde-610	51	11	such	such	ADJ
ejde-610	51	12	that	that	SCONJ
ejde-610	51	13	µf	µf	X
ejde-610	51	14	(	(	PUNCT
ejde-610	51	15	x	x	X
ejde-610	51	16	,	,	PUNCT
ejde-610	51	17	t	t	PROPN
ejde-610	51	18	)	)	PUNCT
ejde-610	51	19	≤	≤	NOUN
ejde-610	51	20	tf(x	tf(x	NUM
ejde-610	51	21	,	,	PUNCT
ejde-610	51	22	t	t	PROPN
ejde-610	51	23	)	)	PUNCT
ejde-610	51	24	,	,	PUNCT
ejde-610	51	25	for	for	ADP
ejde-610	51	26	all	all	DET
ejde-610	51	27	(	(	PUNCT
ejde-610	51	28	x	x	NOUN
ejde-610	51	29	,	,	PUNCT
ejde-610	51	30	t	t	PROPN
ejde-610	51	31	)	)	PUNCT
ejde-610	51	32	∈	∈	PROPN
ejde-610	51	33	r3	r3	PROPN
ejde-610	51	34	×	×	NOUN
ejde-610	51	35	r	r	NOUN
ejde-610	51	36	is	be	AUX
ejde-610	51	37	widely	widely	ADV
ejde-610	51	38	used	use	VERB
ejde-610	51	39	in	in	ADP
ejde-610	51	40	the	the	DET
ejde-610	51	41	studies	study	NOUN
ejde-610	51	42	of	of	ADP
ejde-610	51	43	elliptic	elliptic	ADJ
ejde-610	51	44	problem	problem	NOUN
ejde-610	51	45	by	by	ADP
ejde-610	51	46	variational	variational	ADJ
ejde-610	51	47	methods	method	NOUN
ejde-610	51	48	.	.	PUNCT
ejde-610	52	1	condition	condition	NOUN
ejde-610	52	2	(	(	PUNCT
ejde-610	52	3	ar	ar	NOUN
ejde-610	52	4	)	)	PUNCT
ejde-610	52	5	is	be	AUX
ejde-610	52	6	used	use	VERB
ejde-610	52	7	not	not	PART
ejde-610	52	8	only	only	ADV
ejde-610	52	9	to	to	PART
ejde-610	52	10	prove	prove	VERB
ejde-610	52	11	that	that	SCONJ
ejde-610	52	12	the	the	DET
ejde-610	52	13	euler	euler	PROPN
ejde-610	52	14	-	-	PUNCT
ejde-610	52	15	lagrange	lagrange	PROPN
ejde-610	52	16	function	function	NOUN
ejde-610	52	17	associated	associate	VERB
ejde-610	52	18	has	have	VERB
ejde-610	52	19	a	a	DET
ejde-610	52	20	mountain	mountain	NOUN
ejde-610	52	21	pass	pass	NOUN
ejde-610	52	22	geometry	geometry	NOUN
ejde-610	52	23	,	,	PUNCT
ejde-610	52	24	but	but	CCONJ
ejde-610	52	25	also	also	ADV
ejde-610	52	26	to	to	PART
ejde-610	52	27	guarantee	guarantee	VERB
ejde-610	52	28	that	that	SCONJ
ejde-610	52	29	the	the	DET
ejde-610	52	30	palais	palais	PROPN
ejde-610	52	31	-	-	PUNCT
ejde-610	52	32	smale	smale	ADJ
ejde-610	52	33	sequences	sequence	NOUN
ejde-610	52	34	,	,	PUNCT
ejde-610	52	35	or	or	CCONJ
ejde-610	52	36	cerami	cerami	PROPN
ejde-610	52	37	sequences	sequence	NOUN
ejde-610	52	38	are	be	AUX
ejde-610	52	39	bounded	bound	VERB
ejde-610	52	40	.	.	PUNCT
ejde-610	53	1	obviously	obviously	ADV
ejde-610	53	2	,	,	PUNCT
ejde-610	53	3	we	we	PRON
ejde-610	53	4	can	can	AUX
ejde-610	53	5	observe	observe	VERB
ejde-610	53	6	that	that	SCONJ
ejde-610	53	7	the	the	DET
ejde-610	53	8	condition	condition	NOUN
ejde-610	53	9	(	(	PUNCT
ejde-610	53	10	ar	ar	NOUN
ejde-610	53	11	)	)	PUNCT
ejde-610	53	12	implies	imply	VERB
ejde-610	53	13	the	the	DET
ejde-610	53	14	following	follow	VERB
ejde-610	53	15	condition	condition	NOUN
ejde-610	53	16	:	:	PUNCT
ejde-610	53	17	(	(	PUNCT
ejde-610	53	18	a6	a6	NOUN
ejde-610	53	19	)	)	PUNCT
ejde-610	53	20	there	there	PRON
ejde-610	53	21	exist	exist	VERB
ejde-610	53	22	µ	µ	X
ejde-610	53	23	>	>	ADP
ejde-610	53	24	4	4	NUM
ejde-610	53	25	and	and	CCONJ
ejde-610	53	26	c1	c1	PROPN
ejde-610	53	27	,	,	PUNCT
ejde-610	53	28	c2	c2	PROPN
ejde-610	53	29	>	>	X
ejde-610	53	30	0	0	NUM
ejde-610	54	1	such	such	ADJ
ejde-610	54	2	that	that	SCONJ
ejde-610	54	3	f	f	PROPN
ejde-610	54	4	(	(	PUNCT
ejde-610	54	5	x	x	PROPN
ejde-610	54	6	,	,	PUNCT
ejde-610	54	7	t	t	PROPN
ejde-610	54	8	)	)	PUNCT
ejde-610	54	9	≥	≥	PROPN
ejde-610	54	10	c1|t|µ	c1|t|µ	PROPN
ejde-610	54	11	−	−	PROPN
ejde-610	54	12	c2	c2	PROPN
ejde-610	54	13	,	,	PUNCT
ejde-610	54	14	for	for	ADP
ejde-610	54	15	t	t	PROPN
ejde-610	54	16	sufficiently	sufficiently	ADV
ejde-610	54	17	large	large	ADJ
ejde-610	54	18	.	.	PUNCT
ejde-610	54	19	ejde-2024/18	ejde-2024/18	PROPN
ejde-610	54	20	coupled	couple	VERB
ejde-610	54	21	klein	klein	PROPN
ejde-610	54	22	-	-	PUNCT
ejde-610	54	23	gordon	gordon	PROPN
ejde-610	54	24	and	and	CCONJ
ejde-610	54	25	born	bear	VERB
ejde-610	54	26	-	-	PUNCT
ejde-610	54	27	infeld	infeld	NOUN
ejde-610	54	28	equatons	equaton	NOUN
ejde-610	54	29	3	3	NUM
ejde-610	54	30	moreover	moreover	ADV
ejde-610	54	31	,	,	PUNCT
ejde-610	54	32	the	the	DET
ejde-610	54	33	condition	condition	NOUN
ejde-610	54	34	(	(	PUNCT
ejde-610	54	35	a6	a6	NOUN
ejde-610	54	36	)	)	PUNCT
ejde-610	54	37	implies	imply	VERB
ejde-610	54	38	condition	condition	NOUN
ejde-610	54	39	(	(	PUNCT
ejde-610	54	40	a4	a4	NOUN
ejde-610	54	41	)	)	PUNCT
ejde-610	54	42	.	.	PUNCT
ejde-610	55	1	another	another	DET
ejde-610	55	2	widely	widely	ADV
ejde-610	55	3	employed	employ	VERB
ejde-610	55	4	condition	condition	NOUN
ejde-610	55	5	is	be	AUX
ejde-610	55	6	the	the	DET
ejde-610	55	7	following	follow	VERB
ejde-610	55	8	condition	condition	NOUN
ejde-610	55	9	,	,	PUNCT
ejde-610	55	10	which	which	PRON
ejde-610	55	11	is	be	AUX
ejde-610	55	12	first	first	ADV
ejde-610	55	13	introduced	introduce	VERB
ejde-610	55	14	by	by	ADP
ejde-610	55	15	jeanjean	jeanjean	PROPN
ejde-610	55	16	[	[	X
ejde-610	55	17	18	18	NUM
ejde-610	55	18	]	]	PUNCT
ejde-610	55	19	.	.	PUNCT
ejde-610	56	1	(	(	PUNCT
ejde-610	56	2	a7	a7	PROPN
ejde-610	56	3	)	)	PUNCT
ejde-610	56	4	there	there	PRON
ejde-610	56	5	exist	exist	VERB
ejde-610	56	6	θ0	θ0	PROPN
ejde-610	56	7	≥	≥	NOUN
ejde-610	56	8	1	1	NUM
ejde-610	56	9	such	such	ADJ
ejde-610	56	10	that	that	DET
ejde-610	56	11	θ0f(x	θ0f(x	PROPN
ejde-610	56	12	,	,	PUNCT
ejde-610	56	13	t	t	PROPN
ejde-610	56	14	)	)	PUNCT
ejde-610	56	15	≥	≥	NOUN
ejde-610	56	16	f(x	f(x	PROPN
ejde-610	56	17	,	,	PUNCT
ejde-610	56	18	st	st	PROPN
ejde-610	56	19	)	)	PUNCT
ejde-610	56	20	for	for	ADP
ejde-610	56	21	all	all	DET
ejde-610	56	22	s	s	PART
ejde-610	56	23	∈	∈	NOUN
ejde-610	57	1	[	[	X
ejde-610	57	2	0	0	NUM
ejde-610	57	3	,	,	PUNCT
ejde-610	57	4	1	1	NUM
ejde-610	57	5	]	]	PUNCT
ejde-610	57	6	and	and	CCONJ
ejde-610	57	7	t	t	PROPN
ejde-610	57	8	∈	∈	PROPN
ejde-610	57	9	r	r	NOUN
ejde-610	57	10	,	,	PUNCT
ejde-610	57	11	where	where	SCONJ
ejde-610	57	12	f(x	f(x	PROPN
ejde-610	57	13	,	,	PUNCT
ejde-610	57	14	t	t	PROPN
ejde-610	57	15	)	)	PUNCT
ejde-610	57	16	is	be	AUX
ejde-610	57	17	given	give	VERB
ejde-610	57	18	in	in	ADP
ejde-610	57	19	(	(	PUNCT
ejde-610	57	20	a5	a5	PROPN
ejde-610	57	21	)	)	PUNCT
ejde-610	57	22	.	.	PUNCT
ejde-610	58	1	we	we	PRON
ejde-610	58	2	can	can	AUX
ejde-610	58	3	observe	observe	VERB
ejde-610	58	4	that	that	SCONJ
ejde-610	58	5	when	when	SCONJ
ejde-610	58	6	s	s	VERB
ejde-610	58	7	=	=	SYM
ejde-610	58	8	0	0	NUM
ejde-610	58	9	,	,	PUNCT
ejde-610	58	10	then	then	ADV
ejde-610	58	11	f(x	f(x	PROPN
ejde-610	58	12	,	,	PUNCT
ejde-610	58	13	t	t	PROPN
ejde-610	58	14	)	)	PUNCT
ejde-610	58	15	≥	≥	NOUN
ejde-610	58	16	0	0	NUM
ejde-610	58	17	,	,	PUNCT
ejde-610	58	18	but	but	CCONJ
ejde-610	58	19	for	for	ADP
ejde-610	58	20	our	our	PRON
ejde-610	58	21	condition	condition	NOUN
ejde-610	58	22	(	(	PUNCT
ejde-610	58	23	a5	a5	PROPN
ejde-610	58	24	)	)	PUNCT
ejde-610	58	25	,	,	PUNCT
ejde-610	58	26	f(x	f(x	PROPN
ejde-610	58	27	,	,	PUNCT
ejde-610	58	28	t	t	PROPN
ejde-610	58	29	)	)	PUNCT
ejde-610	58	30	may	may	AUX
ejde-610	58	31	assume	assume	VERB
ejde-610	58	32	negative	negative	ADJ
ejde-610	58	33	values	value	NOUN
ejde-610	58	34	.	.	PUNCT
ejde-610	59	1	therefore	therefore	ADV
ejde-610	59	2	,	,	PUNCT
ejde-610	59	3	it	it	PRON
ejde-610	59	4	is	be	AUX
ejde-610	59	5	interesting	interesting	ADJ
ejde-610	59	6	to	to	PART
ejde-610	59	7	consider	consider	VERB
ejde-610	59	8	2superlinear	2superlinear	NUM
ejde-610	59	9	problems	problem	NOUN
ejde-610	59	10	under	under	ADP
ejde-610	59	11	conditions	condition	NOUN
ejde-610	59	12	(	(	PUNCT
ejde-610	59	13	a4	a4	NOUN
ejde-610	59	14	)	)	PUNCT
ejde-610	59	15	and	and	CCONJ
ejde-610	59	16	(	(	PUNCT
ejde-610	59	17	a5	a5	PROPN
ejde-610	59	18	)	)	PUNCT
ejde-610	59	19	.	.	PUNCT
ejde-610	60	1	condition	condition	NOUN
ejde-610	60	2	(	(	PUNCT
ejde-610	60	3	a5	a5	PROPN
ejde-610	60	4	)	)	PUNCT
ejde-610	60	5	was	be	AUX
ejde-610	60	6	used	use	VERB
ejde-610	60	7	by	by	ADP
ejde-610	60	8	alves	alve	NOUN
ejde-610	60	9	,	,	PUNCT
ejde-610	60	10	soares	soare	NOUN
ejde-610	60	11	and	and	CCONJ
ejde-610	60	12	souto	souto	NOUN
ejde-610	60	13	in	in	ADP
ejde-610	60	14	[	[	X
ejde-610	60	15	2	2	NUM
ejde-610	60	16	]	]	PUNCT
ejde-610	60	17	.	.	PUNCT
ejde-610	61	1	with	with	ADP
ejde-610	61	2	the	the	DET
ejde-610	61	3	additional	additional	ADJ
ejde-610	61	4	conditions	condition	NOUN
ejde-610	61	5	that	that	SCONJ
ejde-610	61	6	α	α	PROPN
ejde-610	61	7	=	=	PROPN
ejde-610	61	8	inf	inf	PROPN
ejde-610	61	9	x∈r3	x∈r3	PROPN
ejde-610	61	10	v	v	PROPN
ejde-610	61	11	(	(	PUNCT
ejde-610	61	12	x	x	X
ejde-610	61	13	)	)	PUNCT
ejde-610	61	14	>	>	X
ejde-610	61	15	0	0	PUNCT
ejde-610	62	1	(	(	PUNCT
ejde-610	62	2	1.3	1.3	NUM
ejde-610	62	3	)	)	PUNCT
ejde-610	62	4	and	and	CCONJ
ejde-610	62	5	b	b	X
ejde-610	62	6	∈	∈	PROPN
ejde-610	63	1	[	[	X
ejde-610	63	2	0	0	NUM
ejde-610	63	3	,	,	PUNCT
ejde-610	63	4	α	α	NOUN
ejde-610	63	5	)	)	PUNCT
ejde-610	63	6	,	,	PUNCT
ejde-610	63	7	they	they	PRON
ejde-610	63	8	proved	prove	VERB
ejde-610	63	9	that	that	SCONJ
ejde-610	63	10	all	all	DET
ejde-610	63	11	cerami	cerami	PROPN
ejde-610	63	12	sequences	sequence	NOUN
ejde-610	63	13	are	be	AUX
ejde-610	63	14	bounded	bound	VERB
ejde-610	63	15	.	.	PUNCT
ejde-610	64	1	under	under	ADP
ejde-610	64	2	the	the	DET
ejde-610	64	3	much	much	ADV
ejde-610	64	4	weaker	weak	ADJ
ejde-610	64	5	condition	condition	NOUN
ejde-610	64	6	(	(	PUNCT
ejde-610	64	7	a5	a5	PROPN
ejde-610	64	8	)	)	PUNCT
ejde-610	64	9	,	,	PUNCT
ejde-610	64	10	we	we	PRON
ejde-610	64	11	can	can	AUX
ejde-610	64	12	obtain	obtain	VERB
ejde-610	64	13	the	the	DET
ejde-610	64	14	boundedness	boundedness	NOUN
ejde-610	64	15	of	of	ADP
ejde-610	64	16	palais	palais	PROPN
ejde-610	64	17	-	-	PUNCT
ejde-610	64	18	smale	smale	ADJ
ejde-610	64	19	sequences	sequence	NOUN
ejde-610	64	20	,	,	PUNCT
ejde-610	64	21	see	see	VERB
ejde-610	64	22	lemma	lemma	PROPN
ejde-610	64	23	2.4	2.4	NUM
ejde-610	64	24	.	.	PUNCT
ejde-610	65	1	in	in	ADP
ejde-610	65	2	2015	2015	NUM
ejde-610	65	3	,	,	PUNCT
ejde-610	65	4	chen	chen	PROPN
ejde-610	65	5	and	and	CCONJ
ejde-610	65	6	liu	liu	PROPN
ejde-610	66	1	[	[	X
ejde-610	66	2	13	13	NUM
ejde-610	66	3	]	]	PUNCT
ejde-610	66	4	also	also	ADV
ejde-610	66	5	used	use	VERB
ejde-610	66	6	conditions	condition	NOUN
ejde-610	66	7	(	(	PUNCT
ejde-610	66	8	a4	a4	NOUN
ejde-610	66	9	)	)	PUNCT
ejde-610	66	10	and	and	CCONJ
ejde-610	66	11	(	(	PUNCT
ejde-610	66	12	a5	a5	PROPN
ejde-610	66	13	)	)	PUNCT
ejde-610	66	14	to	to	PART
ejde-610	66	15	show	show	VERB
ejde-610	66	16	the	the	DET
ejde-610	66	17	existence	existence	NOUN
ejde-610	66	18	of	of	ADP
ejde-610	66	19	infinitely	infinitely	ADV
ejde-610	66	20	many	many	ADJ
ejde-610	66	21	solutions	solution	NOUN
ejde-610	66	22	for	for	ADP
ejde-610	66	23	schrödinger	schrödinger	NOUN
ejde-610	66	24	-	-	PUNCT
ejde-610	66	25	maxwell	maxwell	NOUN
ejde-610	66	26	systems	system	NOUN
ejde-610	66	27	.	.	PUNCT
ejde-610	67	1	in	in	ADP
ejde-610	67	2	our	our	PRON
ejde-610	67	3	case	case	NOUN
ejde-610	67	4	,	,	PUNCT
ejde-610	67	5	many	many	ADJ
ejde-610	67	6	technical	technical	ADJ
ejde-610	67	7	difficulties	difficulty	NOUN
ejde-610	67	8	arise	arise	VERB
ejde-610	67	9	because	because	SCONJ
ejde-610	67	10	of	of	ADP
ejde-610	67	11	the	the	DET
ejde-610	67	12	presence	presence	NOUN
ejde-610	67	13	of	of	ADP
ejde-610	67	14	the	the	DET
ejde-610	67	15	non	non	ADJ
ejde-610	67	16	-	-	ADJ
ejde-610	67	17	local	local	ADJ
ejde-610	67	18	term	term	NOUN
ejde-610	67	19	φ	φ	PROPN
ejde-610	67	20	,	,	PUNCT
ejde-610	67	21	which	which	PRON
ejde-610	67	22	is	be	AUX
ejde-610	67	23	not	not	PART
ejde-610	67	24	homogeneous	homogeneous	ADJ
ejde-610	67	25	as	as	SCONJ
ejde-610	67	26	it	it	PRON
ejde-610	67	27	is	be	AUX
ejde-610	67	28	in	in	ADP
ejde-610	67	29	the	the	DET
ejde-610	67	30	schrödinger	schrödinger	NOUN
ejde-610	67	31	-	-	PUNCT
ejde-610	67	32	maxwell	maxwell	PROPN
ejde-610	67	33	systems	system	NOUN
ejde-610	67	34	.	.	PUNCT
ejde-610	68	1	hence	hence	ADV
ejde-610	68	2	,	,	PUNCT
ejde-610	68	3	a	a	DET
ejde-610	68	4	more	more	ADV
ejde-610	68	5	careful	careful	ADJ
ejde-610	68	6	analysis	analysis	NOUN
ejde-610	68	7	of	of	ADP
ejde-610	68	8	the	the	DET
ejde-610	68	9	interaction	interaction	NOUN
ejde-610	68	10	between	between	ADP
ejde-610	68	11	the	the	DET
ejde-610	68	12	couple	couple	NOUN
ejde-610	68	13	(	(	PUNCT
ejde-610	68	14	u	u	NOUN
ejde-610	68	15	,	,	PUNCT
ejde-610	68	16	φ	φ	NOUN
ejde-610	68	17	)	)	PUNCT
ejde-610	68	18	is	be	AUX
ejde-610	68	19	required	require	VERB
ejde-610	68	20	.	.	PUNCT
ejde-610	69	1	by	by	ADP
ejde-610	69	2	(	(	PUNCT
ejde-610	69	3	a1	a1	NOUN
ejde-610	69	4	)	)	PUNCT
ejde-610	69	5	,	,	PUNCT
ejde-610	69	6	we	we	PRON
ejde-610	69	7	know	know	VERB
ejde-610	69	8	that	that	SCONJ
ejde-610	69	9	v	v	NOUN
ejde-610	69	10	is	be	AUX
ejde-610	69	11	bounded	bound	VERB
ejde-610	69	12	from	from	ADP
ejde-610	69	13	below	below	ADV
ejde-610	69	14	,	,	PUNCT
ejde-610	69	15	hence	hence	ADV
ejde-610	69	16	we	we	PRON
ejde-610	69	17	may	may	AUX
ejde-610	69	18	choose	choose	VERB
ejde-610	69	19	v0	v0	PROPN
ejde-610	69	20	>	>	X
ejde-610	69	21	0	0	NUM
ejde-610	70	1	such	such	ADJ
ejde-610	70	2	that	that	DET
ejde-610	70	3	ṽ	ṽ	PROPN
ejde-610	70	4	(	(	PUNCT
ejde-610	70	5	x	x	NOUN
ejde-610	70	6	)	)	PUNCT
ejde-610	70	7	:	:	PUNCT
ejde-610	70	8	=	=	SYM
ejde-610	70	9	v	v	X
ejde-610	70	10	(	(	PUNCT
ejde-610	70	11	x	x	NOUN
ejde-610	70	12	)	)	PUNCT
ejde-610	70	13	+	+	CCONJ
ejde-610	70	14	v0	v0	X
ejde-610	70	15	>	>	X
ejde-610	70	16	1	1	NUM
ejde-610	70	17	,	,	PUNCT
ejde-610	70	18	∀x	∀x	NOUN
ejde-610	70	19	∈	∈	PROPN
ejde-610	70	20	r3	r3	PROPN
ejde-610	70	21	and	and	CCONJ
ejde-610	70	22	define	define	VERB
ejde-610	70	23	a	a	DET
ejde-610	70	24	hilbert	hilbert	NOUN
ejde-610	70	25	space	space	NOUN
ejde-610	70	26	e	e	NOUN
ejde-610	70	27	:	:	PUNCT
ejde-610	70	28	=	=	SYM
ejde-610	70	29	{	{	PUNCT
ejde-610	70	30	u	u	NOUN
ejde-610	70	31	∈	∈	PROPN
ejde-610	70	32	h1(r3	h1(r3	ADV
ejde-610	70	33	)	)	PUNCT
ejde-610	70	34	:	:	PUNCT
ejde-610	70	35	∫	∫	PROPN
ejde-610	70	36	r3	r3	PROPN
ejde-610	70	37	v	v	X
ejde-610	70	38	(	(	PUNCT
ejde-610	70	39	x)u2	x)u2	PROPN
ejde-610	70	40	dx	dx	X
ejde-610	70	41	<	<	X
ejde-610	70	42	∞	∞	PROPN
ejde-610	70	43	}	}	PUNCT
ejde-610	70	44	with	with	ADP
ejde-610	70	45	the	the	DET
ejde-610	70	46	inner	inner	ADJ
ejde-610	70	47	product	product	NOUN
ejde-610	70	48	〈	〈	PROPN
ejde-610	70	49	u	u	NOUN
ejde-610	70	50	,	,	PUNCT
ejde-610	70	51	v	v	NOUN
ejde-610	70	52	〉	〉	NOUN
ejde-610	70	53	=	=	SYM
ejde-610	70	54	∫	∫	PROPN
ejde-610	70	55	r3	r3	PROPN
ejde-610	70	56	(	(	PUNCT
ejde-610	70	57	∇u	∇u	PROPN
ejde-610	70	58	·	·	PUNCT
ejde-610	70	59	∇v	∇v	PROPN
ejde-610	70	60	+	+	PUNCT
ejde-610	70	61	ṽ	ṽ	PROPN
ejde-610	70	62	(	(	PUNCT
ejde-610	70	63	x)uv	x)uv	PROPN
ejde-610	70	64	)	)	PUNCT
ejde-610	70	65	dx	dx	PROPN
ejde-610	70	66	and	and	CCONJ
ejde-610	70	67	the	the	DET
ejde-610	70	68	norm	norm	NOUN
ejde-610	71	1	‖u‖	‖u‖	PROPN
ejde-610	71	2	=	=	SYM
ejde-610	71	3	〈	〈	PROPN
ejde-610	71	4	u	u	PROPN
ejde-610	71	5	,	,	PUNCT
ejde-610	71	6	u〉1/2	u〉1/2	PROPN
ejde-610	71	7	.	.	PUNCT
ejde-610	72	1	we	we	PRON
ejde-610	72	2	also	also	ADV
ejde-610	72	3	know	know	VERB
ejde-610	72	4	that	that	SCONJ
ejde-610	72	5	if	if	SCONJ
ejde-610	72	6	v	v	NOUN
ejde-610	72	7	is	be	AUX
ejde-610	72	8	coercive	coercive	ADJ
ejde-610	72	9	,	,	PUNCT
ejde-610	72	10	then	then	ADV
ejde-610	72	11	(	(	PUNCT
ejde-610	72	12	a1	a1	NOUN
ejde-610	72	13	)	)	PUNCT
ejde-610	72	14	is	be	AUX
ejde-610	72	15	satisfied	satisfied	ADJ
ejde-610	72	16	.	.	PUNCT
ejde-610	73	1	obviously	obviously	ADV
ejde-610	73	2	,	,	PUNCT
ejde-610	73	3	the	the	DET
ejde-610	73	4	embedding	embed	VERB
ejde-610	73	5	e	e	NOUN
ejde-610	73	6	↪	↪	PROPN
ejde-610	73	7	→	→	SYM
ejde-610	73	8	ls(r3	ls(r3	X
ejde-610	73	9	)	)	PUNCT
ejde-610	73	10	is	be	AUX
ejde-610	73	11	continuous	continuous	ADJ
ejde-610	73	12	,	,	PUNCT
ejde-610	73	13	for	for	ADP
ejde-610	73	14	each	each	DET
ejde-610	73	15	s	s	X
ejde-610	73	16	∈	∈	PROPN
ejde-610	74	1	[	[	X
ejde-610	74	2	2	2	NUM
ejde-610	74	3	,	,	PUNCT
ejde-610	74	4	2∗	2∗	NUM
ejde-610	74	5	]	]	PUNCT
ejde-610	74	6	.	.	PUNCT
ejde-610	75	1	the	the	DET
ejde-610	75	2	norm	norm	NOUN
ejde-610	75	3	on	on	ADP
ejde-610	75	4	ls	ls	PROPN
ejde-610	75	5	=	=	X
ejde-610	75	6	ls(r3	ls(r3	PROPN
ejde-610	75	7	)	)	PUNCT
ejde-610	75	8	with	with	ADP
ejde-610	75	9	1	1	NUM
ejde-610	75	10	<	<	X
ejde-610	75	11	s	s	X
ejde-610	75	12	<	<	X
ejde-610	75	13	∞	∞	PROPN
ejde-610	75	14	is	be	AUX
ejde-610	75	15	|u|s	|u|s	PROPN
ejde-610	75	16	=	=	PRON
ejde-610	75	17	(	(	PUNCT
ejde-610	75	18	∫	∫	PROPN
ejde-610	75	19	r3	r3	PROPN
ejde-610	75	20	|u|s	|u|s	PROPN
ejde-610	75	21	dx)1	dx)1	PROPN
ejde-610	75	22	/	/	SYM
ejde-610	75	23	s.	s.	PROPN
ejde-610	75	24	consequently	consequently	ADV
ejde-610	75	25	,	,	PUNCT
ejde-610	75	26	for	for	ADP
ejde-610	75	27	each	each	DET
ejde-610	75	28	s	s	X
ejde-610	75	29	∈	∈	PROPN
ejde-610	76	1	[	[	X
ejde-610	76	2	2	2	NUM
ejde-610	76	3	,	,	PUNCT
ejde-610	76	4	6	6	NUM
ejde-610	76	5	]	]	PUNCT
ejde-610	76	6	,	,	PUNCT
ejde-610	76	7	there	there	PRON
ejde-610	76	8	exists	exist	VERB
ejde-610	76	9	a	a	DET
ejde-610	76	10	constant	constant	ADJ
ejde-610	76	11	ds	ds	NOUN
ejde-610	76	12	>	>	X
ejde-610	76	13	0	0	NUM
ejde-610	76	14	such	such	ADJ
ejde-610	76	15	that	that	SCONJ
ejde-610	76	16	|u|s	|u|s	PROPN
ejde-610	76	17	≤	≤	PROPN
ejde-610	76	18	ds‖u‖	ds‖u‖	PROPN
ejde-610	76	19	,	,	PUNCT
ejde-610	76	20	∀u	∀u	PROPN
ejde-610	76	21	∈	∈	PROPN
ejde-610	76	22	e.	e.	PROPN
ejde-610	76	23	(	(	PUNCT
ejde-610	76	24	1.4	1.4	NUM
ejde-610	76	25	)	)	PUNCT
ejde-610	76	26	d(r3	d(r3	NOUN
ejde-610	76	27	)	)	PUNCT
ejde-610	76	28	is	be	AUX
ejde-610	76	29	the	the	DET
ejde-610	76	30	completion	completion	NOUN
ejde-610	76	31	of	of	ADP
ejde-610	76	32	c∞0	c∞0	PROPN
ejde-610	76	33	(	(	PUNCT
ejde-610	76	34	r3	r3	PROPN
ejde-610	76	35	)	)	PUNCT
ejde-610	76	36	with	with	ADP
ejde-610	76	37	respect	respect	NOUN
ejde-610	76	38	to	to	ADP
ejde-610	76	39	the	the	DET
ejde-610	76	40	norm	norm	NOUN
ejde-610	76	41	‖u‖d	‖u‖d	NOUN
ejde-610	76	42	:	:	PUNCT
ejde-610	76	43	=	=	SYM
ejde-610	76	44	|∇u|2	|∇u|2	X
ejde-610	76	45	+	+	X
ejde-610	76	46	|∇u|4	|∇u|4	NOUN
ejde-610	76	47	.	.	PUNCT
ejde-610	76	48	d(r3	d(r3	NOUN
ejde-610	76	49	)	)	PUNCT
ejde-610	76	50	is	be	AUX
ejde-610	76	51	continuously	continuously	ADV
ejde-610	76	52	embedded	embed	VERB
ejde-610	76	53	in	in	ADP
ejde-610	76	54	d1,2(r3	d1,2(r3	PROPN
ejde-610	76	55	)	)	PUNCT
ejde-610	76	56	.	.	PUNCT
ejde-610	77	1	by	by	ADP
ejde-610	77	2	the	the	DET
ejde-610	77	3	sobolev	sobolev	NOUN
ejde-610	77	4	inequality	inequality	NOUN
ejde-610	77	5	,	,	PUNCT
ejde-610	77	6	we	we	PRON
ejde-610	77	7	know	know	VERB
ejde-610	77	8	that	that	SCONJ
ejde-610	77	9	d1,2(r3	d1,2(r3	VERB
ejde-610	77	10	)	)	PUNCT
ejde-610	77	11	is	be	AUX
ejde-610	77	12	continuously	continuously	ADV
ejde-610	77	13	embedded	embed	VERB
ejde-610	77	14	in	in	ADP
ejde-610	77	15	l6	l6	PROPN
ejde-610	77	16	=	=	SYM
ejde-610	77	17	l6(r3	l6(r3	NOUN
ejde-610	77	18	)	)	PUNCT
ejde-610	77	19	and	and	CCONJ
ejde-610	77	20	d(r3	d(r3	NOUN
ejde-610	77	21	)	)	PUNCT
ejde-610	77	22	is	be	AUX
ejde-610	77	23	continuously	continuously	ADV
ejde-610	77	24	embedded	embed	VERB
ejde-610	77	25	in	in	ADP
ejde-610	77	26	l∞	l∞	NOUN
ejde-610	77	27	=	=	SYM
ejde-610	78	1	l∞(r3	l∞(r3	PROPN
ejde-610	78	2	)	)	PUNCT
ejde-610	78	3	.	.	PUNCT
ejde-610	79	1	system	system	NOUN
ejde-610	79	2	(	(	PUNCT
ejde-610	79	3	1.1	1.1	NUM
ejde-610	79	4	)	)	PUNCT
ejde-610	79	5	has	have	VERB
ejde-610	79	6	a	a	DET
ejde-610	79	7	variational	variational	ADJ
ejde-610	79	8	structure	structure	NOUN
ejde-610	79	9	.	.	PUNCT
ejde-610	80	1	in	in	ADP
ejde-610	80	2	fact	fact	NOUN
ejde-610	80	3	,	,	PUNCT
ejde-610	80	4	we	we	PRON
ejde-610	80	5	consider	consider	VERB
ejde-610	80	6	the	the	DET
ejde-610	80	7	functional	functional	ADJ
ejde-610	80	8	j	j	NOUN
ejde-610	80	9	:	:	PUNCT
ejde-610	81	1	e	e	X
ejde-610	81	2	×d(r3)→	×d(r3)→	NUM
ejde-610	81	3	r	r	NOUN
ejde-610	81	4	defined	define	VERB
ejde-610	81	5	by	by	ADP
ejde-610	81	6	j	j	PROPN
ejde-610	81	7	(	(	PUNCT
ejde-610	81	8	u	u	PROPN
ejde-610	81	9	,	,	PUNCT
ejde-610	81	10	φ	φ	NUM
ejde-610	81	11	)	)	PUNCT
ejde-610	81	12	=	=	SYM
ejde-610	81	13	1	1	NUM
ejde-610	81	14	2	2	NUM
ejde-610	81	15	∫	∫	NOUN
ejde-610	81	16	r3	r3	PROPN
ejde-610	81	17	(	(	PUNCT
ejde-610	81	18	|∇u|2	|∇u|2	X
ejde-610	81	19	+	+	CCONJ
ejde-610	81	20	v	v	X
ejde-610	81	21	(	(	PUNCT
ejde-610	81	22	x)u2	x)u2	PROPN
ejde-610	81	23	−	−	PROPN
ejde-610	81	24	(	(	PUNCT
ejde-610	81	25	2ω	2ω	NOUN
ejde-610	81	26	+	+	CCONJ
ejde-610	81	27	φ)φu2	φ)φu2	ADJ
ejde-610	81	28	)	)	PUNCT
ejde-610	81	29	dx−	dx−	SYM
ejde-610	81	30	1	1	NUM
ejde-610	81	31	8π	8π	NUM
ejde-610	81	32	∫	∫	PROPN
ejde-610	81	33	r3	r3	PROPN
ejde-610	81	34	|∇φ|2	|∇φ|2	NUM
ejde-610	81	35	dx	dx	PROPN
ejde-610	81	36	4	4	NUM
ejde-610	81	37	l.	l.	PROPN
ejde-610	81	38	wang	wang	PROPN
ejde-610	81	39	,	,	PUNCT
ejde-610	81	40	p.	p.	PROPN
ejde-610	81	41	zhao	zhao	PROPN
ejde-610	81	42	,	,	PUNCT
ejde-610	81	43	d.	d.	PROPN
ejde-610	81	44	zhang	zhang	PROPN
ejde-610	81	45	ejde-2024/18	ejde-2024/18	PROPN
ejde-610	82	1	−	−	PROPN
ejde-610	82	2	β	β	X
ejde-610	82	3	16π	16π	PROPN
ejde-610	82	4	∫	∫	PROPN
ejde-610	82	5	r3	r3	PROPN
ejde-610	82	6	|∇φ|4	|∇φ|4	ADV
ejde-610	82	7	dx−	dx−	NUM
ejde-610	82	8	∫	∫	PROPN
ejde-610	82	9	r3	r3	PROPN
ejde-610	82	10	f	f	PROPN
ejde-610	82	11	(	(	PUNCT
ejde-610	82	12	x	x	X
ejde-610	82	13	,	,	PUNCT
ejde-610	82	14	u	u	NOUN
ejde-610	82	15	)	)	PUNCT
ejde-610	82	16	dx	dx	PROPN
ejde-610	82	17	.	.	PUNCT
ejde-610	83	1	solutions	solution	NOUN
ejde-610	83	2	(	(	PUNCT
ejde-610	83	3	u	u	NOUN
ejde-610	83	4	,	,	PUNCT
ejde-610	83	5	φ	φ	NOUN
ejde-610	83	6	)	)	PUNCT
ejde-610	83	7	∈	∈	PROPN
ejde-610	83	8	e	e	X
ejde-610	83	9	×d(r3	×d(r3	PROPN
ejde-610	83	10	)	)	PUNCT
ejde-610	83	11	of	of	ADP
ejde-610	83	12	system	system	NOUN
ejde-610	83	13	(	(	PUNCT
ejde-610	83	14	1.1	1.1	NUM
ejde-610	83	15	)	)	PUNCT
ejde-610	83	16	are	be	AUX
ejde-610	83	17	the	the	DET
ejde-610	83	18	critical	critical	ADJ
ejde-610	83	19	points	point	NOUN
ejde-610	83	20	of	of	ADP
ejde-610	83	21	j	j	PROPN
ejde-610	83	22	.	.	PUNCT
ejde-610	84	1	as	as	SCONJ
ejde-610	84	2	it	it	PRON
ejde-610	84	3	is	be	AUX
ejde-610	84	4	pointed	point	VERB
ejde-610	84	5	in	in	ADP
ejde-610	84	6	[	[	X
ejde-610	84	7	14	14	NUM
ejde-610	84	8	]	]	PUNCT
ejde-610	84	9	,	,	PUNCT
ejde-610	84	10	the	the	DET
ejde-610	84	11	functional	functional	ADJ
ejde-610	84	12	j	j	PROPN
ejde-610	84	13	is	be	AUX
ejde-610	84	14	strongly	strongly	ADV
ejde-610	84	15	indefinite	indefinite	ADJ
ejde-610	84	16	and	and	CCONJ
ejde-610	84	17	is	be	AUX
ejde-610	84	18	difficult	difficult	ADJ
ejde-610	84	19	to	to	PART
ejde-610	84	20	investigate	investigate	VERB
ejde-610	84	21	.	.	PUNCT
ejde-610	85	1	by	by	ADP
ejde-610	85	2	the	the	DET
ejde-610	85	3	reduction	reduction	NOUN
ejde-610	85	4	method	method	NOUN
ejde-610	85	5	described	describe	VERB
ejde-610	85	6	in	in	ADP
ejde-610	85	7	[	[	X
ejde-610	85	8	6	6	NUM
ejde-610	85	9	]	]	PUNCT
ejde-610	85	10	,	,	PUNCT
ejde-610	85	11	we	we	PRON
ejde-610	85	12	are	be	AUX
ejde-610	85	13	led	lead	VERB
ejde-610	85	14	to	to	ADP
ejde-610	85	15	the	the	DET
ejde-610	85	16	study	study	NOUN
ejde-610	85	17	of	of	ADP
ejde-610	85	18	a	a	DET
ejde-610	85	19	new	new	ADJ
ejde-610	85	20	functional	functional	ADJ
ejde-610	85	21	i	i	PRON
ejde-610	85	22	:	:	PUNCT
ejde-610	86	1	e	e	X
ejde-610	86	2	→	→	SYM
ejde-610	86	3	r	r	NOUN
ejde-610	86	4	defined	define	VERB
ejde-610	86	5	by	by	ADP
ejde-610	86	6	i(u	i(u	PROPN
ejde-610	86	7	)	)	PUNCT
ejde-610	87	1	=	=	SYM
ejde-610	87	2	j	j	PROPN
ejde-610	87	3	(	(	PUNCT
ejde-610	87	4	u	u	NOUN
ejde-610	87	5	,	,	PUNCT
ejde-610	87	6	φu	φu	NOUN
ejde-610	87	7	)	)	PUNCT
ejde-610	87	8	.	.	PUNCT
ejde-610	88	1	by	by	ADP
ejde-610	88	2	proposition	proposition	NOUN
ejde-610	88	3	2.1	2.1	NUM
ejde-610	88	4	below	below	ADV
ejde-610	88	5	,	,	PUNCT
ejde-610	88	6	i(u	i(u	PROPN
ejde-610	88	7	)	)	PUNCT
ejde-610	88	8	as	as	SCONJ
ejde-610	88	9	defined	define	VERB
ejde-610	88	10	next	next	ADV
ejde-610	88	11	does	do	AUX
ejde-610	88	12	not	not	PART
ejde-610	88	13	present	present	VERB
ejde-610	88	14	such	such	ADJ
ejde-610	88	15	strongly	strongly	ADV
ejde-610	88	16	indefinite	indefinite	ADJ
ejde-610	88	17	nature	nature	NOUN
ejde-610	88	18	.	.	PUNCT
ejde-610	89	1	now	now	ADV
ejde-610	89	2	we	we	PRON
ejde-610	89	3	can	can	AUX
ejde-610	89	4	state	state	VERB
ejde-610	89	5	our	our	PRON
ejde-610	89	6	main	main	ADJ
ejde-610	89	7	result	result	NOUN
ejde-610	89	8	.	.	PUNCT
ejde-610	90	1	theorem	theorem	VERB
ejde-610	90	2	1.1	1.1	NUM
ejde-610	90	3	.	.	PUNCT
ejde-610	91	1	assume	assume	VERB
ejde-610	91	2	that	that	SCONJ
ejde-610	91	3	(	(	PUNCT
ejde-610	91	4	a1)—(a5	a1)—(a5	ADV
ejde-610	91	5	)	)	PUNCT
ejde-610	91	6	are	be	AUX
ejde-610	91	7	satisfied	satisfied	ADJ
ejde-610	91	8	,	,	PUNCT
ejde-610	91	9	and	and	CCONJ
ejde-610	91	10	f	f	PROPN
ejde-610	91	11	is	be	AUX
ejde-610	91	12	odd	odd	ADJ
ejde-610	91	13	in	in	ADP
ejde-610	91	14	u.	u.	PROPN
ejde-610	91	15	if	if	SCONJ
ejde-610	91	16	0	0	NUM
ejde-610	91	17	is	be	AUX
ejde-610	91	18	not	not	PART
ejde-610	91	19	an	an	DET
ejde-610	91	20	eigenvalue	eigenvalue	NOUN
ejde-610	91	21	of	of	ADP
ejde-610	91	22	(	(	PUNCT
ejde-610	91	23	2.2	2.2	NUM
ejde-610	91	24	)	)	PUNCT
ejde-610	91	25	,	,	PUNCT
ejde-610	91	26	then	then	ADV
ejde-610	91	27	(	(	PUNCT
ejde-610	91	28	1.1	1.1	NUM
ejde-610	91	29	)	)	PUNCT
ejde-610	91	30	has	have	VERB
ejde-610	91	31	a	a	DET
ejde-610	91	32	sequence	sequence	NOUN
ejde-610	91	33	of	of	ADP
ejde-610	91	34	solutions	solution	NOUN
ejde-610	91	35	(	(	PUNCT
ejde-610	91	36	un	un	PROPN
ejde-610	91	37	,	,	PUNCT
ejde-610	91	38	φn	φn	NOUN
ejde-610	91	39	)	)	PUNCT
ejde-610	91	40	∈	∈	PROPN
ejde-610	91	41	e×d(r3	e×d(r3	PROPN
ejde-610	91	42	)	)	PUNCT
ejde-610	91	43	such	such	ADJ
ejde-610	91	44	that	that	SCONJ
ejde-610	91	45	the	the	DET
ejde-610	91	46	energy	energy	NOUN
ejde-610	91	47	j	j	PROPN
ejde-610	91	48	(	(	PUNCT
ejde-610	91	49	un	un	PROPN
ejde-610	91	50	,	,	PUNCT
ejde-610	91	51	φn)→	φn)→	ADP
ejde-610	91	52	+	+	NOUN
ejde-610	91	53	∞.	∞.	PROPN
ejde-610	91	54	we	we	PRON
ejde-610	91	55	emphasize	emphasize	VERB
ejde-610	91	56	that	that	SCONJ
ejde-610	91	57	unlike	unlike	ADP
ejde-610	91	58	all	all	DET
ejde-610	91	59	previous	previous	ADJ
ejde-610	91	60	results	result	NOUN
ejde-610	91	61	about	about	ADP
ejde-610	91	62	system	system	NOUN
ejde-610	91	63	(	(	PUNCT
ejde-610	91	64	1.1	1.1	NUM
ejde-610	91	65	)	)	PUNCT
ejde-610	91	66	,	,	PUNCT
ejde-610	91	67	see	see	VERB
ejde-610	91	68	e.g.	e.g.	ADV
ejde-610	91	69	[	[	X
ejde-610	91	70	1	1	NUM
ejde-610	91	71	,	,	PUNCT
ejde-610	91	72	11	11	NUM
ejde-610	91	73	,	,	PUNCT
ejde-610	91	74	14	14	NUM
ejde-610	91	75	,	,	PUNCT
ejde-610	91	76	23	23	NUM
ejde-610	91	77	,	,	PUNCT
ejde-610	91	78	25	25	NUM
ejde-610	91	79	]	]	PUNCT
ejde-610	91	80	,	,	PUNCT
ejde-610	91	81	we	we	PRON
ejde-610	91	82	do	do	AUX
ejde-610	91	83	not	not	PART
ejde-610	91	84	assume	assume	VERB
ejde-610	91	85	that	that	SCONJ
ejde-610	91	86	the	the	DET
ejde-610	91	87	potential	potential	NOUN
ejde-610	91	88	is	be	AUX
ejde-610	91	89	the	the	DET
ejde-610	91	90	positive	positive	ADJ
ejde-610	91	91	constant	constant	ADJ
ejde-610	91	92	v	v	NOUN
ejde-610	91	93	(	(	PUNCT
ejde-610	91	94	x	x	NOUN
ejde-610	91	95	)	)	PUNCT
ejde-610	91	96	=	=	SYM
ejde-610	91	97	m2	m2	PROPN
ejde-610	91	98	−	−	PROPN
ejde-610	91	99	ω2	ω2	PROPN
ejde-610	91	100	.	.	PUNCT
ejde-610	92	1	we	we	PRON
ejde-610	92	2	allow	allow	VERB
ejde-610	92	3	the	the	DET
ejde-610	92	4	potential	potential	NOUN
ejde-610	92	5	v	v	AUX
ejde-610	92	6	be	be	AUX
ejde-610	92	7	sign	sign	NOUN
ejde-610	92	8	changing	change	VERB
ejde-610	92	9	.	.	PUNCT
ejde-610	93	1	the	the	DET
ejde-610	93	2	author	author	NOUN
ejde-610	93	3	[	[	X
ejde-610	93	4	20	20	NUM
ejde-610	93	5	]	]	PUNCT
ejde-610	93	6	considered	consider	VERB
ejde-610	93	7	the	the	DET
ejde-610	93	8	multiplicity	multiplicity	NOUN
ejde-610	93	9	of	of	ADP
ejde-610	93	10	solutions	solution	NOUN
ejde-610	93	11	for	for	ADP
ejde-610	93	12	klein	klein	PROPN
ejde-610	93	13	-	-	PUNCT
ejde-610	93	14	gordon	gordon	PROPN
ejde-610	93	15	-	-	PUNCT
ejde-610	93	16	maxwell	maxwell	PROPN
ejde-610	93	17	system	system	NOUN
ejde-610	93	18	.	.	PUNCT
ejde-610	94	1	there	there	ADV
ejde-610	94	2	the	the	DET
ejde-610	94	3	author	author	NOUN
ejde-610	94	4	assumed	assume	VERB
ejde-610	94	5	in	in	ADP
ejde-610	94	6	addition	addition	NOUN
ejde-610	94	7	that	that	SCONJ
ejde-610	94	8	α	α	PRON
ejde-610	94	9	=	=	SYM
ejde-610	94	10	infx∈r3	infx∈r3	PROPN
ejde-610	94	11	v	v	NOUN
ejde-610	94	12	(	(	PUNCT
ejde-610	94	13	x	x	X
ejde-610	94	14	)	)	PUNCT
ejde-610	94	15	>	>	X
ejde-610	94	16	0	0	NUM
ejde-610	94	17	,	,	PUNCT
ejde-610	94	18	and	and	CCONJ
ejde-610	94	19	(	(	PUNCT
ejde-610	94	20	ar	ar	NOUN
ejde-610	94	21	)	)	PUNCT
ejde-610	94	22	or	or	CCONJ
ejde-610	94	23	(	(	PUNCT
ejde-610	94	24	a6	a6	NOUN
ejde-610	94	25	)	)	PUNCT
ejde-610	94	26	.	.	PUNCT
ejde-610	95	1	when	when	SCONJ
ejde-610	95	2	v	v	NOUN
ejde-610	95	3	is	be	AUX
ejde-610	95	4	positive	positive	ADJ
ejde-610	95	5	,	,	PUNCT
ejde-610	95	6	the	the	DET
ejde-610	95	7	quadratic	quadratic	ADJ
ejde-610	95	8	part	part	NOUN
ejde-610	95	9	of	of	ADP
ejde-610	95	10	the	the	DET
ejde-610	95	11	functional	functional	ADJ
ejde-610	95	12	i	i	PRON
ejde-610	95	13	(	(	PUNCT
ejde-610	95	14	see	see	VERB
ejde-610	95	15	(	(	PUNCT
ejde-610	95	16	1.3	1.3	NUM
ejde-610	95	17	)	)	PUNCT
ejde-610	95	18	)	)	PUNCT
ejde-610	95	19	is	be	AUX
ejde-610	95	20	positively	positively	ADV
ejde-610	95	21	definite	definite	ADJ
ejde-610	95	22	,	,	PUNCT
ejde-610	95	23	and	and	CCONJ
ejde-610	95	24	i	i	PRON
ejde-610	95	25	has	have	VERB
ejde-610	95	26	a	a	DET
ejde-610	95	27	mountain	mountain	NOUN
ejde-610	95	28	pass	pass	NOUN
ejde-610	95	29	geometry	geometry	NOUN
ejde-610	95	30	.	.	PUNCT
ejde-610	96	1	therefore	therefore	ADV
ejde-610	96	2	,	,	PUNCT
ejde-610	96	3	the	the	DET
ejde-610	96	4	mountain	mountain	NOUN
ejde-610	96	5	pass	pass	VERB
ejde-610	96	6	lemma	lemma	PROPN
ejde-610	96	7	[	[	X
ejde-610	96	8	22	22	NUM
ejde-610	96	9	]	]	PUNCT
ejde-610	96	10	can	can	AUX
ejde-610	96	11	be	be	AUX
ejde-610	96	12	applied	apply	VERB
ejde-610	96	13	.	.	PUNCT
ejde-610	97	1	in	in	ADP
ejde-610	97	2	our	our	PRON
ejde-610	97	3	case	case	NOUN
ejde-610	97	4	,	,	PUNCT
ejde-610	97	5	the	the	DET
ejde-610	97	6	quadratic	quadratic	ADJ
ejde-610	97	7	part	part	NOUN
ejde-610	97	8	may	may	AUX
ejde-610	97	9	possesses	possess	VERB
ejde-610	97	10	a	a	DET
ejde-610	97	11	nontrivial	nontrivial	ADJ
ejde-610	97	12	negative	negative	ADJ
ejde-610	97	13	space	space	NOUN
ejde-610	97	14	e−	e−	PROPN
ejde-610	97	15	,	,	PUNCT
ejde-610	97	16	so	so	SCONJ
ejde-610	97	17	i	i	PRON
ejde-610	97	18	no	no	ADV
ejde-610	97	19	longer	long	ADV
ejde-610	97	20	possesses	possess	VERB
ejde-610	97	21	the	the	DET
ejde-610	97	22	mountain	mountain	NOUN
ejde-610	97	23	pass	pass	NOUN
ejde-610	97	24	geometry	geometry	NOUN
ejde-610	97	25	.	.	PUNCT
ejde-610	98	1	therefore	therefore	ADV
ejde-610	98	2	the	the	DET
ejde-610	98	3	methods	method	NOUN
ejde-610	98	4	in	in	ADP
ejde-610	98	5	[	[	X
ejde-610	98	6	20	20	NUM
ejde-610	98	7	]	]	PUNCT
ejde-610	98	8	can	can	AUX
ejde-610	98	9	not	not	PART
ejde-610	98	10	be	be	AUX
ejde-610	98	11	applied	apply	VERB
ejde-610	98	12	here	here	ADV
ejde-610	98	13	.	.	PUNCT
ejde-610	99	1	to	to	PART
ejde-610	99	2	obtain	obtain	VERB
ejde-610	99	3	our	our	PRON
ejde-610	99	4	result	result	NOUN
ejde-610	99	5	,	,	PUNCT
ejde-610	99	6	we	we	PRON
ejde-610	99	7	adopt	adopt	VERB
ejde-610	99	8	a	a	DET
ejde-610	99	9	technique	technique	NOUN
ejde-610	99	10	developed	develop	VERB
ejde-610	99	11	in	in	ADP
ejde-610	99	12	[	[	X
ejde-610	99	13	13	13	NUM
ejde-610	99	14	]	]	PUNCT
ejde-610	99	15	.	.	PUNCT
ejde-610	100	1	we	we	PRON
ejde-610	100	2	denote	denote	VERB
ejde-610	100	3	by	by	ADP
ejde-610	100	4	”	"	PUNCT
ejde-610	100	5	⇀	⇀	PROPN
ejde-610	100	6	”	"	PUNCT
ejde-610	100	7	weak	weak	ADJ
ejde-610	100	8	convergence	convergence	NOUN
ejde-610	100	9	,	,	PUNCT
ejde-610	100	10	and	and	CCONJ
ejde-610	100	11	by	by	ADP
ejde-610	100	12	”	"	PUNCT
ejde-610	100	13	→	→	SYM
ejde-610	100	14	”	"	PUNCT
ejde-610	100	15	strong	strong	ADJ
ejde-610	100	16	convergence	convergence	NOUN
ejde-610	100	17	.	.	PUNCT
ejde-610	101	1	also	also	ADV
ejde-610	101	2	if	if	SCONJ
ejde-610	101	3	we	we	PRON
ejde-610	101	4	take	take	VERB
ejde-610	101	5	a	a	DET
ejde-610	101	6	subsequence	subsequence	NOUN
ejde-610	101	7	of	of	ADP
ejde-610	101	8	a	a	DET
ejde-610	101	9	sequence	sequence	NOUN
ejde-610	101	10	{	{	PUNCT
ejde-610	101	11	un	un	PROPN
ejde-610	101	12	}	}	PUNCT
ejde-610	101	13	,	,	PUNCT
ejde-610	101	14	we	we	PRON
ejde-610	101	15	shall	shall	AUX
ejde-610	101	16	denote	denote	VERB
ejde-610	101	17	it	it	PRON
ejde-610	101	18	again	again	ADV
ejde-610	101	19	{	{	PUNCT
ejde-610	101	20	un	un	PROPN
ejde-610	101	21	}	}	PUNCT
ejde-610	101	22	.	.	PUNCT
ejde-610	102	1	2	2	X
ejde-610	102	2	.	.	X
ejde-610	102	3	variational	variational	ADJ
ejde-610	102	4	setting	setting	NOUN
ejde-610	102	5	and	and	CCONJ
ejde-610	102	6	compactness	compactness	NOUN
ejde-610	102	7	condition	condition	NOUN
ejde-610	102	8	evidently	evidently	ADV
ejde-610	102	9	,	,	PUNCT
ejde-610	102	10	the	the	DET
ejde-610	102	11	properties	property	NOUN
ejde-610	102	12	of	of	ADP
ejde-610	102	13	φu	φu	PRON
ejde-610	102	14	play	play	VERB
ejde-610	102	15	an	an	DET
ejde-610	102	16	important	important	ADJ
ejde-610	102	17	role	role	NOUN
ejde-610	102	18	in	in	ADP
ejde-610	102	19	the	the	DET
ejde-610	102	20	study	study	NOUN
ejde-610	102	21	of	of	ADP
ejde-610	102	22	j	j	PROPN
ejde-610	102	23	.	.	PUNCT
ejde-610	103	1	so	so	ADV
ejde-610	103	2	we	we	PRON
ejde-610	103	3	need	need	VERB
ejde-610	103	4	the	the	DET
ejde-610	103	5	following	follow	VERB
ejde-610	103	6	technical	technical	ADJ
ejde-610	103	7	results	result	NOUN
ejde-610	103	8	.	.	PUNCT
ejde-610	104	1	proposition	proposition	NOUN
ejde-610	104	2	2.1	2.1	NUM
ejde-610	104	3	.	.	PUNCT
ejde-610	105	1	for	for	ADP
ejde-610	105	2	each	each	DET
ejde-610	105	3	u	u	NOUN
ejde-610	105	4	∈	∈	PROPN
ejde-610	105	5	h1(r3	h1(r3	NOUN
ejde-610	105	6	)	)	PUNCT
ejde-610	105	7	,	,	PUNCT
ejde-610	105	8	there	there	PRON
ejde-610	105	9	exists	exist	VERB
ejde-610	105	10	a	a	DET
ejde-610	105	11	unique	unique	ADJ
ejde-610	105	12	φ	φ	NOUN
ejde-610	105	13	=	=	SYM
ejde-610	105	14	φu	φu	PROPN
ejde-610	105	15	∈	∈	PROPN
ejde-610	105	16	d(r3	d(r3	NOUN
ejde-610	105	17	)	)	PUNCT
ejde-610	105	18	which	which	PRON
ejde-610	105	19	satisfies	satisfy	VERB
ejde-610	105	20	∆φ+	∆φ+	PROPN
ejde-610	105	21	β∆4φ	β∆4φ	PUNCT
ejde-610	106	1	=	=	PUNCT
ejde-610	107	1	4π(φ+	4π(φ+	NUM
ejde-610	107	2	ω)u2	ω)u2	PROPN
ejde-610	107	3	in	in	ADP
ejde-610	107	4	r3	r3	PROPN
ejde-610	107	5	.	.	PUNCT
ejde-610	108	1	moreover	moreover	ADV
ejde-610	108	2	,	,	PUNCT
ejde-610	108	3	the	the	DET
ejde-610	108	4	map	map	NOUN
ejde-610	108	5	φ	φ	X
ejde-610	108	6	:	:	PUNCT
ejde-610	108	7	u	u	PROPN
ejde-610	108	8	∈	∈	PROPN
ejde-610	108	9	h1(r3	h1(r3	AUX
ejde-610	108	10	)	)	PUNCT
ejde-610	108	11	7→	7→	NUM
ejde-610	108	12	φu	φu	ADP
ejde-610	108	13	∈	∈	PROPN
ejde-610	108	14	d(r3	d(r3	NOUN
ejde-610	108	15	)	)	PUNCT
ejde-610	108	16	is	be	AUX
ejde-610	108	17	continuously	continuously	ADV
ejde-610	108	18	differentiable	differentiable	ADJ
ejde-610	108	19	,	,	PUNCT
ejde-610	108	20	and	and	CCONJ
ejde-610	108	21	(	(	PUNCT
ejde-610	108	22	i	i	NOUN
ejde-610	108	23	)	)	PUNCT
ejde-610	108	24	−ω	−ω	ADJ
ejde-610	108	25	≤	≤	PROPN
ejde-610	108	26	φu	φu	NOUN
ejde-610	108	27	≤	≤	NOUN
ejde-610	108	28	0	0	NUM
ejde-610	108	29	on	on	ADP
ejde-610	108	30	the	the	DET
ejde-610	108	31	set	set	NOUN
ejde-610	108	32	{	{	PUNCT
ejde-610	108	33	x	x	SYM
ejde-610	108	34	∈	∈	NOUN
ejde-610	108	35	r3|u(x	r3|u(x	VERB
ejde-610	108	36	)	)	PUNCT
ejde-610	108	37	6=	6=	ADP
ejde-610	108	38	0	0	NUM
ejde-610	108	39	}	}	PUNCT
ejde-610	108	40	;	;	PUNCT
ejde-610	108	41	(	(	PUNCT
ejde-610	108	42	ii	ii	NOUN
ejde-610	108	43	)	)	PUNCT
ejde-610	108	44	∫	∫	PROPN
ejde-610	109	1	r3(|∇φu|2	r3(|∇φu|2	PROPN
ejde-610	109	2	+	+	SYM
ejde-610	109	3	β|∇φu|4	β|∇φu|4	NUM
ejde-610	109	4	)	)	PUNCT
ejde-610	109	5	dx	dx	PROPN
ejde-610	109	6	≤	≤	ADJ
ejde-610	109	7	4πω2|u|22	4πω2|u|22	NOUN
ejde-610	109	8	.	.	PUNCT
ejde-610	110	1	the	the	DET
ejde-610	110	2	first	first	ADJ
ejde-610	110	3	part	part	NOUN
ejde-610	110	4	of	of	ADP
ejde-610	110	5	proposition	proposition	NOUN
ejde-610	110	6	2.1	2.1	NUM
ejde-610	110	7	was	be	AUX
ejde-610	110	8	proved	prove	VERB
ejde-610	110	9	in	in	ADP
ejde-610	110	10	[	[	X
ejde-610	110	11	14	14	NUM
ejde-610	110	12	]	]	PUNCT
ejde-610	110	13	,	,	PUNCT
ejde-610	110	14	and	and	CCONJ
ejde-610	110	15	the	the	DET
ejde-610	110	16	second	second	ADJ
ejde-610	110	17	part	part	NOUN
ejde-610	110	18	in	in	ADP
ejde-610	110	19	[	[	X
ejde-610	110	20	21	21	NUM
ejde-610	110	21	]	]	PUNCT
ejde-610	110	22	.	.	PUNCT
ejde-610	111	1	after	after	ADP
ejde-610	111	2	multiplying	multiply	VERB
ejde-610	111	3	∆φ+	∆φ+	PROPN
ejde-610	111	4	β∆4φ	β∆4φ	PUNCT
ejde-610	112	1	=	=	PUNCT
ejde-610	113	1	4π(φ+	4π(φ+	NUM
ejde-610	113	2	ω)u2	ω)u2	PROPN
ejde-610	113	3	by	by	ADP
ejde-610	113	4	φu	φu	NOUN
ejde-610	113	5	and	and	CCONJ
ejde-610	113	6	integrating	integrate	VERB
ejde-610	113	7	by	by	ADP
ejde-610	113	8	parts	part	NOUN
ejde-610	113	9	,	,	PUNCT
ejde-610	113	10	by	by	ADP
ejde-610	113	11	the	the	DET
ejde-610	113	12	condition	condition	NOUN
ejde-610	113	13	(	(	PUNCT
ejde-610	113	14	i	i	NOUN
ejde-610	113	15	)	)	PUNCT
ejde-610	113	16	,	,	PUNCT
ejde-610	113	17	we	we	PRON
ejde-610	113	18	obtain	obtain	VERB
ejde-610	113	19	that∫	that∫	NOUN
ejde-610	113	20	r3	r3	PROPN
ejde-610	113	21	(	(	PUNCT
ejde-610	113	22	|∇φu|2	|∇φu|2	NOUN
ejde-610	113	23	+	+	CCONJ
ejde-610	113	24	β|∇φu|4	β|∇φu|4	NUM
ejde-610	113	25	)	)	PUNCT
ejde-610	113	26	dx	dx	PROPN
ejde-610	114	1	=	=	SYM
ejde-610	114	2	−4π	−4π	PROPN
ejde-610	114	3	∫	∫	PROPN
ejde-610	114	4	r3	r3	PROPN
ejde-610	114	5	(	(	PUNCT
ejde-610	114	6	φu	φu	NOUN
ejde-610	114	7	+	+	CCONJ
ejde-610	114	8	ω)φuu	ω)φuu	PROPN
ejde-610	114	9	2	2	NUM
ejde-610	114	10	dx	dx	PROPN
ejde-610	114	11	≤	≤	NUM
ejde-610	114	12	−4πω	−4πω	PROPN
ejde-610	114	13	∫	∫	PROPN
ejde-610	114	14	r3	r3	PROPN
ejde-610	114	15	φuu	φuu	VERB
ejde-610	114	16	2	2	NUM
ejde-610	114	17	dx	dx	PROPN
ejde-610	114	18	≤	≤	ADJ
ejde-610	114	19	4πω2|u|22	4πω2|u|22	NOUN
ejde-610	114	20	.	.	PUNCT
ejde-610	115	1	ejde-2024/18	ejde-2024/18	PROPN
ejde-610	115	2	coupled	couple	VERB
ejde-610	115	3	klein	klein	PROPN
ejde-610	115	4	-	-	PUNCT
ejde-610	115	5	gordon	gordon	PROPN
ejde-610	115	6	and	and	CCONJ
ejde-610	115	7	born	bear	VERB
ejde-610	115	8	-	-	PUNCT
ejde-610	115	9	infeld	infeld	NOUN
ejde-610	115	10	equatons	equaton	NOUN
ejde-610	115	11	5	5	NUM
ejde-610	115	12	by	by	ADP
ejde-610	115	13	proposition	proposition	NOUN
ejde-610	115	14	2.1	2.1	NUM
ejde-610	115	15	and	and	CCONJ
ejde-610	115	16	(	(	PUNCT
ejde-610	115	17	1.1	1.1	NUM
ejde-610	115	18	)	)	PUNCT
ejde-610	115	19	,	,	PUNCT
ejde-610	115	20	if	if	SCONJ
ejde-610	115	21	u	u	PROPN
ejde-610	115	22	∈	∈	PROPN
ejde-610	115	23	e	e	NOUN
ejde-610	115	24	is	be	AUX
ejde-610	115	25	a	a	DET
ejde-610	115	26	critical	critical	ADJ
ejde-610	115	27	point	point	NOUN
ejde-610	115	28	of	of	ADP
ejde-610	115	29	i	i	PRON
ejde-610	115	30	,	,	PUNCT
ejde-610	115	31	then	then	ADV
ejde-610	115	32	(	(	PUNCT
ejde-610	115	33	u	u	NOUN
ejde-610	115	34	,	,	PUNCT
ejde-610	115	35	φu	φu	NOUN
ejde-610	115	36	)	)	PUNCT
ejde-610	115	37	∈	∈	PROPN
ejde-610	115	38	e	e	X
ejde-610	115	39	×	×	NOUN
ejde-610	115	40	d(r3	d(r3	NOUN
ejde-610	115	41	)	)	PUNCT
ejde-610	115	42	is	be	AUX
ejde-610	115	43	a	a	DET
ejde-610	115	44	critical	critical	ADJ
ejde-610	115	45	point	point	NOUN
ejde-610	115	46	of	of	ADP
ejde-610	115	47	j	j	PROPN
ejde-610	115	48	,	,	PUNCT
ejde-610	115	49	that	that	ADV
ejde-610	115	50	is	is	ADV
ejde-610	115	51	,	,	PUNCT
ejde-610	115	52	(	(	PUNCT
ejde-610	115	53	u	u	NOUN
ejde-610	115	54	,	,	PUNCT
ejde-610	115	55	φu	φu	NOUN
ejde-610	115	56	)	)	PUNCT
ejde-610	115	57	∈	∈	PROPN
ejde-610	115	58	e	e	X
ejde-610	115	59	×	×	NOUN
ejde-610	115	60	d(r3	d(r3	NOUN
ejde-610	115	61	)	)	PUNCT
ejde-610	115	62	is	be	AUX
ejde-610	115	63	a	a	DET
ejde-610	115	64	solution	solution	NOUN
ejde-610	115	65	of	of	ADP
ejde-610	115	66	(	(	PUNCT
ejde-610	115	67	1.1	1.1	NUM
ejde-610	115	68	)	)	PUNCT
ejde-610	115	69	.	.	PUNCT
ejde-610	116	1	we	we	PRON
ejde-610	116	2	can	can	AUX
ejde-610	116	3	obtain	obtain	VERB
ejde-610	116	4	a	a	DET
ejde-610	116	5	c1	c1	NOUN
ejde-610	116	6	functional	functional	ADJ
ejde-610	117	1	i	i	PRON
ejde-610	117	2	:	:	PUNCT
ejde-610	117	3	e	e	X
ejde-610	117	4	→	→	SYM
ejde-610	117	5	r	r	NOUN
ejde-610	117	6	given	give	VERB
ejde-610	117	7	by	by	ADP
ejde-610	117	8	i(u	i(u	PROPN
ejde-610	117	9	)	)	PUNCT
ejde-610	118	1	=	=	SYM
ejde-610	118	2	j	j	PROPN
ejde-610	118	3	(	(	PUNCT
ejde-610	118	4	u	u	NOUN
ejde-610	118	5	,	,	PUNCT
ejde-610	118	6	φu	φu	NOUN
ejde-610	118	7	)	)	PUNCT
ejde-610	118	8	=	=	SYM
ejde-610	118	9	1	1	NUM
ejde-610	118	10	2	2	NUM
ejde-610	118	11	∫	∫	NOUN
ejde-610	118	12	r3	r3	PROPN
ejde-610	118	13	[	[	X
ejde-610	118	14	|∇u|2	|∇u|2	NOUN
ejde-610	118	15	+	+	SYM
ejde-610	118	16	v	v	X
ejde-610	118	17	(	(	PUNCT
ejde-610	118	18	x)u2	x)u2	PROPN
ejde-610	118	19	−	−	PROPN
ejde-610	118	20	(	(	PUNCT
ejde-610	118	21	2ω	2ω	PROPN
ejde-610	118	22	+	+	CCONJ
ejde-610	118	23	φu)φuu	φu)φuu	NOUN
ejde-610	118	24	2	2	NUM
ejde-610	118	25	]	]	PUNCT
ejde-610	118	26	dx	dx	PROPN
ejde-610	118	27	−	−	PROPN
ejde-610	118	28	1	1	NUM
ejde-610	118	29	8π	8π	NUM
ejde-610	118	30	∫	∫	PROPN
ejde-610	118	31	r3	r3	PROPN
ejde-610	118	32	|∇φu|2	|∇φu|2	NOUN
ejde-610	118	33	dx−	dx−	PROPN
ejde-610	118	34	β	β	PROPN
ejde-610	118	35	16π	16π	PROPN
ejde-610	118	36	∫	∫	PROPN
ejde-610	118	37	r3	r3	PROPN
ejde-610	118	38	|∇φu|4	|∇φu|4	NOUN
ejde-610	119	1	dx−	dx−	NUM
ejde-610	119	2	∫	∫	PROPN
ejde-610	119	3	r3	r3	PROPN
ejde-610	119	4	f	f	PROPN
ejde-610	119	5	(	(	PUNCT
ejde-610	119	6	x	x	X
ejde-610	119	7	,	,	PUNCT
ejde-610	119	8	u	u	NOUN
ejde-610	119	9	)	)	PUNCT
ejde-610	119	10	dx	dx	PROPN
ejde-610	120	1	=	=	SYM
ejde-610	120	2	1	1	NUM
ejde-610	120	3	2	2	NUM
ejde-610	120	4	∫	∫	NOUN
ejde-610	120	5	r3	r3	PROPN
ejde-610	120	6	(	(	PUNCT
ejde-610	120	7	|∇u|2	|∇u|2	X
ejde-610	120	8	+	+	CCONJ
ejde-610	120	9	v	v	X
ejde-610	120	10	(	(	PUNCT
ejde-610	120	11	x)u2	x)u2	PROPN
ejde-610	120	12	+	+	CCONJ
ejde-610	120	13	φ2	φ2	PROPN
ejde-610	120	14	uu	uu	INTJ
ejde-610	120	15	2	2	X
ejde-610	120	16	)	)	PUNCT
ejde-610	120	17	dx	dx	PROPN
ejde-610	121	1	+	+	CCONJ
ejde-610	121	2	1	1	NUM
ejde-610	121	3	8π	8π	NUM
ejde-610	121	4	∫	∫	PROPN
ejde-610	121	5	r3	r3	PROPN
ejde-610	121	6	|∇φu|2	|∇φu|2	PROPN
ejde-610	121	7	dx+	dx+	PROPN
ejde-610	121	8	3β	3β	PROPN
ejde-610	121	9	16π	16π	PROPN
ejde-610	122	1	∫	∫	PROPN
ejde-610	122	2	r3	r3	PROPN
ejde-610	122	3	|∇φu|4	|∇φu|4	NOUN
ejde-610	122	4	dx−	dx−	NUM
ejde-610	123	1	∫	∫	PROPN
ejde-610	123	2	r3	r3	PROPN
ejde-610	123	3	f	f	PROPN
ejde-610	123	4	(	(	PUNCT
ejde-610	123	5	x	x	X
ejde-610	123	6	,	,	PUNCT
ejde-610	123	7	u	u	NOUN
ejde-610	123	8	)	)	PUNCT
ejde-610	123	9	dx	dx	PROPN
ejde-610	123	10	=	=	SYM
ejde-610	123	11	1	1	NUM
ejde-610	123	12	2	2	NUM
ejde-610	123	13	∫	∫	NOUN
ejde-610	123	14	r3	r3	PROPN
ejde-610	123	15	(	(	PUNCT
ejde-610	123	16	|∇u|2	|∇u|2	X
ejde-610	123	17	+	+	CCONJ
ejde-610	123	18	v	v	X
ejde-610	123	19	(	(	PUNCT
ejde-610	123	20	x)u2	x)u2	PROPN
ejde-610	123	21	−	−	NUM
ejde-610	123	22	ωφuu2	ωφuu2	NOUN
ejde-610	123	23	)	)	PUNCT
ejde-610	123	24	dx+	dx+	NOUN
ejde-610	123	25	β	β	X
ejde-610	123	26	16π	16π	PROPN
ejde-610	123	27	∫	∫	PROPN
ejde-610	123	28	r3	r3	PROPN
ejde-610	123	29	|∇φu|4	|∇φu|4	NOUN
ejde-610	123	30	dx	dx	PROPN
ejde-610	123	31	−	−	PROPN
ejde-610	123	32	∫	∫	PROPN
ejde-610	123	33	r3	r3	PROPN
ejde-610	123	34	f	f	PROPN
ejde-610	123	35	(	(	PUNCT
ejde-610	123	36	x	x	X
ejde-610	123	37	,	,	PUNCT
ejde-610	123	38	u	u	NOUN
ejde-610	123	39	)	)	PUNCT
ejde-610	123	40	dx	dx	PROPN
ejde-610	123	41	.	.	PUNCT
ejde-610	124	1	(	(	PUNCT
ejde-610	124	2	2.1	2.1	NUM
ejde-610	124	3	)	)	PUNCT
ejde-610	124	4	we	we	PRON
ejde-610	124	5	consider	consider	VERB
ejde-610	124	6	the	the	DET
ejde-610	124	7	map	map	NOUN
ejde-610	124	8	φ	φ	X
ejde-610	124	9	:	:	PUNCT
ejde-610	125	1	e	e	X
ejde-610	125	2	→	→	SYM
ejde-610	125	3	d	d	PROPN
ejde-610	125	4	,	,	PUNCT
ejde-610	125	5	u→	u→	PROPN
ejde-610	125	6	φu	φu	NOUN
ejde-610	125	7	.	.	PUNCT
ejde-610	126	1	by	by	ADP
ejde-610	126	2	standard	standard	ADJ
ejde-610	126	3	arguments	argument	NOUN
ejde-610	126	4	,	,	PUNCT
ejde-610	126	5	φ	φ	PROPN
ejde-610	126	6	∈	∈	PROPN
ejde-610	126	7	c1(e	c1(e	PROPN
ejde-610	126	8	,	,	PUNCT
ejde-610	126	9	d	d	NOUN
ejde-610	126	10	)	)	PUNCT
ejde-610	126	11	.	.	PUNCT
ejde-610	127	1	the	the	DET
ejde-610	127	2	gateaux	gateaux	ADV
ejde-610	127	3	derivative	derivative	NOUN
ejde-610	127	4	of	of	ADP
ejde-610	127	5	i	i	PRON
ejde-610	127	6	is	be	AUX
ejde-610	127	7	〈	〈	PROPN
ejde-610	127	8	i	i	PRON
ejde-610	127	9	′(u	′(u	NOUN
ejde-610	127	10	)	)	PUNCT
ejde-610	127	11	,	,	PUNCT
ejde-610	127	12	v	v	NOUN
ejde-610	127	13	〉	〉	NOUN
ejde-610	127	14	=	=	SYM
ejde-610	127	15	∫	∫	PROPN
ejde-610	127	16	r3	r3	PROPN
ejde-610	127	17	(	(	PUNCT
ejde-610	127	18	∇u	∇u	PROPN
ejde-610	127	19	·	·	PUNCT
ejde-610	128	1	∇v	∇v	PROPN
ejde-610	128	2	+	+	NUM
ejde-610	128	3	v	v	X
ejde-610	128	4	(	(	PUNCT
ejde-610	128	5	x)uv	x)uv	PROPN
ejde-610	128	6	−	−	PROPN
ejde-610	128	7	(	(	PUNCT
ejde-610	128	8	2ω	2ω	PROPN
ejde-610	128	9	+	+	CCONJ
ejde-610	128	10	φu)φuuv	φu)φuuv	PROPN
ejde-610	128	11	)	)	PUNCT
ejde-610	128	12	dx−	dx−	NUM
ejde-610	128	13	∫	∫	PROPN
ejde-610	128	14	r3	r3	PROPN
ejde-610	128	15	f(x	f(x	PROPN
ejde-610	128	16	,	,	PUNCT
ejde-610	128	17	u)v	u)v	X
ejde-610	128	18	dx	dx	VERB
ejde-610	128	19	for	for	ADP
ejde-610	128	20	all	all	DET
ejde-610	128	21	u	u	NOUN
ejde-610	128	22	,	,	PUNCT
ejde-610	128	23	v	v	PROPN
ejde-610	128	24	∈	∈	PROPN
ejde-610	128	25	e.	e.	PROPN
ejde-610	128	26	furthermore	furthermore	ADV
ejde-610	128	27	,	,	PUNCT
ejde-610	128	28	under	under	ADP
ejde-610	128	29	the	the	DET
ejde-610	128	30	condition	condition	NOUN
ejde-610	128	31	(	(	PUNCT
ejde-610	128	32	a1	a1	NOUN
ejde-610	128	33	)	)	PUNCT
ejde-610	128	34	,	,	PUNCT
ejde-610	128	35	the	the	DET
ejde-610	128	36	embedding	embed	VERB
ejde-610	128	37	e	e	PROPN
ejde-610	128	38	↪	↪	PROPN
ejde-610	128	39	→	→	SYM
ejde-610	128	40	ls(r3	ls(r3	X
ejde-610	128	41	)	)	PUNCT
ejde-610	128	42	is	be	AUX
ejde-610	128	43	compact	compact	ADJ
ejde-610	128	44	for	for	ADP
ejde-610	128	45	any	any	DET
ejde-610	128	46	s	s	X
ejde-610	128	47	∈	∈	NOUN
ejde-610	129	1	[	[	X
ejde-610	129	2	2	2	NUM
ejde-610	129	3	,	,	PUNCT
ejde-610	129	4	6	6	NUM
ejde-610	129	5	)	)	PUNCT
ejde-610	129	6	(	(	PUNCT
ejde-610	129	7	see	see	VERB
ejde-610	129	8	[	[	X
ejde-610	129	9	3	3	NUM
ejde-610	129	10	]	]	NUM
ejde-610	129	11	)	)	PUNCT
ejde-610	129	12	.	.	PUNCT
ejde-610	130	1	by	by	ADP
ejde-610	130	2	the	the	DET
ejde-610	130	3	compact	compact	ADJ
ejde-610	130	4	embedding	embed	VERB
ejde-610	130	5	e	e	X
ejde-610	130	6	↪	↪	PROPN
ejde-610	130	7	→	→	SYM
ejde-610	130	8	l2(r3	l2(r3	NUM
ejde-610	130	9	)	)	PUNCT
ejde-610	130	10	and	and	CCONJ
ejde-610	130	11	the	the	DET
ejde-610	130	12	standard	standard	ADJ
ejde-610	130	13	elliptic	elliptic	ADJ
ejde-610	130	14	theory	theory	NOUN
ejde-610	130	15	[	[	X
ejde-610	130	16	33	33	NUM
ejde-610	130	17	]	]	PUNCT
ejde-610	130	18	,	,	PUNCT
ejde-610	130	19	it	it	PRON
ejde-610	130	20	is	be	AUX
ejde-610	130	21	easy	easy	ADJ
ejde-610	130	22	to	to	PART
ejde-610	130	23	see	see	VERB
ejde-610	130	24	that	that	SCONJ
ejde-610	130	25	the	the	DET
ejde-610	130	26	eigenvalue	eigenvalue	PROPN
ejde-610	130	27	problem	problem	NOUN
ejde-610	130	28	−4u+	−4u+	NOUN
ejde-610	130	29	v	v	X
ejde-610	130	30	(	(	PUNCT
ejde-610	130	31	x)u	x)u	PUNCT
ejde-610	130	32	=	=	PUNCT
ejde-610	130	33	λu	λu	X
ejde-610	130	34	,	,	PUNCT
ejde-610	130	35	u	u	PROPN
ejde-610	130	36	∈	∈	PROPN
ejde-610	130	37	e	e	X
ejde-610	130	38	(	(	PUNCT
ejde-610	130	39	2.2	2.2	NUM
ejde-610	130	40	)	)	PUNCT
ejde-610	130	41	possesses	possess	VERB
ejde-610	130	42	a	a	DET
ejde-610	130	43	complete	complete	ADJ
ejde-610	130	44	sequence	sequence	NOUN
ejde-610	130	45	of	of	ADP
ejde-610	130	46	eigenvalues	eigenvalue	NOUN
ejde-610	130	47	−∞	−∞	PUNCT
ejde-610	130	48	<	<	X
ejde-610	130	49	λ1	λ1	ADJ
ejde-610	130	50	≤	≤	NUM
ejde-610	130	51	λ2	λ2	NOUN
ejde-610	130	52	≤	≤	NUM
ejde-610	130	53	λ3	λ3	PROPN
ejde-610	130	54	≤	≤	PROPN
ejde-610	130	55	.	.	PUNCT
ejde-610	130	56	.	.	PUNCT
ejde-610	131	1	.	.	PUNCT
ejde-610	132	1	,	,	PUNCT
ejde-610	132	2	λj	λj	PROPN
ejde-610	132	3	→	→	SYM
ejde-610	132	4	+	+	NOUN
ejde-610	132	5	∞.	∞.	PROPN
ejde-610	132	6	each	each	PRON
ejde-610	132	7	λj	λj	PROPN
ejde-610	132	8	has	have	AUX
ejde-610	132	9	finite	finite	VERB
ejde-610	132	10	multiplicity	multiplicity	NOUN
ejde-610	132	11	and	and	CCONJ
ejde-610	132	12	|λj	|λj	NUM
ejde-610	132	13	|2	|2	NUM
ejde-610	132	14	=	=	SYM
ejde-610	132	15	1	1	X
ejde-610	132	16	.	.	X
ejde-610	132	17	denote	denote	VERB
ejde-610	132	18	ej	ej	PROPN
ejde-610	132	19	be	be	AUX
ejde-610	132	20	the	the	DET
ejde-610	132	21	eigenfunction	eigenfunction	NOUN
ejde-610	132	22	of	of	ADP
ejde-610	132	23	λj	λj	PROPN
ejde-610	132	24	.	.	PUNCT
ejde-610	133	1	e−	e−	PROPN
ejde-610	133	2	is	be	AUX
ejde-610	133	3	spanned	span	VERB
ejde-610	133	4	by	by	ADP
ejde-610	133	5	the	the	DET
ejde-610	133	6	eigenfunctions	eigenfunction	NOUN
ejde-610	133	7	corresponding	correspond	VERB
ejde-610	133	8	to	to	ADP
ejde-610	133	9	negative	negative	ADJ
ejde-610	133	10	eigenvalues	eigenvalue	NOUN
ejde-610	133	11	.	.	PUNCT
ejde-610	134	1	note	note	VERB
ejde-610	134	2	that	that	SCONJ
ejde-610	134	3	the	the	DET
ejde-610	134	4	negative	negative	ADJ
ejde-610	134	5	space	space	NOUN
ejde-610	134	6	e−	e−	PROPN
ejde-610	134	7	of	of	ADP
ejde-610	134	8	the	the	DET
ejde-610	134	9	quadratic	quadratic	ADJ
ejde-610	134	10	part	part	NOUN
ejde-610	134	11	of	of	ADP
ejde-610	134	12	i	i	PRON
ejde-610	134	13	is	be	AUX
ejde-610	134	14	nontrivial	nontrivial	ADJ
ejde-610	134	15	if	if	SCONJ
ejde-610	134	16	and	and	CCONJ
ejde-610	134	17	only	only	ADV
ejde-610	135	1	if	if	SCONJ
ejde-610	135	2	some	some	PRON
ejde-610	135	3	λj	λj	PROPN
ejde-610	135	4	is	be	AUX
ejde-610	135	5	negative	negative	ADJ
ejde-610	135	6	.	.	PUNCT
ejde-610	136	1	if	if	SCONJ
ejde-610	136	2	λ1	λ1	PROPN
ejde-610	136	3	>	>	X
ejde-610	136	4	0	0	PROPN
ejde-610	136	5	,	,	PUNCT
ejde-610	136	6	we	we	PRON
ejde-610	136	7	can	can	AUX
ejde-610	136	8	easy	easy	VERB
ejde-610	136	9	to	to	PART
ejde-610	136	10	prove	prove	VERB
ejde-610	136	11	that	that	SCONJ
ejde-610	136	12	i	i	PRON
ejde-610	136	13	has	have	VERB
ejde-610	136	14	the	the	DET
ejde-610	136	15	mountain	mountain	NOUN
ejde-610	136	16	pass	pass	NOUN
ejde-610	136	17	geometry	geometry	NOUN
ejde-610	136	18	,	,	PUNCT
ejde-610	136	19	so	so	ADV
ejde-610	136	20	we	we	PRON
ejde-610	136	21	omit	omit	VERB
ejde-610	136	22	this	this	DET
ejde-610	136	23	case	case	NOUN
ejde-610	136	24	.	.	PUNCT
ejde-610	137	1	since	since	SCONJ
ejde-610	137	2	0	0	NUM
ejde-610	137	3	is	be	AUX
ejde-610	137	4	not	not	PART
ejde-610	137	5	an	an	DET
ejde-610	137	6	eigenvalue	eigenvalue	NOUN
ejde-610	137	7	of	of	ADP
ejde-610	137	8	(	(	PUNCT
ejde-610	137	9	2.2	2.2	NUM
ejde-610	137	10	)	)	PUNCT
ejde-610	137	11	,	,	PUNCT
ejde-610	137	12	we	we	PRON
ejde-610	137	13	assume	assume	VERB
ejde-610	137	14	that	that	SCONJ
ejde-610	137	15	there	there	PRON
ejde-610	137	16	exists	exist	VERB
ejde-610	137	17	l	l	PROPN
ejde-610	137	18	≥	≥	NUM
ejde-610	137	19	1	1	NUM
ejde-610	137	20	such	such	ADJ
ejde-610	137	21	that	that	DET
ejde-610	137	22	0	0	NUM
ejde-610	137	23	∈	∈	NOUN
ejde-610	137	24	(	(	PUNCT
ejde-610	137	25	λl	λl	PROPN
ejde-610	137	26	,	,	PUNCT
ejde-610	137	27	λl+1	λl+1	PROPN
ejde-610	137	28	)	)	PUNCT
ejde-610	137	29	.	.	PUNCT
ejde-610	138	1	set	set	VERB
ejde-610	138	2	e−	e−	PROPN
ejde-610	138	3	=	=	SYM
ejde-610	138	4	span{e1	span{e1	NOUN
ejde-610	138	5	,	,	PUNCT
ejde-610	138	6	.	.	PUNCT
ejde-610	138	7	.	.	PUNCT
ejde-610	139	1	.	.	PUNCT
ejde-610	140	1	,	,	PUNCT
ejde-610	140	2	el	el	PROPN
ejde-610	140	3	}	}	PUNCT
ejde-610	140	4	,	,	PUNCT
ejde-610	140	5	e+	e+	VERB
ejde-610	140	6	=	=	SYM
ejde-610	140	7	(	(	PUNCT
ejde-610	140	8	e−)⊥.	e−)⊥.	INTJ
ejde-610	140	9	(	(	PUNCT
ejde-610	140	10	2.3	2.3	NUM
ejde-610	140	11	)	)	PUNCT
ejde-610	140	12	then	then	ADV
ejde-610	140	13	e−	e−	PROPN
ejde-610	140	14	and	and	CCONJ
ejde-610	140	15	e+	e+	NUM
ejde-610	140	16	are	be	AUX
ejde-610	140	17	the	the	DET
ejde-610	140	18	negative	negative	ADJ
ejde-610	140	19	space	space	NOUN
ejde-610	140	20	and	and	CCONJ
ejde-610	140	21	positive	positive	ADJ
ejde-610	140	22	space	space	NOUN
ejde-610	140	23	of	of	ADP
ejde-610	140	24	the	the	DET
ejde-610	140	25	quadratic	quadratic	ADJ
ejde-610	140	26	form	form	NOUN
ejde-610	140	27	n(u	n(u	PROPN
ejde-610	140	28	)	)	PUNCT
ejde-610	140	29	=	=	SYM
ejde-610	140	30	1	1	NUM
ejde-610	140	31	2	2	NUM
ejde-610	140	32	∫	∫	NOUN
ejde-610	140	33	r3	r3	PROPN
ejde-610	140	34	(	(	PUNCT
ejde-610	140	35	|∇u|2	|∇u|2	X
ejde-610	140	36	+	+	CCONJ
ejde-610	140	37	v	v	X
ejde-610	140	38	(	(	PUNCT
ejde-610	140	39	x)u2	x)u2	PROPN
ejde-610	140	40	)	)	PUNCT
ejde-610	140	41	dx	dx	PROPN
ejde-610	140	42	respectively	respectively	ADV
ejde-610	140	43	,	,	PUNCT
ejde-610	140	44	and	and	CCONJ
ejde-610	140	45	dime−	dime−	VERB
ejde-610	140	46	<	<	PRON
ejde-610	140	47	∞.	∞.	PROPN
ejde-610	140	48	moreover	moreover	ADV
ejde-610	140	49	,	,	PUNCT
ejde-610	140	50	there	there	PRON
ejde-610	140	51	is	be	VERB
ejde-610	140	52	a	a	DET
ejde-610	140	53	positive	positive	ADJ
ejde-610	140	54	constant	constant	ADJ
ejde-610	140	55	b	b	NOUN
ejde-610	140	56	such	such	ADJ
ejde-610	140	57	that	that	DET
ejde-610	140	58	±n(u	±n(u	PROPN
ejde-610	140	59	)	)	PUNCT
ejde-610	140	60	≥	≥	NOUN
ejde-610	141	1	b‖u‖2	b‖u‖2	PROPN
ejde-610	141	2	,	,	PUNCT
ejde-610	141	3	u	u	PROPN
ejde-610	141	4	∈	∈	PROPN
ejde-610	141	5	e±.	e±.	X
ejde-610	141	6	(	(	PUNCT
ejde-610	141	7	2.4	2.4	NUM
ejde-610	141	8	)	)	PUNCT
ejde-610	141	9	to	to	PART
ejde-610	141	10	prove	prove	VERB
ejde-610	141	11	theorem	theorem	VERB
ejde-610	141	12	1.1	1.1	NUM
ejde-610	141	13	,	,	PUNCT
ejde-610	141	14	we	we	PRON
ejde-610	141	15	shall	shall	AUX
ejde-610	141	16	use	use	VERB
ejde-610	141	17	the	the	DET
ejde-610	141	18	fountain	fountain	NOUN
ejde-610	141	19	theorem	theorem	VERB
ejde-610	141	20	by	by	ADP
ejde-610	141	21	bartsch	bartsch	NOUN
ejde-610	141	22	[	[	X
ejde-610	141	23	4	4	NUM
ejde-610	141	24	]	]	PUNCT
ejde-610	141	25	;	;	PUNCT
ejde-610	141	26	see	see	VERB
ejde-610	141	27	also	also	ADV
ejde-610	141	28	[	[	X
ejde-610	141	29	29	29	NUM
ejde-610	141	30	,	,	PUNCT
ejde-610	141	31	theorem	theorem	VERB
ejde-610	141	32	3.6	3.6	NUM
ejde-610	141	33	]	]	PUNCT
ejde-610	141	34	.	.	PUNCT
ejde-610	142	1	for	for	ADP
ejde-610	142	2	k	k	PROPN
ejde-610	142	3	=	=	SYM
ejde-610	142	4	1	1	NUM
ejde-610	142	5	,	,	PUNCT
ejde-610	142	6	2	2	NUM
ejde-610	142	7	,	,	PUNCT
ejde-610	142	8	.	.	PUNCT
ejde-610	142	9	.	.	PUNCT
ejde-610	142	10	.	.	PUNCT
ejde-610	143	1	,	,	PUNCT
ejde-610	143	2	set	set	VERB
ejde-610	143	3	yk	yk	NOUN
ejde-610	143	4	=	=	SYM
ejde-610	143	5	span{e1	span{e1	PROPN
ejde-610	143	6	,	,	PUNCT
ejde-610	143	7	.	.	PUNCT
ejde-610	143	8	.	.	PUNCT
ejde-610	144	1	.	.	PUNCT
ejde-610	145	1	,	,	PUNCT
ejde-610	145	2	ek	ek	PROPN
ejde-610	145	3	}	}	PUNCT
ejde-610	145	4	,	,	PUNCT
ejde-610	145	5	zk	zk	PROPN
ejde-610	145	6	=	=	PUNCT
ejde-610	145	7	span{ek+1	span{ek+1	PROPN
ejde-610	145	8	,	,	PUNCT
ejde-610	145	9	.	.	PUNCT
ejde-610	145	10	.	.	PUNCT
ejde-610	146	1	.	.	PUNCT
ejde-610	147	1	,	,	PUNCT
ejde-610	147	2	}	}	PUNCT
ejde-610	147	3	.	.	PUNCT
ejde-610	148	1	(	(	PUNCT
ejde-610	148	2	2.5	2.5	NUM
ejde-610	148	3	)	)	PUNCT
ejde-610	148	4	6	6	NUM
ejde-610	148	5	l.	l.	PROPN
ejde-610	148	6	wang	wang	PROPN
ejde-610	148	7	,	,	PUNCT
ejde-610	148	8	p.	p.	PROPN
ejde-610	148	9	zhao	zhao	PROPN
ejde-610	148	10	,	,	PUNCT
ejde-610	148	11	d.	d.	PROPN
ejde-610	148	12	zhang	zhang	PROPN
ejde-610	148	13	ejde-2024/18	ejde-2024/18	PROPN
ejde-610	148	14	proposition	proposition	NOUN
ejde-610	148	15	2.2	2.2	NUM
ejde-610	148	16	(	(	PUNCT
ejde-610	148	17	fountain	fountain	NOUN
ejde-610	148	18	theorem	theorem	NOUN
ejde-610	148	19	)	)	PUNCT
ejde-610	148	20	.	.	PUNCT
ejde-610	149	1	assume	assume	VERB
ejde-610	149	2	the	the	DET
ejde-610	149	3	even	even	ADV
ejde-610	149	4	functional	functional	ADJ
ejde-610	149	5	i	i	NOUN
ejde-610	149	6	∈	∈	PROPN
ejde-610	149	7	c1(e	c1(e	PROPN
ejde-610	149	8	,	,	PUNCT
ejde-610	149	9	r	r	NOUN
ejde-610	149	10	)	)	PUNCT
ejde-610	149	11	satisfies	satisfy	VERB
ejde-610	149	12	the	the	DET
ejde-610	149	13	(	(	PUNCT
ejde-610	149	14	ps	ps	NOUN
ejde-610	149	15	)	)	PUNCT
ejde-610	149	16	condition	condition	NOUN
ejde-610	149	17	.	.	PUNCT
ejde-610	150	1	if	if	SCONJ
ejde-610	150	2	there	there	PRON
ejde-610	150	3	is	be	VERB
ejde-610	150	4	a	a	DET
ejde-610	150	5	positive	positive	ADJ
ejde-610	150	6	constant	constant	ADJ
ejde-610	150	7	k	k	NOUN
ejde-610	150	8	such	such	ADJ
ejde-610	150	9	that	that	PRON
ejde-610	150	10	for	for	ADP
ejde-610	150	11	any	any	DET
ejde-610	150	12	k	k	PROPN
ejde-610	150	13	≥	≥	NOUN
ejde-610	150	14	k	k	NOUN
ejde-610	150	15	there	there	PRON
ejde-610	150	16	exist	exist	VERB
ejde-610	150	17	ρk	ρk	ADP
ejde-610	150	18	>	>	X
ejde-610	150	19	rk	rk	PROPN
ejde-610	150	20	>	>	X
ejde-610	150	21	0	0	NUM
ejde-610	151	1	such	such	ADJ
ejde-610	151	2	that	that	SCONJ
ejde-610	151	3	(	(	PUNCT
ejde-610	151	4	i	i	NOUN
ejde-610	151	5	)	)	PUNCT
ejde-610	151	6	ak	ak	PROPN
ejde-610	151	7	=	=	PROPN
ejde-610	151	8	maxu∈yk	maxu∈yk	PROPN
ejde-610	151	9	,	,	PUNCT
ejde-610	151	10	‖u‖=ρk	‖u‖=ρk	VERB
ejde-610	151	11	i(u	i(u	NOUN
ejde-610	151	12	)	)	PUNCT
ejde-610	151	13	≤	≤	NOUN
ejde-610	151	14	0	0	NUM
ejde-610	151	15	,	,	PUNCT
ejde-610	151	16	(	(	PUNCT
ejde-610	151	17	ii	ii	NOUN
ejde-610	151	18	)	)	PUNCT
ejde-610	151	19	bk	bk	PROPN
ejde-610	151	20	=	=	SYM
ejde-610	151	21	infu∈zk	infu∈zk	PROPN
ejde-610	151	22	,	,	PUNCT
ejde-610	151	23	‖u‖=rk	‖u‖=rk	NOUN
ejde-610	151	24	i(u)→	i(u)→	NOUN
ejde-610	152	1	+	+	NOUN
ejde-610	152	2	∞	∞	PROPN
ejde-610	152	3	as	as	ADP
ejde-610	152	4	k	k	PROPN
ejde-610	152	5	→	→	SYM
ejde-610	152	6	+	+	PROPN
ejde-610	152	7	∞	∞	PROPN
ejde-610	152	8	,	,	PUNCT
ejde-610	152	9	then	then	ADV
ejde-610	152	10	i	i	PRON
ejde-610	152	11	has	have	VERB
ejde-610	152	12	a	a	DET
ejde-610	152	13	sequence	sequence	NOUN
ejde-610	152	14	of	of	ADP
ejde-610	152	15	critical	critical	ADJ
ejde-610	152	16	points	point	NOUN
ejde-610	152	17	{	{	PUNCT
ejde-610	152	18	uk	uk	PROPN
ejde-610	152	19	}	}	PUNCT
ejde-610	153	1	such	such	DET
ejde-610	153	2	that	that	DET
ejde-610	153	3	i(uk)→	i(uk)→	NUM
ejde-610	154	1	+	+	NOUN
ejde-610	154	2	∞.	∞.	PROPN
ejde-610	154	3	proposition	proposition	NOUN
ejde-610	154	4	2.3	2.3	NUM
ejde-610	154	5	.	.	PUNCT
ejde-610	155	1	assume	assume	VERB
ejde-610	155	2	that	that	SCONJ
ejde-610	155	3	p1	p1	NOUN
ejde-610	155	4	,	,	PUNCT
ejde-610	155	5	p2	p2	PROPN
ejde-610	155	6	>	>	X
ejde-610	155	7	1	1	NUM
ejde-610	155	8	,	,	PUNCT
ejde-610	155	9	r	r	NOUN
ejde-610	155	10	,	,	PUNCT
ejde-610	155	11	q	q	X
ejde-610	155	12	≥	≥	NOUN
ejde-610	155	13	1	1	NUM
ejde-610	155	14	and	and	CCONJ
ejde-610	155	15	ω	ω	NUM
ejde-610	155	16	⊂	⊂	PROPN
ejde-610	155	17	rn	rn	PROPN
ejde-610	155	18	.	.	PUNCT
ejde-610	156	1	let	let	VERB
ejde-610	156	2	g	g	PRON
ejde-610	156	3	be	be	AUX
ejde-610	156	4	a	a	DET
ejde-610	156	5	caratheodory	caratheodory	ADJ
ejde-610	156	6	function	function	NOUN
ejde-610	156	7	on	on	ADP
ejde-610	156	8	ω×	ω×	PUNCT
ejde-610	156	9	r	r	NOUN
ejde-610	156	10	that	that	PRON
ejde-610	156	11	satisfies	satisfy	VERB
ejde-610	156	12	|g(x	|g(x	PROPN
ejde-610	156	13	,	,	PUNCT
ejde-610	156	14	t)|	t)|	ADJ
ejde-610	156	15	≤	≤	NOUN
ejde-610	157	1	a1|t|(p1−1)/r	a1|t|(p1−1)/r	PROPN
ejde-610	158	1	+	+	CCONJ
ejde-610	158	2	a2|t|(p2−1)/r	a2|t|(p2−1)/r	PROPN
ejde-610	158	3	,	,	PUNCT
ejde-610	158	4	for	for	ADP
ejde-610	158	5	all	all	DET
ejde-610	158	6	(	(	PUNCT
ejde-610	158	7	x	x	NOUN
ejde-610	158	8	,	,	PUNCT
ejde-610	158	9	t	t	PROPN
ejde-610	158	10	)	)	PUNCT
ejde-610	158	11	∈	∈	PROPN
ejde-610	158	12	ω×	ω×	PUNCT
ejde-610	158	13	r	r	NOUN
ejde-610	158	14	,	,	PUNCT
ejde-610	158	15	where	where	SCONJ
ejde-610	158	16	a1	a1	NOUN
ejde-610	158	17	,	,	PUNCT
ejde-610	158	18	a2	a2	PROPN
ejde-610	158	19	≥	≥	NOUN
ejde-610	158	20	0	0	NUM
ejde-610	158	21	.	.	PUNCT
ejde-610	159	1	if	if	SCONJ
ejde-610	159	2	un	un	PROPN
ejde-610	159	3	→	→	SYM
ejde-610	159	4	u	u	PROPN
ejde-610	159	5	in	in	ADP
ejde-610	159	6	lp1(ω	lp1(ω	NOUN
ejde-610	159	7	)	)	PUNCT
ejde-610	159	8	∩	∩	ADJ
ejde-610	159	9	lp2(ω	lp2(ω	PROPN
ejde-610	159	10	)	)	PUNCT
ejde-610	159	11	,	,	PUNCT
ejde-610	159	12	and	and	CCONJ
ejde-610	159	13	un	un	PROPN
ejde-610	159	14	→	→	SYM
ejde-610	159	15	u	u	X
ejde-610	159	16	a.e	a.e	PROPN
ejde-610	159	17	.	.	PROPN
ejde-610	159	18	x	x	SYM
ejde-610	159	19	∈	∈	PROPN
ejde-610	159	20	ω	ω	PROPN
ejde-610	159	21	,	,	PUNCT
ejde-610	159	22	then	then	ADV
ejde-610	159	23	for	for	ADP
ejde-610	159	24	each	each	DET
ejde-610	159	25	v	v	X
ejde-610	159	26	∈	∈	PROPN
ejde-610	159	27	lp1q(ω	lp1q(ω	NOUN
ejde-610	159	28	)	)	PUNCT
ejde-610	159	29	∩	∩	NOUN
ejde-610	159	30	lp2q(ω	lp2q(ω	NOUN
ejde-610	159	31	)	)	PUNCT
ejde-610	159	32	,	,	PUNCT
ejde-610	159	33	we	we	PRON
ejde-610	159	34	have	have	VERB
ejde-610	159	35	lim	lim	PROPN
ejde-610	159	36	n→∞	n→∞	NUM
ejde-610	159	37	∫	∫	PROPN
ejde-610	159	38	ω	ω	PROPN
ejde-610	159	39	|g(x	|g(x	PROPN
ejde-610	159	40	,	,	PUNCT
ejde-610	159	41	un)−	un)−	NOUN
ejde-610	159	42	g(x	g(x	NOUN
ejde-610	159	43	,	,	PUNCT
ejde-610	159	44	u)|r|v|q	u)|r|v|q	PROPN
ejde-610	159	45	dx	dx	PROPN
ejde-610	160	1	=	=	NOUN
ejde-610	160	2	0	0	PROPN
ejde-610	160	3	.	.	PUNCT
ejde-610	160	4	to	to	PART
ejde-610	160	5	study	study	VERB
ejde-610	160	6	the	the	DET
ejde-610	160	7	functional	functional	ADJ
ejde-610	160	8	i	i	PRON
ejde-610	160	9	,	,	PUNCT
ejde-610	160	10	we	we	PRON
ejde-610	160	11	will	will	AUX
ejde-610	160	12	write	write	VERB
ejde-610	160	13	the	the	DET
ejde-610	160	14	functional	functional	ADJ
ejde-610	160	15	i	i	PRON
ejde-610	160	16	in	in	ADP
ejde-610	160	17	a	a	DET
ejde-610	160	18	form	form	NOUN
ejde-610	160	19	in	in	ADP
ejde-610	160	20	which	which	PRON
ejde-610	160	21	the	the	DET
ejde-610	160	22	quadratic	quadratic	ADJ
ejde-610	160	23	part	part	NOUN
ejde-610	160	24	is	be	AUX
ejde-610	160	25	‖u‖2	‖u‖2	PROPN
ejde-610	160	26	.	.	PUNCT
ejde-610	161	1	let	let	VERB
ejde-610	161	2	h(x	h(x	PROPN
ejde-610	161	3	,	,	PUNCT
ejde-610	161	4	t	t	PROPN
ejde-610	161	5	)	)	PUNCT
ejde-610	161	6	=	=	SYM
ejde-610	161	7	f(x	f(x	PROPN
ejde-610	161	8	,	,	PUNCT
ejde-610	161	9	t)+v0	t)+v0	ADJ
ejde-610	161	10	t.	t.	PROPN
ejde-610	161	11	then	then	ADV
ejde-610	161	12	,	,	PUNCT
ejde-610	161	13	by	by	ADP
ejde-610	161	14	(	(	PUNCT
ejde-610	161	15	a5	a5	PROPN
ejde-610	161	16	)	)	PUNCT
ejde-610	161	17	and	and	CCONJ
ejde-610	161	18	computations	computation	NOUN
ejde-610	161	19	,	,	PUNCT
ejde-610	161	20	we	we	PRON
ejde-610	161	21	obtain	obtain	VERB
ejde-610	161	22	that	that	SCONJ
ejde-610	161	23	h(x	h(x	PROPN
ejde-610	161	24	,	,	PUNCT
ejde-610	161	25	t	t	PROPN
ejde-610	161	26	)	)	PUNCT
ejde-610	161	27	:	:	PUNCT
ejde-610	162	1	=	=	SYM
ejde-610	162	2	∫	∫	PROPN
ejde-610	162	3	t	t	PROPN
ejde-610	162	4	0	0	NUM
ejde-610	162	5	h(x	h(x	PROPN
ejde-610	162	6	,	,	PUNCT
ejde-610	162	7	s)ds	s)ds	PROPN
ejde-610	162	8	≤	≤	PROPN
ejde-610	162	9	t	t	PROPN
ejde-610	162	10	θ	θ	PROPN
ejde-610	162	11	h(x	h(x	PROPN
ejde-610	162	12	,	,	PUNCT
ejde-610	162	13	t	t	PROPN
ejde-610	162	14	)	)	PUNCT
ejde-610	162	15	+	+	CCONJ
ejde-610	163	1	ṽ0	ṽ0	PROPN
ejde-610	163	2	t	t	NOUN
ejde-610	163	3	2	2	NUM
ejde-610	163	4	,	,	PUNCT
ejde-610	163	5	ṽ0	ṽ0	NOUN
ejde-610	163	6	:	:	PUNCT
ejde-610	163	7	=	=	NUM
ejde-610	163	8	b+	b+	PUNCT
ejde-610	163	9	v0	v0	PROPN
ejde-610	163	10	2	2	NUM
ejde-610	163	11	−	−	PROPN
ejde-610	163	12	v0	v0	NOUN
ejde-610	163	13	θ	θ	PROPN
ejde-610	163	14	>	>	X
ejde-610	163	15	0	0	NUM
ejde-610	163	16	.	.	PUNCT
ejde-610	164	1	(	(	PUNCT
ejde-610	164	2	2.6	2.6	NUM
ejde-610	164	3	)	)	PUNCT
ejde-610	164	4	by	by	ADP
ejde-610	164	5	(	(	PUNCT
ejde-610	164	6	a4	a4	NOUN
ejde-610	164	7	)	)	PUNCT
ejde-610	164	8	we	we	PRON
ejde-610	164	9	have	have	VERB
ejde-610	164	10	lim	lim	PROPN
ejde-610	164	11	|t|→∞	|t|→∞	PROPN
ejde-610	164	12	h(x	h(x	PROPN
ejde-610	164	13	,	,	PUNCT
ejde-610	164	14	t)t	t)t	ADJ
ejde-610	164	15	t2	t2	NOUN
ejde-610	164	16	=	=	PUNCT
ejde-610	165	1	+	+	NUM
ejde-610	165	2	∞.	∞.	PROPN
ejde-610	165	3	(	(	PUNCT
ejde-610	165	4	2.7	2.7	NUM
ejde-610	165	5	)	)	PUNCT
ejde-610	165	6	furthermore	furthermore	ADV
ejde-610	165	7	,	,	PUNCT
ejde-610	165	8	by	by	ADP
ejde-610	165	9	(	(	PUNCT
ejde-610	165	10	a3	a3	NOUN
ejde-610	165	11	)	)	PUNCT
ejde-610	165	12	we	we	PRON
ejde-610	165	13	obtain	obtain	VERB
ejde-610	165	14	lim	lim	PROPN
ejde-610	165	15	|t|→0	|t|→0	PROPN
ejde-610	165	16	h(x	h(x	PROPN
ejde-610	165	17	,	,	PUNCT
ejde-610	165	18	t)t	t)t	PUNCT
ejde-610	165	19	tθ	tθ	ADP
ejde-610	165	20	=	=	SYM
ejde-610	165	21	lim	lim	PROPN
ejde-610	165	22	|t|→0	|t|→0	PROPN
ejde-610	165	23	(	(	PUNCT
ejde-610	165	24	t2	t2	NOUN
ejde-610	165	25	tθ	tθ	NOUN
ejde-610	165	26	·	·	PUNCT
ejde-610	165	27	f(x	f(x	PROPN
ejde-610	165	28	,	,	PUNCT
ejde-610	165	29	t)t+	t)t+	NUM
ejde-610	165	30	v0	v0	NOUN
ejde-610	165	31	t	t	NOUN
ejde-610	165	32	2	2	NUM
ejde-610	165	33	t2	t2	NOUN
ejde-610	165	34	)	)	PUNCT
ejde-610	165	35	=	=	PUNCT
ejde-610	166	1	+	+	NUM
ejde-610	166	2	∞.	∞.	PROPN
ejde-610	166	3	hence	hence	ADV
ejde-610	166	4	there	there	PRON
ejde-610	166	5	exists	exist	VERB
ejde-610	166	6	m	m	VERB
ejde-610	166	7	>	>	X
ejde-610	166	8	0	0	NUM
ejde-610	167	1	such	such	ADJ
ejde-610	167	2	that	that	SCONJ
ejde-610	167	3	h(x	h(x	PROPN
ejde-610	167	4	,	,	PUNCT
ejde-610	167	5	t)t	t)t	X
ejde-610	167	6	≥	≥	NUM
ejde-610	167	7	−mtθ	−mtθ	NOUN
ejde-610	167	8	,	,	PUNCT
ejde-610	167	9	∀t	∀t	PROPN
ejde-610	167	10	∈	∈	PROPN
ejde-610	167	11	r.	r.	NOUN
ejde-610	167	12	(	(	PUNCT
ejde-610	167	13	2.8	2.8	NUM
ejde-610	167	14	)	)	PUNCT
ejde-610	167	15	with	with	ADP
ejde-610	167	16	the	the	DET
ejde-610	167	17	modified	modify	VERB
ejde-610	167	18	nonlinearity	nonlinearity	NOUN
ejde-610	167	19	h	h	NOUN
ejde-610	167	20	,	,	PUNCT
ejde-610	167	21	the	the	DET
ejde-610	167	22	functional	functional	ADJ
ejde-610	167	23	i	i	X
ejde-610	167	24	:	:	PUNCT
ejde-610	167	25	e	e	X
ejde-610	167	26	→	→	SYM
ejde-610	167	27	r	r	NOUN
ejde-610	167	28	can	can	AUX
ejde-610	167	29	be	be	AUX
ejde-610	167	30	rewritten	rewrite	VERB
ejde-610	167	31	in	in	ADP
ejde-610	167	32	the	the	DET
ejde-610	167	33	form	form	NOUN
ejde-610	167	34	i(u	i(u	PROPN
ejde-610	167	35	)	)	PUNCT
ejde-610	167	36	=	=	SYM
ejde-610	167	37	1	1	NUM
ejde-610	167	38	2	2	NUM
ejde-610	167	39	‖u‖2	‖u‖2	ADJ
ejde-610	167	40	−	−	PROPN
ejde-610	167	41	ω	ω	NUM
ejde-610	167	42	2	2	NUM
ejde-610	167	43	∫	∫	NOUN
ejde-610	167	44	r3	r3	PROPN
ejde-610	167	45	φuu	φuu	VERB
ejde-610	167	46	2	2	NUM
ejde-610	167	47	dx+	dx+	NOUN
ejde-610	167	48	β	β	X
ejde-610	167	49	16π	16π	PROPN
ejde-610	167	50	∫	∫	PROPN
ejde-610	167	51	r3	r3	PROPN
ejde-610	167	52	|∇φu|4	|∇φu|4	NOUN
ejde-610	167	53	dx−	dx−	NUM
ejde-610	167	54	∫	∫	PROPN
ejde-610	167	55	r3	r3	PROPN
ejde-610	167	56	h(x	h(x	PROPN
ejde-610	167	57	,	,	PUNCT
ejde-610	167	58	u	u	NOUN
ejde-610	167	59	)	)	PUNCT
ejde-610	167	60	dx	dx	PROPN
ejde-610	167	61	(	(	PUNCT
ejde-610	167	62	2.9	2.9	NUM
ejde-610	167	63	)	)	PUNCT
ejde-610	167	64	with	with	ADP
ejde-610	167	65	derivative	derivative	ADJ
ejde-610	167	66	〈	〈	PROPN
ejde-610	167	67	i	i	PRON
ejde-610	167	68	′(u	′(u	NOUN
ejde-610	167	69	)	)	PUNCT
ejde-610	167	70	,	,	PUNCT
ejde-610	167	71	v	v	NOUN
ejde-610	167	72	〉	〉	NOUN
ejde-610	167	73	=	=	SYM
ejde-610	168	1	〈	〈	PROPN
ejde-610	168	2	u	u	NOUN
ejde-610	168	3	,	,	PUNCT
ejde-610	168	4	v	v	PROPN
ejde-610	168	5	〉	〉	NUM
ejde-610	168	6	−	−	PROPN
ejde-610	168	7	∫	∫	PROPN
ejde-610	168	8	r3	r3	PROPN
ejde-610	168	9	(	(	PUNCT
ejde-610	168	10	2ω	2ω	NOUN
ejde-610	168	11	+	+	CCONJ
ejde-610	168	12	φu)φuuv	φu)φuuv	NOUN
ejde-610	168	13	dx−	dx−	NUM
ejde-610	168	14	∫	∫	PROPN
ejde-610	168	15	r3	r3	PROPN
ejde-610	168	16	h(x	h(x	PROPN
ejde-610	168	17	,	,	PUNCT
ejde-610	168	18	u)v	u)v	X
ejde-610	168	19	dx	dx	PROPN
ejde-610	168	20	.	.	PUNCT
ejde-610	169	1	lemma	lemma	PROPN
ejde-610	169	2	2.4	2.4	NUM
ejde-610	169	3	.	.	PUNCT
ejde-610	170	1	assume	assume	VERB
ejde-610	170	2	(	(	PUNCT
ejde-610	170	3	a1)—(a5	a1)—(a5	ADV
ejde-610	170	4	)	)	PUNCT
ejde-610	170	5	are	be	AUX
ejde-610	170	6	satisfied	satisfied	ADJ
ejde-610	170	7	,	,	PUNCT
ejde-610	170	8	then	then	ADV
ejde-610	170	9	the	the	DET
ejde-610	170	10	function	function	NOUN
ejde-610	170	11	i	i	PRON
ejde-610	170	12	satisfies	satisfy	VERB
ejde-610	170	13	the	the	DET
ejde-610	170	14	(	(	PUNCT
ejde-610	170	15	ps	ps	NOUN
ejde-610	170	16	)	)	PUNCT
ejde-610	170	17	condition	condition	NOUN
ejde-610	170	18	.	.	PUNCT
ejde-610	171	1	proof	proof	NOUN
ejde-610	171	2	.	.	PUNCT
ejde-610	172	1	it	it	PRON
ejde-610	172	2	follows	follow	VERB
ejde-610	172	3	from	from	ADP
ejde-610	172	4	1	1	NUM
ejde-610	172	5	θ	θ	NOUN
ejde-610	172	6	tf(x	tf(x	NUM
ejde-610	172	7	,	,	PUNCT
ejde-610	172	8	t)−	t)−	PROPN
ejde-610	172	9	f	f	X
ejde-610	172	10	(	(	PUNCT
ejde-610	172	11	x	x	PROPN
ejde-610	172	12	,	,	PUNCT
ejde-610	172	13	t	t	PROPN
ejde-610	172	14	)	)	PUNCT
ejde-610	172	15	≥	≥	NOUN
ejde-610	172	16	−bt2	−bt2	ADP
ejde-610	172	17	that	that	DET
ejde-610	172	18	condition	condition	NOUN
ejde-610	172	19	(	(	PUNCT
ejde-610	172	20	a4	a4	NOUN
ejde-610	172	21	)	)	PUNCT
ejde-610	172	22	is	be	AUX
ejde-610	172	23	equivalent	equivalent	ADJ
ejde-610	172	24	to	to	ADP
ejde-610	172	25	lim	lim	PROPN
ejde-610	172	26	|t|→+∞	|t|→+∞	PROPN
ejde-610	172	27	h(x	h(x	PROPN
ejde-610	172	28	,	,	PUNCT
ejde-610	172	29	t	t	PROPN
ejde-610	172	30	)	)	PUNCT
ejde-610	172	31	tθ	tθ	NOUN
ejde-610	172	32	=	=	PUNCT
ejde-610	173	1	+	+	PROPN
ejde-610	173	2	∞.	∞.	PROPN
ejde-610	173	3	let	let	AUX
ejde-610	173	4	{	{	PUNCT
ejde-610	173	5	un	un	AUX
ejde-610	173	6	}	}	PUNCT
ejde-610	173	7	be	be	AUX
ejde-610	173	8	a	a	DET
ejde-610	173	9	(	(	PUNCT
ejde-610	173	10	ps	ps	NOUN
ejde-610	173	11	)	)	PUNCT
ejde-610	173	12	sequence	sequence	NOUN
ejde-610	173	13	,	,	PUNCT
ejde-610	173	14	i.e.	i.e.	X
ejde-610	173	15	,	,	PUNCT
ejde-610	173	16	i(un)→	i(un)→	ADP
ejde-610	173	17	c	c	PROPN
ejde-610	173	18	>	>	X
ejde-610	173	19	0	0	PROPN
ejde-610	173	20	,	,	PUNCT
ejde-610	173	21	〈	〈	PROPN
ejde-610	173	22	i	i	PRON
ejde-610	173	23	′(un	′(un	PROPN
ejde-610	173	24	)	)	PUNCT
ejde-610	173	25	,	,	PUNCT
ejde-610	173	26	un	un	PROPN
ejde-610	173	27	〉	〉	PROPN
ejde-610	173	28	→	→	SYM
ejde-610	173	29	0	0	NUM
ejde-610	173	30	.	.	NOUN
ejde-610	173	31	ejde-2024/18	ejde-2024/18	NOUN
ejde-610	173	32	coupled	couple	VERB
ejde-610	173	33	klein	klein	PROPN
ejde-610	173	34	-	-	PUNCT
ejde-610	173	35	gordon	gordon	PROPN
ejde-610	173	36	and	and	CCONJ
ejde-610	173	37	born	bear	VERB
ejde-610	173	38	-	-	PUNCT
ejde-610	173	39	infeld	infeld	NOUN
ejde-610	173	40	equatons	equaton	NOUN
ejde-610	173	41	7	7	NUM
ejde-610	173	42	we	we	PRON
ejde-610	173	43	first	first	ADV
ejde-610	173	44	prove	prove	VERB
ejde-610	173	45	that	that	SCONJ
ejde-610	173	46	{	{	PUNCT
ejde-610	173	47	un	un	PROPN
ejde-610	173	48	}	}	PUNCT
ejde-610	173	49	is	be	AUX
ejde-610	173	50	bounded	bound	VERB
ejde-610	173	51	in	in	ADP
ejde-610	173	52	e.	e.	PROPN
ejde-610	173	53	arguing	argue	VERB
ejde-610	173	54	by	by	ADP
ejde-610	173	55	contradiction	contradiction	NOUN
ejde-610	173	56	,	,	PUNCT
ejde-610	173	57	suppose	suppose	VERB
ejde-610	173	58	that	that	SCONJ
ejde-610	173	59	{	{	PUNCT
ejde-610	173	60	un	un	PROPN
ejde-610	173	61	}	}	PUNCT
ejde-610	173	62	is	be	AUX
ejde-610	173	63	unbounded	unbounde	VERB
ejde-610	173	64	,	,	PUNCT
ejde-610	173	65	passing	pass	VERB
ejde-610	173	66	to	to	ADP
ejde-610	173	67	a	a	DET
ejde-610	173	68	subsequence	subsequence	NOUN
ejde-610	173	69	,	,	PUNCT
ejde-610	173	70	by	by	ADP
ejde-610	173	71	(	(	PUNCT
ejde-610	173	72	2.6	2.6	NUM
ejde-610	173	73	)	)	PUNCT
ejde-610	173	74	,	,	PUNCT
ejde-610	173	75	we	we	PRON
ejde-610	173	76	obtain	obtain	VERB
ejde-610	173	77	θ	θ	PROPN
ejde-610	173	78	sup	sup	NOUN
ejde-610	173	79	n	n	PRON
ejde-610	173	80	i(un	i(un	NUM
ejde-610	173	81	)	)	PUNCT
ejde-610	173	82	+	+	CCONJ
ejde-610	173	83	‖un‖	‖un‖	NOUN
ejde-610	173	84	≥	≥	NOUN
ejde-610	173	85	θi(un)−	θi(un)−	PUNCT
ejde-610	173	86	〈	〈	PROPN
ejde-610	173	87	i	i	PRON
ejde-610	173	88	′(un	′(un	PROPN
ejde-610	173	89	)	)	PUNCT
ejde-610	173	90	,	,	PUNCT
ejde-610	173	91	un	un	PROPN
ejde-610	173	92	〉	〉	NOUN
ejde-610	173	93	=	=	SYM
ejde-610	173	94	(	(	PUNCT
ejde-610	173	95	θ	θ	NOUN
ejde-610	173	96	2	2	NUM
ejde-610	173	97	−	−	NOUN
ejde-610	173	98	1	1	NUM
ejde-610	173	99	)	)	PUNCT
ejde-610	174	1	‖un‖2	‖un‖2	ADP
ejde-610	174	2	−	−	NOUN
ejde-610	174	3	ωθ	ωθ	NUM
ejde-610	174	4	2	2	NUM
ejde-610	174	5	∫	∫	NOUN
ejde-610	174	6	r3	r3	PROPN
ejde-610	174	7	φun	φun	NOUN
ejde-610	174	8	u2	u2	PROPN
ejde-610	174	9	n	n	PRON
ejde-610	174	10	dx+	dx+	NOUN
ejde-610	174	11	θβ	θβ	ADP
ejde-610	174	12	16π	16π	X
ejde-610	175	1	∫	∫	PROPN
ejde-610	175	2	r3	r3	PROPN
ejde-610	175	3	|∇φun	|∇φun	NUM
ejde-610	175	4	|4	|4	NUM
ejde-610	175	5	dx	dx	PROPN
ejde-610	176	1	+	+	CCONJ
ejde-610	177	1	∫	∫	PROPN
ejde-610	177	2	r3	r3	PROPN
ejde-610	177	3	(	(	PUNCT
ejde-610	177	4	2ω	2ω	NOUN
ejde-610	177	5	+	+	CCONJ
ejde-610	177	6	φun	φun	NOUN
ejde-610	177	7	)	)	PUNCT
ejde-610	177	8	φun	φun	VERB
ejde-610	177	9	u2	u2	NOUN
ejde-610	177	10	n	n	PRON
ejde-610	177	11	dx+	dx+	ADJ
ejde-610	177	12	∫	∫	PROPN
ejde-610	177	13	r3	r3	PROPN
ejde-610	177	14	(	(	PUNCT
ejde-610	177	15	h(x	h(x	PROPN
ejde-610	177	16	,	,	PUNCT
ejde-610	177	17	un)un	un)un	X
ejde-610	177	18	−	−	PROPN
ejde-610	177	19	θh(x	θh(x	PROPN
ejde-610	177	20	,	,	PUNCT
ejde-610	177	21	un	un	PROPN
ejde-610	177	22	)	)	PUNCT
ejde-610	177	23	)	)	PUNCT
ejde-610	178	1	dx	dx	PROPN
ejde-610	178	2	≥	≥	NUM
ejde-610	178	3	(	(	PUNCT
ejde-610	178	4	θ	θ	NOUN
ejde-610	178	5	2	2	NUM
ejde-610	178	6	−	−	NOUN
ejde-610	178	7	1	1	NUM
ejde-610	178	8	)	)	PUNCT
ejde-610	179	1	‖un‖2	‖un‖2	ADP
ejde-610	179	2	−	−	PROPN
ejde-610	179	3	ṽ0	ṽ0	ADJ
ejde-610	179	4	∫	∫	PROPN
ejde-610	179	5	r3	r3	PROPN
ejde-610	179	6	u2	u2	PROPN
ejde-610	179	7	n	n	PROPN
ejde-610	179	8	dx	dx	PROPN
ejde-610	179	9	.	.	PUNCT
ejde-610	180	1	(	(	PUNCT
ejde-610	180	2	2.10	2.10	NUM
ejde-610	180	3	)	)	PUNCT
ejde-610	180	4	let	let	VERB
ejde-610	180	5	vn	vn	VERB
ejde-610	180	6	=	=	PUNCT
ejde-610	180	7	un/‖un‖.	un/‖un‖.	X
ejde-610	180	8	then	then	ADV
ejde-610	180	9	,	,	PUNCT
ejde-610	180	10	going	go	VERB
ejde-610	180	11	if	if	SCONJ
ejde-610	180	12	necessary	necessary	ADJ
ejde-610	180	13	to	to	ADP
ejde-610	180	14	a	a	DET
ejde-610	180	15	subsequence	subsequence	NOUN
ejde-610	180	16	,	,	PUNCT
ejde-610	180	17	by	by	ADP
ejde-610	180	18	the	the	DET
ejde-610	180	19	compact	compact	ADJ
ejde-610	180	20	embedding	embed	VERB
ejde-610	180	21	e	e	X
ejde-610	180	22	↪	↪	PROPN
ejde-610	180	23	→	→	SYM
ejde-610	180	24	l2(r3	l2(r3	NUM
ejde-610	180	25	)	)	PUNCT
ejde-610	180	26	we	we	PRON
ejde-610	180	27	can	can	AUX
ejde-610	180	28	assume	assume	VERB
ejde-610	180	29	that	that	SCONJ
ejde-610	180	30	vn	vn	PROPN
ejde-610	180	31	⇀	⇀	NUM
ejde-610	180	32	v0	v0	NOUN
ejde-610	180	33	in	in	ADP
ejde-610	180	34	e	e	PROPN
ejde-610	180	35	,	,	PUNCT
ejde-610	180	36	vn	vn	PROPN
ejde-610	180	37	→	→	SYM
ejde-610	180	38	v0	v0	NOUN
ejde-610	180	39	in	in	ADP
ejde-610	180	40	l2(r3	l2(r3	NUM
ejde-610	180	41	)	)	PUNCT
ejde-610	180	42	,	,	PUNCT
ejde-610	181	1	vn(x)→	vn(x)→	PROPN
ejde-610	181	2	v0(x	v0(x	NUM
ejde-610	181	3	)	)	PUNCT
ejde-610	181	4	a.	a.	PROPN
ejde-610	181	5	e.	e.	PROPN
ejde-610	181	6	in	in	ADP
ejde-610	181	7	r3	r3	PROPN
ejde-610	181	8	.	.	PUNCT
ejde-610	182	1	dividing	divide	VERB
ejde-610	182	2	both	both	DET
ejde-610	182	3	sides	side	NOUN
ejde-610	182	4	of	of	ADP
ejde-610	182	5	(	(	PUNCT
ejde-610	182	6	2.10	2.10	NUM
ejde-610	182	7	)	)	PUNCT
ejde-610	182	8	by	by	ADP
ejde-610	182	9	‖un‖2	‖un‖2	PROPN
ejde-610	182	10	,	,	PUNCT
ejde-610	182	11	we	we	PRON
ejde-610	182	12	have	have	VERB
ejde-610	182	13	ṽ0	ṽ0	PROPN
ejde-610	182	14	∫	∫	PROPN
ejde-610	182	15	r3	r3	PROPN
ejde-610	182	16	v2	v2	PROPN
ejde-610	182	17	0	0	NUM
ejde-610	182	18	dx	dx	PROPN
ejde-610	182	19	≥	≥	PROPN
ejde-610	182	20	1	1	NUM
ejde-610	182	21	as	as	ADP
ejde-610	182	22	n→∞.	n→∞.	PROPN
ejde-610	182	23	consequently	consequently	ADV
ejde-610	182	24	,	,	PUNCT
ejde-610	182	25	we	we	PRON
ejde-610	182	26	have	have	VERB
ejde-610	182	27	that	that	DET
ejde-610	182	28	v0	v0	NOUN
ejde-610	182	29	6=	6=	PRON
ejde-610	182	30	0	0	NUM
ejde-610	182	31	.	.	PUNCT
ejde-610	183	1	by	by	ADP
ejde-610	183	2	(	(	PUNCT
ejde-610	183	3	1.4	1.4	NUM
ejde-610	183	4	)	)	PUNCT
ejde-610	183	5	and	and	CCONJ
ejde-610	183	6	(	(	PUNCT
ejde-610	183	7	2.8	2.8	NUM
ejde-610	183	8	)	)	PUNCT
ejde-610	183	9	,	,	PUNCT
ejde-610	183	10	we	we	PRON
ejde-610	183	11	have∫	have∫	VERB
ejde-610	183	12	v0=0	v0=0	PROPN
ejde-610	183	13	h(x	h(x	PROPN
ejde-610	183	14	,	,	PUNCT
ejde-610	183	15	un)un	un)un	X
ejde-610	184	1	‖un‖θ	‖un‖θ	PROPN
ejde-610	184	2	dx	dx	PROPN
ejde-610	184	3	=	=	SYM
ejde-610	184	4	∫	∫	PROPN
ejde-610	184	5	v0=0	v0=0	PROPN
ejde-610	184	6	h(x	h(x	PROPN
ejde-610	184	7	,	,	PUNCT
ejde-610	184	8	un)un	un)un	X
ejde-610	184	9	uθn	uθn	PROPN
ejde-610	184	10	vθn	vθn	VERB
ejde-610	184	11	dx	dx	PROPN
ejde-610	184	12	≥	≥	PROPN
ejde-610	184	13	−m	−m	PROPN
ejde-610	184	14	∫	∫	PROPN
ejde-610	185	1	v0=0	v0=0	PROPN
ejde-610	185	2	vθn	vθn	VERB
ejde-610	185	3	dx	dx	PROPN
ejde-610	185	4	≥	≥	PROPN
ejde-610	185	5	−m	−m	PROPN
ejde-610	185	6	∫	∫	PROPN
ejde-610	185	7	r3	r3	PROPN
ejde-610	185	8	vθn	vθn	VERB
ejde-610	185	9	dx	dx	PROPN
ejde-610	185	10	=	=	SYM
ejde-610	185	11	−m	−m	PROPN
ejde-610	185	12	|vn|θθ	|vn|θθ	NOUN
ejde-610	185	13	≥	≥	NOUN
ejde-610	185	14	−mdθθ	−mdθθ	X
ejde-610	185	15	>	>	X
ejde-610	185	16	−∞.	−∞.	PROPN
ejde-610	185	17	(	(	PUNCT
ejde-610	185	18	2.11	2.11	NUM
ejde-610	185	19	)	)	PUNCT
ejde-610	185	20	for	for	ADP
ejde-610	185	21	x	x	PROPN
ejde-610	185	22	∈	∈	PROPN
ejde-610	185	23	{	{	PUNCT
ejde-610	185	24	x	x	SYM
ejde-610	185	25	∈	∈	PROPN
ejde-610	185	26	r3|v0	r3|v0	NOUN
ejde-610	185	27	6=	6=	NOUN
ejde-610	185	28	0	0	NUM
ejde-610	185	29	}	}	PUNCT
ejde-610	185	30	,	,	PUNCT
ejde-610	185	31	we	we	PRON
ejde-610	185	32	have	have	VERB
ejde-610	185	33	|un(x)|	|un(x)|	NOUN
ejde-610	185	34	→	→	SYM
ejde-610	185	35	+	+	ADJ
ejde-610	185	36	∞	∞	PROPN
ejde-610	185	37	as	as	ADP
ejde-610	185	38	n→∞.	n→∞.	VERB
ejde-610	185	39	by	by	ADP
ejde-610	185	40	(	(	PUNCT
ejde-610	185	41	2.7	2.7	NUM
ejde-610	185	42	)	)	PUNCT
ejde-610	185	43	we	we	PRON
ejde-610	185	44	have	have	VERB
ejde-610	185	45	h(x	h(x	PROPN
ejde-610	185	46	,	,	PUNCT
ejde-610	185	47	un(x))un(x	un(x))un(x	PROPN
ejde-610	185	48	)	)	PUNCT
ejde-610	185	49	‖un‖2	‖un‖2	PROPN
ejde-610	185	50	=	=	SYM
ejde-610	185	51	h(x	h(x	PROPN
ejde-610	185	52	,	,	PUNCT
ejde-610	185	53	un(x))un(x	un(x))un(x	PROPN
ejde-610	185	54	)	)	PUNCT
ejde-610	185	55	u2	u2	NOUN
ejde-610	185	56	n(x	n(x	PROPN
ejde-610	185	57	)	)	PUNCT
ejde-610	185	58	v2	v2	VERB
ejde-610	185	59	n(x)→	n(x)→	NUM
ejde-610	186	1	+	+	SYM
ejde-610	186	2	∞.	∞.	PROPN
ejde-610	186	3	(	(	PUNCT
ejde-610	186	4	2.12	2.12	NUM
ejde-610	186	5	)	)	PUNCT
ejde-610	186	6	hence	hence	ADV
ejde-610	186	7	,	,	PUNCT
ejde-610	186	8	by	by	ADP
ejde-610	186	9	(	(	PUNCT
ejde-610	186	10	2.11	2.11	NUM
ejde-610	186	11	)	)	PUNCT
ejde-610	186	12	,	,	PUNCT
ejde-610	186	13	(	(	PUNCT
ejde-610	186	14	2.12	2.12	NUM
ejde-610	186	15	)	)	PUNCT
ejde-610	186	16	and	and	CCONJ
ejde-610	186	17	fatou	fatou	PROPN
ejde-610	186	18	’s	’s	PART
ejde-610	186	19	lemma	lemma	PROPN
ejde-610	186	20	we	we	PRON
ejde-610	186	21	obtain∫	obtain∫	VERB
ejde-610	186	22	r3	r3	PROPN
ejde-610	186	23	h(x	h(x	PROPN
ejde-610	186	24	,	,	PUNCT
ejde-610	186	25	un)un	un)un	PROPN
ejde-610	186	26	‖un‖θ	‖un‖θ	PROPN
ejde-610	186	27	dx	dx	PROPN
ejde-610	186	28	≥	≥	PROPN
ejde-610	186	29	∫	∫	PROPN
ejde-610	186	30	v0	v0	PROPN
ejde-610	186	31	6=0	6=0	NUM
ejde-610	186	32	h(x	h(x	PROPN
ejde-610	186	33	,	,	PUNCT
ejde-610	186	34	un)un	un)un	X
ejde-610	186	35	uθn	uθn	PROPN
ejde-610	186	36	vθn(x	vθn(x	PROPN
ejde-610	186	37	)	)	PUNCT
ejde-610	186	38	dx−mdθθ	dx−mdθθ	X
ejde-610	186	39	→	→	PUNCT
ejde-610	187	1	+	+	ADJ
ejde-610	187	2	∞.	∞.	PROPN
ejde-610	187	3	(	(	PUNCT
ejde-610	187	4	2.13	2.13	NUM
ejde-610	187	5	)	)	PUNCT
ejde-610	187	6	hence	hence	ADV
ejde-610	187	7	∫	∫	PROPN
ejde-610	187	8	r3	r3	PROPN
ejde-610	187	9	h(x	h(x	PROPN
ejde-610	187	10	,	,	PUNCT
ejde-610	187	11	un	un	PROPN
ejde-610	187	12	)	)	PUNCT
ejde-610	187	13	‖un‖θ	‖un‖θ	NOUN
ejde-610	187	14	dx→	dx→	NOUN
ejde-610	188	1	+	+	VERB
ejde-610	188	2	∞.	∞.	PROPN
ejde-610	188	3	(	(	PUNCT
ejde-610	188	4	2.14	2.14	NUM
ejde-610	188	5	)	)	PUNCT
ejde-610	188	6	8	8	NUM
ejde-610	188	7	l.	l.	PROPN
ejde-610	188	8	wang	wang	PROPN
ejde-610	188	9	,	,	PUNCT
ejde-610	188	10	p.	p.	PROPN
ejde-610	188	11	zhao	zhao	PROPN
ejde-610	188	12	,	,	PUNCT
ejde-610	188	13	d.	d.	PROPN
ejde-610	188	14	zhang	zhang	PROPN
ejde-610	188	15	ejde-2024/18	ejde-2024/18	PROPN
ejde-610	188	16	since	since	SCONJ
ejde-610	188	17	{	{	PUNCT
ejde-610	188	18	un	un	VERB
ejde-610	188	19	}	}	PUNCT
ejde-610	188	20	is	be	AUX
ejde-610	188	21	a	a	DET
ejde-610	188	22	(	(	PUNCT
ejde-610	188	23	ps	ps	NOUN
ejde-610	188	24	)	)	PUNCT
ejde-610	188	25	sequence	sequence	NOUN
ejde-610	188	26	,	,	PUNCT
ejde-610	188	27	using	use	VERB
ejde-610	188	28	proposition	proposition	NOUN
ejde-610	188	29	2.1	2.1	NUM
ejde-610	188	30	and	and	CCONJ
ejde-610	188	31	(	(	PUNCT
ejde-610	188	32	2.13	2.13	NUM
ejde-610	188	33	)	)	PUNCT
ejde-610	188	34	,	,	PUNCT
ejde-610	188	35	for	for	ADP
ejde-610	188	36	n	n	CCONJ
ejde-610	188	37	large	large	ADJ
ejde-610	188	38	enough	enough	ADV
ejde-610	188	39	,	,	PUNCT
ejde-610	188	40	we	we	PRON
ejde-610	188	41	have	have	AUX
ejde-610	188	42	cω	cω	NOUN
ejde-610	188	43	+	+	ADJ
ejde-610	188	44	1	1	NUM
ejde-610	188	45	≥	≥	NOUN
ejde-610	188	46	1	1	NUM
ejde-610	189	1	‖un‖θ	‖un‖θ	NOUN
ejde-610	189	2	(	(	PUNCT
ejde-610	189	3	1	1	NUM
ejde-610	189	4	2	2	NUM
ejde-610	189	5	‖un‖2	‖un‖2	NUM
ejde-610	189	6	−	−	PROPN
ejde-610	189	7	ω	ω	NUM
ejde-610	189	8	2	2	NUM
ejde-610	189	9	∫	∫	NOUN
ejde-610	189	10	r3	r3	PROPN
ejde-610	189	11	φun	φun	NOUN
ejde-610	189	12	u2	u2	PROPN
ejde-610	189	13	n	n	PRON
ejde-610	189	14	dx+	dx+	NOUN
ejde-610	189	15	3β	3β	NUM
ejde-610	189	16	16π	16π	PROPN
ejde-610	190	1	∫	∫	PROPN
ejde-610	190	2	r3	r3	PROPN
ejde-610	190	3	|∇φun	|∇φun	NUM
ejde-610	190	4	|4	|4	NUM
ejde-610	190	5	dx−	dx−	NUM
ejde-610	190	6	i(un	i(un	NUM
ejde-610	190	7	)	)	PUNCT
ejde-610	190	8	)	)	PUNCT
ejde-610	191	1	=	=	SYM
ejde-610	191	2	∫	∫	PROPN
ejde-610	191	3	r3	r3	PROPN
ejde-610	191	4	h(x	h(x	PROPN
ejde-610	191	5	,	,	PUNCT
ejde-610	191	6	un	un	PROPN
ejde-610	191	7	)	)	PUNCT
ejde-610	191	8	‖un‖θ	‖un‖θ	NOUN
ejde-610	191	9	dx→	dx→	NOUN
ejde-610	192	1	+	+	NOUN
ejde-610	192	2	∞	∞	PROPN
ejde-610	192	3	,	,	PUNCT
ejde-610	192	4	(	(	PUNCT
ejde-610	192	5	2.15	2.15	NUM
ejde-610	192	6	)	)	PUNCT
ejde-610	192	7	which	which	PRON
ejde-610	192	8	is	be	AUX
ejde-610	192	9	a	a	DET
ejde-610	192	10	contradiction	contradiction	NOUN
ejde-610	192	11	.	.	PUNCT
ejde-610	193	1	it	it	PRON
ejde-610	193	2	follows	follow	VERB
ejde-610	193	3	that	that	SCONJ
ejde-610	193	4	{	{	PUNCT
ejde-610	193	5	un	un	PROPN
ejde-610	193	6	}	}	PUNCT
ejde-610	193	7	is	be	AUX
ejde-610	193	8	bounded	bound	VERB
ejde-610	193	9	in	in	ADP
ejde-610	193	10	e.	e.	PROPN
ejde-610	193	11	next	next	ADV
ejde-610	193	12	we	we	PRON
ejde-610	193	13	shall	shall	AUX
ejde-610	193	14	prove	prove	VERB
ejde-610	193	15	{	{	PUNCT
ejde-610	193	16	un	un	PROPN
ejde-610	193	17	}	}	PUNCT
ejde-610	193	18	contains	contain	VERB
ejde-610	193	19	a	a	DET
ejde-610	193	20	convergent	convergent	NOUN
ejde-610	193	21	subsequence	subsequence	NOUN
ejde-610	193	22	.	.	PUNCT
ejde-610	194	1	without	without	ADP
ejde-610	194	2	loss	loss	NOUN
ejde-610	194	3	of	of	ADP
ejde-610	194	4	generality	generality	NOUN
ejde-610	194	5	,	,	PUNCT
ejde-610	194	6	passing	pass	VERB
ejde-610	194	7	to	to	ADP
ejde-610	194	8	a	a	DET
ejde-610	194	9	subsequence	subsequence	NOUN
ejde-610	194	10	if	if	SCONJ
ejde-610	194	11	necessary	necessary	ADJ
ejde-610	194	12	,	,	PUNCT
ejde-610	194	13	there	there	PRON
ejde-610	194	14	exists	exist	VERB
ejde-610	194	15	u	u	NOUN
ejde-610	194	16	∈	∈	PROPN
ejde-610	194	17	e	e	NOUN
ejde-610	194	18	such	such	ADJ
ejde-610	194	19	that	that	DET
ejde-610	194	20	un	un	PROPN
ejde-610	194	21	⇀	⇀	PROPN
ejde-610	194	22	u	u	PROPN
ejde-610	194	23	in	in	ADP
ejde-610	194	24	e.	e.	PROPN
ejde-610	194	25	by	by	ADP
ejde-610	194	26	using	use	VERB
ejde-610	194	27	the	the	DET
ejde-610	194	28	embedding	embed	VERB
ejde-610	194	29	e	e	X
ejde-610	194	30	↪	↪	PROPN
ejde-610	194	31	→	→	SYM
ejde-610	194	32	ls(r3	ls(r3	X
ejde-610	194	33	)	)	PUNCT
ejde-610	194	34	are	be	AUX
ejde-610	194	35	compact	compact	ADJ
ejde-610	194	36	for	for	ADP
ejde-610	194	37	any	any	DET
ejde-610	194	38	s	s	X
ejde-610	194	39	∈	∈	NOUN
ejde-610	195	1	[	[	X
ejde-610	195	2	2	2	NUM
ejde-610	195	3	,	,	PUNCT
ejde-610	195	4	6	6	NUM
ejde-610	195	5	)	)	PUNCT
ejde-610	195	6	,	,	PUNCT
ejde-610	195	7	un	un	PROPN
ejde-610	195	8	→	→	SYM
ejde-610	195	9	u	u	PROPN
ejde-610	195	10	in	in	ADP
ejde-610	195	11	ls(r3	ls(r3	PROPN
ejde-610	195	12	)	)	PUNCT
ejde-610	195	13	for	for	ADP
ejde-610	195	14	2	2	NUM
ejde-610	195	15	≤	≤	NOUN
ejde-610	195	16	s	s	PART
ejde-610	195	17	<	<	X
ejde-610	195	18	6	6	NUM
ejde-610	195	19	and	and	CCONJ
ejde-610	195	20	un(x	un(x	NUM
ejde-610	195	21	)	)	PUNCT
ejde-610	195	22	→	→	SYM
ejde-610	195	23	u(x	u(x	PROPN
ejde-610	195	24	)	)	PUNCT
ejde-610	195	25	a.e	a.e	PROPN
ejde-610	195	26	.	.	PUNCT
ejde-610	195	27	x	x	SYM
ejde-610	195	28	∈	∈	PROPN
ejde-610	195	29	r3	r3	PROPN
ejde-610	195	30	.	.	PUNCT
ejde-610	196	1	by	by	ADP
ejde-610	196	2	(	(	PUNCT
ejde-610	196	3	1.3	1.3	NUM
ejde-610	196	4	)	)	PUNCT
ejde-610	196	5	and	and	CCONJ
ejde-610	196	6	the	the	DET
ejde-610	196	7	gateaux	gateaux	ADV
ejde-610	196	8	derivative	derivative	NOUN
ejde-610	196	9	of	of	ADP
ejde-610	196	10	i	i	PRON
ejde-610	196	11	,	,	PUNCT
ejde-610	196	12	we	we	PRON
ejde-610	196	13	can	can	AUX
ejde-610	196	14	obtain	obtain	VERB
ejde-610	196	15	that	that	PRON
ejde-610	196	16	‖un	‖un	PROPN
ejde-610	196	17	−	−	PROPN
ejde-610	197	1	u‖2	u‖2	NOUN
ejde-610	198	1	=	=	PUNCT
ejde-610	199	1	〈	〈	PROPN
ejde-610	199	2	i	i	PRON
ejde-610	199	3	′(un)−	′(un)−	NOUN
ejde-610	199	4	i	i	PRON
ejde-610	199	5	′(u	′(u	NOUN
ejde-610	199	6	)	)	PUNCT
ejde-610	199	7	,	,	PUNCT
ejde-610	199	8	un	un	PROPN
ejde-610	199	9	−	−	PROPN
ejde-610	199	10	u〉+	u〉+	PROPN
ejde-610	199	11	v0	v0	PROPN
ejde-610	199	12	∫	∫	PROPN
ejde-610	199	13	r3	r3	PROPN
ejde-610	199	14	(	(	PUNCT
ejde-610	199	15	un	un	PROPN
ejde-610	199	16	−	−	PROPN
ejde-610	199	17	u)2	u)2	ADV
ejde-610	199	18	dx+	dx+	PROPN
ejde-610	199	19	2ω	2ω	PROPN
ejde-610	199	20	∫	∫	PROPN
ejde-610	199	21	r3	r3	PROPN
ejde-610	199	22	(	(	PUNCT
ejde-610	199	23	φunun	φunun	PROPN
ejde-610	199	24	−	−	PROPN
ejde-610	199	25	φuu)(un	φuu)(un	PROPN
ejde-610	199	26	−	−	PROPN
ejde-610	199	27	u	u	NOUN
ejde-610	199	28	)	)	PUNCT
ejde-610	199	29	dx	dx	PROPN
ejde-610	200	1	+	+	CCONJ
ejde-610	200	2	∫	∫	PROPN
ejde-610	200	3	r3	r3	PROPN
ejde-610	200	4	(	(	PUNCT
ejde-610	200	5	h(x	h(x	PROPN
ejde-610	200	6	,	,	PUNCT
ejde-610	200	7	un)−	un)−	PART
ejde-610	200	8	h(x	h(x	PROPN
ejde-610	200	9	,	,	PUNCT
ejde-610	200	10	u))(un	u))(un	PROPN
ejde-610	200	11	−	−	PROPN
ejde-610	200	12	u	u	NOUN
ejde-610	200	13	)	)	PUNCT
ejde-610	200	14	dx+	dx+	ADJ
ejde-610	200	15	∫	∫	PROPN
ejde-610	200	16	r3	r3	PROPN
ejde-610	200	17	(	(	PUNCT
ejde-610	200	18	φ2	φ2	PROPN
ejde-610	200	19	un	un	PROPN
ejde-610	200	20	un	un	PROPN
ejde-610	200	21	−	−	PROPN
ejde-610	200	22	φ2	φ2	PROPN
ejde-610	200	23	uu)(un	uu)(un	PROPN
ejde-610	200	24	−	−	PROPN
ejde-610	200	25	u	u	NOUN
ejde-610	200	26	)	)	PUNCT
ejde-610	200	27	dx	dx	PROPN
ejde-610	200	28	by	by	ADP
ejde-610	200	29	an	an	DET
ejde-610	200	30	easy	easy	ADJ
ejde-610	200	31	computation	computation	NOUN
ejde-610	200	32	,	,	PUNCT
ejde-610	200	33	we	we	PRON
ejde-610	200	34	obtain	obtain	VERB
ejde-610	200	35	that	that	SCONJ
ejde-610	200	36	〈	〈	PROPN
ejde-610	200	37	i	i	PRON
ejde-610	200	38	′(un)−	′(un)−	NOUN
ejde-610	200	39	i	i	PRON
ejde-610	200	40	′(u	′(u	NOUN
ejde-610	200	41	)	)	PUNCT
ejde-610	200	42	,	,	PUNCT
ejde-610	200	43	un	un	PROPN
ejde-610	200	44	−	−	PROPN
ejde-610	200	45	u	u	PROPN
ejde-610	200	46	〉	〉	PROPN
ejde-610	200	47	→	→	SYM
ejde-610	200	48	0	0	PUNCT
ejde-610	200	49	as	as	ADP
ejde-610	200	50	n→∞,∫	n→∞,∫	X
ejde-610	200	51	r3	r3	PROPN
ejde-610	201	1	[	[	X
ejde-610	201	2	(	(	PUNCT
ejde-610	201	3	φun	φun	PROPN
ejde-610	201	4	un	un	PROPN
ejde-610	201	5	−	−	PROPN
ejde-610	201	6	φuu)(un	φuu)(un	PROPN
ejde-610	201	7	−	−	PROPN
ejde-610	201	8	u	u	NOUN
ejde-610	201	9	)	)	PUNCT
ejde-610	201	10	dx+	dx+	ADJ
ejde-610	201	11	∫	∫	PROPN
ejde-610	201	12	r3	r3	PROPN
ejde-610	201	13	[	[	X
ejde-610	201	14	(	(	PUNCT
ejde-610	201	15	φ2	φ2	PROPN
ejde-610	201	16	un	un	PROPN
ejde-610	201	17	un	un	PROPN
ejde-610	201	18	−	−	PROPN
ejde-610	201	19	φ2	φ2	PROPN
ejde-610	201	20	uu)(un	uu)(un	PROPN
ejde-610	201	21	−	−	PROPN
ejde-610	201	22	u	u	NOUN
ejde-610	201	23	)	)	PUNCT
ejde-610	201	24	dx→	dx→	NOUN
ejde-610	201	25	0	0	PUNCT
ejde-610	202	1	as	as	SCONJ
ejde-610	202	2	n	n	PRON
ejde-610	202	3	→	→	PUNCT
ejde-610	202	4	+	+	PROPN
ejde-610	202	5	∞.	∞.	PROPN
ejde-610	202	6	indeed	indeed	ADV
ejde-610	202	7	,	,	PUNCT
ejde-610	202	8	by	by	ADP
ejde-610	202	9	the	the	DET
ejde-610	202	10	hölder	hölder	NOUN
ejde-610	202	11	inequality	inequality	NOUN
ejde-610	202	12	,	,	PUNCT
ejde-610	202	13	the	the	DET
ejde-610	202	14	sobolev	sobolev	NOUN
ejde-610	202	15	inequality	inequality	NOUN
ejde-610	202	16	and	and	CCONJ
ejde-610	202	17	proposition	proposition	NOUN
ejde-610	202	18	2.1	2.1	NUM
ejde-610	202	19	,	,	PUNCT
ejde-610	202	20	we	we	PRON
ejde-610	202	21	obtain∣∣	obtain∣∣	NOUN
ejde-610	202	22	∫	∫	PROPN
ejde-610	202	23	r3	r3	PROPN
ejde-610	202	24	(	(	PUNCT
ejde-610	202	25	φun	φun	NOUN
ejde-610	202	26	−	−	PROPN
ejde-610	202	27	φu)(un	φu)(un	PUNCT
ejde-610	202	28	−	−	PROPN
ejde-610	202	29	u)un	u)un	PROPN
ejde-610	202	30	dx	dx	PROPN
ejde-610	202	31	∣∣	∣∣	NUM
ejde-610	202	32	≤	≤	NUM
ejde-610	202	33	|(φun	|(φun	NUM
ejde-610	202	34	−	−	PROPN
ejde-610	202	35	φu)(un	φu)(un	PUNCT
ejde-610	202	36	−	−	PROPN
ejde-610	202	37	u)|2|un|2	u)|2|un|2	PROPN
ejde-610	202	38	≤	≤	PROPN
ejde-610	202	39	|φun	|φun	VERB
ejde-610	202	40	−	−	NOUN
ejde-610	202	41	φu|6|un	φu|6|un	SYM
ejde-610	202	42	−	−	PRON
ejde-610	202	43	u|3|un|2	u|3|un|2	ADJ
ejde-610	202	44	≤	≤	NOUN
ejde-610	202	45	c‖φun	c‖φun	NOUN
ejde-610	202	46	−	−	NOUN
ejde-610	202	47	φu‖|un	φu‖|un	ADP
ejde-610	202	48	−	−	PROPN
ejde-610	202	49	u|3|un|2	u|3|un|2	ADJ
ejde-610	202	50	,	,	PUNCT
ejde-610	202	51	where	where	SCONJ
ejde-610	202	52	c	c	PROPN
ejde-610	202	53	is	be	AUX
ejde-610	202	54	a	a	DET
ejde-610	202	55	positive	positive	ADJ
ejde-610	202	56	constant	constant	NOUN
ejde-610	202	57	.	.	PUNCT
ejde-610	203	1	since	since	SCONJ
ejde-610	203	2	un	un	PROPN
ejde-610	203	3	→	→	SYM
ejde-610	203	4	u	u	PROPN
ejde-610	203	5	in	in	ADP
ejde-610	203	6	ls(r3	ls(r3	PROPN
ejde-610	203	7	)	)	PUNCT
ejde-610	203	8	for	for	ADP
ejde-610	203	9	2	2	NUM
ejde-610	203	10	≤	≤	NOUN
ejde-610	203	11	s	s	PART
ejde-610	203	12	<	<	X
ejde-610	203	13	6	6	NUM
ejde-610	203	14	,	,	PUNCT
ejde-610	203	15	we	we	PRON
ejde-610	203	16	obtain∣∣	obtain∣∣	NOUN
ejde-610	203	17	∫	∫	PROPN
ejde-610	203	18	r3	r3	PROPN
ejde-610	203	19	(	(	PUNCT
ejde-610	203	20	φun	φun	NOUN
ejde-610	203	21	−	−	PROPN
ejde-610	203	22	φu)(un	φu)(un	PUNCT
ejde-610	203	23	−	−	PROPN
ejde-610	203	24	u)un	u)un	PROPN
ejde-610	203	25	dx	dx	VERB
ejde-610	203	26	∣∣→	∣∣→	ADV
ejde-610	203	27	0	0	PUNCT
ejde-610	203	28	as	as	ADP
ejde-610	203	29	n→	n→	ADV
ejde-610	203	30	+	+	ADJ
ejde-610	203	31	∞,∣∣	∞,∣∣	NOUN
ejde-610	203	32	∫	∫	PROPN
ejde-610	203	33	r3	r3	PROPN
ejde-610	203	34	φu(un	φu(un	PROPN
ejde-610	203	35	−	−	PROPN
ejde-610	203	36	u)(un	u)(un	PROPN
ejde-610	203	37	−	−	PROPN
ejde-610	203	38	u	u	NOUN
ejde-610	203	39	)	)	PUNCT
ejde-610	203	40	dx	dx	PROPN
ejde-610	204	1	∣∣	∣∣	NUM
ejde-610	204	2	≤	≤	PUNCT
ejde-610	204	3	|φu|6|un	|φu|6|un	ADV
ejde-610	204	4	−	−	NOUN
ejde-610	204	5	u|3|un	u|3|un	ADP
ejde-610	204	6	−	−	PROPN
ejde-610	204	7	u|2	u|2	PROPN
ejde-610	204	8	→	→	SYM
ejde-610	204	9	0	0	NUM
ejde-610	204	10	as	as	ADP
ejde-610	204	11	n→	n→	ADV
ejde-610	205	1	+	+	PROPN
ejde-610	205	2	∞.	∞.	PROPN
ejde-610	205	3	thus	thus	ADV
ejde-610	205	4	we	we	PRON
ejde-610	205	5	obtain∫	obtain∫	VERB
ejde-610	205	6	r3	r3	PROPN
ejde-610	205	7	[	[	X
ejde-610	205	8	(	(	PUNCT
ejde-610	205	9	φun	φun	PROPN
ejde-610	205	10	un	un	PROPN
ejde-610	205	11	−	−	PROPN
ejde-610	205	12	φuu)(un	φuu)(un	PROPN
ejde-610	205	13	−	−	PROPN
ejde-610	205	14	u	u	NOUN
ejde-610	205	15	)	)	PUNCT
ejde-610	205	16	dx	dx	PROPN
ejde-610	205	17	=	=	SYM
ejde-610	205	18	∫	∫	PROPN
ejde-610	205	19	r3	r3	PROPN
ejde-610	205	20	(	(	PUNCT
ejde-610	205	21	φun	φun	NOUN
ejde-610	205	22	−	−	PROPN
ejde-610	205	23	φu)(un	φu)(un	PUNCT
ejde-610	205	24	−	−	PROPN
ejde-610	205	25	u)un	u)un	PROPN
ejde-610	206	1	dx+	dx+	ADJ
ejde-610	206	2	∫	∫	PROPN
ejde-610	206	3	r3	r3	PROPN
ejde-610	206	4	φu(un	φu(un	PROPN
ejde-610	206	5	−	−	PROPN
ejde-610	206	6	u)(un	u)(un	PROPN
ejde-610	206	7	−	−	PROPN
ejde-610	206	8	u	u	NOUN
ejde-610	206	9	)	)	PUNCT
ejde-610	206	10	dx→	dx→	NOUN
ejde-610	206	11	0	0	NUM
ejde-610	206	12	as	as	ADP
ejde-610	206	13	n→	n→	ADV
ejde-610	206	14	+	+	PROPN
ejde-610	206	15	∞.	∞.	PROPN
ejde-610	206	16	since	since	SCONJ
ejde-610	206	17	the	the	DET
ejde-610	206	18	sequence	sequence	NOUN
ejde-610	206	19	{	{	PUNCT
ejde-610	206	20	φ2	φ2	PROPN
ejde-610	206	21	un	un	PROPN
ejde-610	206	22	un	un	PROPN
ejde-610	206	23	}	}	PUNCT
ejde-610	206	24	is	be	AUX
ejde-610	206	25	bounded	bound	VERB
ejde-610	206	26	in	in	ADP
ejde-610	206	27	l3/2(r3	l3/2(r3	PROPN
ejde-610	206	28	)	)	PUNCT
ejde-610	206	29	,	,	PUNCT
ejde-610	206	30	we	we	PRON
ejde-610	206	31	have	have	AUX
ejde-610	206	32	|φ2	|φ2	ADJ
ejde-610	206	33	un	un	PROPN
ejde-610	206	34	un|3/2	un|3/2	ADJ
ejde-610	206	35	≤	≤	PUNCT
ejde-610	206	36	|φun	|φun	VERB
ejde-610	206	37	|26|un|3	|26|un|3	NOUN
ejde-610	206	38	,	,	PUNCT
ejde-610	207	1	ejde-2024/18	ejde-2024/18	NOUN
ejde-610	207	2	coupled	couple	VERB
ejde-610	207	3	klein	klein	PROPN
ejde-610	207	4	-	-	PUNCT
ejde-610	207	5	gordon	gordon	PROPN
ejde-610	207	6	and	and	CCONJ
ejde-610	207	7	born	bear	VERB
ejde-610	207	8	-	-	PUNCT
ejde-610	207	9	infeld	infeld	NOUN
ejde-610	207	10	equatons	equaton	NOUN
ejde-610	207	11	9	9	NUM
ejde-610	207	12	so	so	CCONJ
ejde-610	207	13	∣∣	∣∣	NUM
ejde-610	207	14	∫	∫	PROPN
ejde-610	207	15	r3	r3	PROPN
ejde-610	207	16	[	[	X
ejde-610	207	17	(	(	PUNCT
ejde-610	207	18	φ2	φ2	PROPN
ejde-610	207	19	un	un	PROPN
ejde-610	207	20	un	un	PROPN
ejde-610	207	21	−	−	PROPN
ejde-610	207	22	φ2	φ2	PROPN
ejde-610	207	23	uu)(un	uu)(un	PROPN
ejde-610	208	1	−	−	PROPN
ejde-610	208	2	u	u	NOUN
ejde-610	208	3	)	)	PUNCT
ejde-610	208	4	dx	dx	PROPN
ejde-610	208	5	∣∣	∣∣	NUM
ejde-610	208	6	≤	≤	NUM
ejde-610	208	7	|φ2	|φ2	VERB
ejde-610	208	8	un	un	PROPN
ejde-610	208	9	un	un	PROPN
ejde-610	208	10	−	−	PROPN
ejde-610	208	11	φ2	φ2	PROPN
ejde-610	208	12	uu|3/2|un	uu|3/2|un	NOUN
ejde-610	209	1	−	−	PROPN
ejde-610	209	2	u|3	u|3	PROPN
ejde-610	209	3	≤	≤	PROPN
ejde-610	209	4	(	(	PUNCT
ejde-610	209	5	|φ2	|φ2	VERB
ejde-610	209	6	un	un	PROPN
ejde-610	209	7	un|3/2	un|3/2	NOUN
ejde-610	209	8	+	+	CCONJ
ejde-610	209	9	|φ2	|φ2	ADJ
ejde-610	209	10	uu|3/2)|un	uu|3/2)|un	ADV
ejde-610	209	11	−	−	PROPN
ejde-610	209	12	u|3	u|3	PROPN
ejde-610	209	13	→	→	X
ejde-610	209	14	0	0	NUM
ejde-610	209	15	,	,	PUNCT
ejde-610	209	16	as	as	ADP
ejde-610	209	17	n→	n→	ADV
ejde-610	209	18	+	+	ADJ
ejde-610	209	19	∞.	∞.	PROPN
ejde-610	209	20	by	by	ADP
ejde-610	209	21	proposition	proposition	NOUN
ejde-610	209	22	2.3	2.3	NUM
ejde-610	209	23	and	and	CCONJ
ejde-610	209	24	un	un	PROPN
ejde-610	209	25	→	→	SYM
ejde-610	209	26	u	u	PROPN
ejde-610	209	27	in	in	ADP
ejde-610	209	28	ls(r3	ls(r3	PROPN
ejde-610	209	29	)	)	PUNCT
ejde-610	209	30	for	for	ADP
ejde-610	209	31	2	2	NUM
ejde-610	209	32	≤	≤	NOUN
ejde-610	209	33	s	s	PART
ejde-610	209	34	<	<	X
ejde-610	209	35	6	6	NUM
ejde-610	209	36	,	,	PUNCT
ejde-610	209	37	we	we	PRON
ejde-610	209	38	have∫	have∫	VERB
ejde-610	209	39	r3	r3	PROPN
ejde-610	209	40	(	(	PUNCT
ejde-610	209	41	h(x	h(x	PROPN
ejde-610	209	42	,	,	PUNCT
ejde-610	209	43	un)−	un)−	PART
ejde-610	209	44	h(x	h(x	PROPN
ejde-610	209	45	,	,	PUNCT
ejde-610	209	46	u))(un	u))(un	PROPN
ejde-610	209	47	−	−	PROPN
ejde-610	209	48	u	u	NOUN
ejde-610	209	49	)	)	PUNCT
ejde-610	209	50	dx→	dx→	NOUN
ejde-610	209	51	0	0	NUM
ejde-610	209	52	as	as	ADP
ejde-610	209	53	n→	n→	ADV
ejde-610	209	54	+	+	PROPN
ejde-610	209	55	∞.	∞.	PROPN
ejde-610	209	56	since	since	SCONJ
ejde-610	209	57	un	un	PROPN
ejde-610	209	58	→	→	SYM
ejde-610	209	59	u	u	PROPN
ejde-610	209	60	in	in	ADP
ejde-610	209	61	l2(r3	l2(r3	NUM
ejde-610	209	62	)	)	PUNCT
ejde-610	209	63	,	,	PUNCT
ejde-610	209	64	we	we	PRON
ejde-610	209	65	obtain	obtain	VERB
ejde-610	209	66	that	that	DET
ejde-610	209	67	v0	v0	NOUN
ejde-610	209	68	∫	∫	NOUN
ejde-610	210	1	r3(un	r3(un	PROPN
ejde-610	210	2	−	−	PROPN
ejde-610	210	3	u)2	u)2	ADJ
ejde-610	210	4	dx→	dx→	NOUN
ejde-610	210	5	0	0	PUNCT
ejde-610	210	6	as	as	ADP
ejde-610	210	7	n→	n→	ADV
ejde-610	210	8	+	+	PROPN
ejde-610	210	9	∞.	∞.	PROPN
ejde-610	210	10	therefore	therefore	ADV
ejde-610	210	11	‖un	‖un	PROPN
ejde-610	210	12	−	−	PROPN
ejde-610	210	13	u‖	u‖	NOUN
ejde-610	210	14	→	→	SYM
ejde-610	210	15	0	0	NUM
ejde-610	210	16	in	in	ADP
ejde-610	210	17	e	e	PROPN
ejde-610	210	18	as	as	ADP
ejde-610	210	19	n→∞.	n→∞.	NUM
ejde-610	210	20	the	the	DET
ejde-610	210	21	proof	proof	NOUN
ejde-610	210	22	is	be	AUX
ejde-610	210	23	complete	complete	ADJ
ejde-610	210	24	.	.	PUNCT
ejde-610	211	1	�	�	PROPN
ejde-610	211	2	lemma	lemma	PROPN
ejde-610	211	3	2.5	2.5	NUM
ejde-610	211	4	.	.	PUNCT
ejde-610	212	1	let	let	VERB
ejde-610	212	2	x	x	PRON
ejde-610	212	3	be	be	AUX
ejde-610	212	4	a	a	DET
ejde-610	212	5	finite	finite	ADJ
ejde-610	212	6	dimensional	dimensional	ADJ
ejde-610	212	7	subspace	subspace	NOUN
ejde-610	212	8	of	of	ADP
ejde-610	212	9	e	e	NOUN
ejde-610	212	10	,	,	PUNCT
ejde-610	212	11	then	then	ADV
ejde-610	212	12	i	i	PRON
ejde-610	212	13	is	be	AUX
ejde-610	212	14	anti	anti	ADJ
ejde-610	212	15	-	-	ADJ
ejde-610	212	16	coercive	coercive	ADJ
ejde-610	212	17	on	on	ADP
ejde-610	212	18	x	x	SYM
ejde-610	212	19	,	,	PUNCT
ejde-610	212	20	i.e.	i.e.	X
ejde-610	212	21	i(u)→	i(u)→	PROPN
ejde-610	212	22	−∞	−∞	NOUN
ejde-610	212	23	,	,	PUNCT
ejde-610	212	24	as	as	ADP
ejde-610	212	25	‖u‖	‖u‖	PROPN
ejde-610	212	26	→	→	SYM
ejde-610	212	27	∞	∞	PROPN
ejde-610	212	28	,	,	PUNCT
ejde-610	212	29	u	u	PROPN
ejde-610	212	30	∈	∈	NOUN
ejde-610	212	31	x.	x.	NOUN
ejde-610	212	32	proof	proof	NOUN
ejde-610	212	33	.	.	PUNCT
ejde-610	213	1	if	if	SCONJ
ejde-610	213	2	this	this	PRON
ejde-610	213	3	were	be	AUX
ejde-610	213	4	not	not	PART
ejde-610	213	5	true	true	ADJ
ejde-610	213	6	,	,	PUNCT
ejde-610	213	7	we	we	PRON
ejde-610	213	8	can	can	AUX
ejde-610	213	9	choose	choose	VERB
ejde-610	213	10	a	a	DET
ejde-610	213	11	sequence	sequence	NOUN
ejde-610	213	12	{	{	PUNCT
ejde-610	213	13	un	un	PROPN
ejde-610	213	14	}	}	PUNCT
ejde-610	213	15	⊂	⊂	NOUN
ejde-610	213	16	x	x	X
ejde-610	213	17	and	and	CCONJ
ejde-610	213	18	ξ	ξ	PROPN
ejde-610	213	19	is	be	AUX
ejde-610	213	20	a	a	DET
ejde-610	213	21	real	real	ADJ
ejde-610	213	22	number	number	NOUN
ejde-610	213	23	such	such	ADJ
ejde-610	213	24	that	that	SCONJ
ejde-610	213	25	‖un‖	‖un‖	NOUN
ejde-610	213	26	→	→	SYM
ejde-610	213	27	∞	∞	PROPN
ejde-610	213	28	,	,	PUNCT
ejde-610	213	29	i(un	i(un	NUM
ejde-610	213	30	)	)	PUNCT
ejde-610	213	31	≥	≥	NUM
ejde-610	213	32	ξ	ξ	X
ejde-610	213	33	.	.	PUNCT
ejde-610	214	1	(	(	PUNCT
ejde-610	214	2	2.16	2.16	NUM
ejde-610	214	3	)	)	PUNCT
ejde-610	214	4	let	let	VERB
ejde-610	214	5	vn	vn	PROPN
ejde-610	214	6	=	=	PROPN
ejde-610	214	7	un	un	PROPN
ejde-610	214	8	‖un‖	‖un‖	PROPN
ejde-610	214	9	.	.	PUNCT
ejde-610	215	1	since	since	SCONJ
ejde-610	215	2	dimx	dimx	NOUN
ejde-610	215	3	<	<	X
ejde-610	215	4	∞	∞	PROPN
ejde-610	215	5	,	,	PUNCT
ejde-610	215	6	going	go	VERB
ejde-610	215	7	if	if	SCONJ
ejde-610	215	8	necessary	necessary	ADJ
ejde-610	215	9	to	to	ADP
ejde-610	215	10	a	a	DET
ejde-610	215	11	subsequence	subsequence	NOUN
ejde-610	215	12	we	we	PRON
ejde-610	215	13	have	have	VERB
ejde-610	215	14	‖vn	‖vn	PROPN
ejde-610	216	1	−	−	PROPN
ejde-610	216	2	v0‖	v0‖	PROPN
ejde-610	216	3	→	→	SYM
ejde-610	216	4	0	0	NUM
ejde-610	216	5	,	,	PUNCT
ejde-610	216	6	vn(x)→	vn(x)→	PROPN
ejde-610	216	7	v0(x	v0(x	SYM
ejde-610	216	8	)	)	PUNCT
ejde-610	216	9	a.e	a.e	PROPN
ejde-610	216	10	.	.	PROPN
ejde-610	217	1	in	in	ADP
ejde-610	217	2	r3	r3	PROPN
ejde-610	217	3	for	for	ADP
ejde-610	217	4	every	every	DET
ejde-610	217	5	v0	v0	NOUN
ejde-610	217	6	∈	∈	PROPN
ejde-610	217	7	x	x	X
ejde-610	217	8	,	,	PUNCT
ejde-610	217	9	with	with	ADP
ejde-610	217	10	‖v0‖	‖v0‖	PROPN
ejde-610	217	11	=	=	SYM
ejde-610	217	12	1	1	X
ejde-610	217	13	.	.	PUNCT
ejde-610	218	1	since	since	SCONJ
ejde-610	218	2	v0	v0	PROPN
ejde-610	218	3	6=	6=	PROPN
ejde-610	218	4	0	0	NUM
ejde-610	218	5	,	,	PUNCT
ejde-610	218	6	similar	similar	ADJ
ejde-610	218	7	to	to	ADP
ejde-610	218	8	(	(	PUNCT
ejde-610	218	9	2.14	2.14	NUM
ejde-610	218	10	)	)	PUNCT
ejde-610	218	11	we	we	PRON
ejde-610	218	12	obtain	obtain	VERB
ejde-610	218	13	that∫	that∫	PROPN
ejde-610	218	14	r3	r3	PROPN
ejde-610	218	15	h(x	h(x	PROPN
ejde-610	218	16	,	,	PUNCT
ejde-610	218	17	un	un	PROPN
ejde-610	218	18	)	)	PUNCT
ejde-610	218	19	‖un‖θ	‖un‖θ	NOUN
ejde-610	218	20	dx→	dx→	NOUN
ejde-610	219	1	+	+	NOUN
ejde-610	219	2	∞.	∞.	PROPN
ejde-610	219	3	arguing	argue	VERB
ejde-610	219	4	similar	similar	ADJ
ejde-610	219	5	to	to	ADP
ejde-610	219	6	(	(	PUNCT
ejde-610	219	7	2.15	2.15	NUM
ejde-610	219	8	)	)	PUNCT
ejde-610	219	9	,	,	PUNCT
ejde-610	219	10	it	it	PRON
ejde-610	219	11	follows	follow	VERB
ejde-610	219	12	from	from	ADP
ejde-610	219	13	supn	supn	NOUN
ejde-610	219	14	|i(un)|	|i(un)|	PRON
ejde-610	219	15	<	<	X
ejde-610	219	16	∞	∞	PROPN
ejde-610	219	17	that	that	DET
ejde-610	219	18	i(un	i(un	NUM
ejde-610	219	19	)	)	PUNCT
ejde-610	219	20	=	=	PUNCT
ejde-610	220	1	‖un‖θ	‖un‖θ	NOUN
ejde-610	220	2	(	(	PUNCT
ejde-610	220	3	‖un‖2	‖un‖2	PROPN
ejde-610	220	4	2‖un‖θ	2‖un‖θ	NUM
ejde-610	220	5	−	−	PROPN
ejde-610	220	6	ω	ω	NUM
ejde-610	220	7	2‖un‖θ	2‖un‖θ	NUM
ejde-610	220	8	∫	∫	PROPN
ejde-610	220	9	r3	r3	PROPN
ejde-610	220	10	φun	φun	PROPN
ejde-610	220	11	u2	u2	PROPN
ejde-610	220	12	n	n	PROPN
ejde-610	220	13	dx	dx	PROPN
ejde-610	220	14	+	+	X
ejde-610	220	15	β	β	PROPN
ejde-610	220	16	16π‖un‖θ	16π‖un‖θ	NUM
ejde-610	220	17	∫	∫	PROPN
ejde-610	220	18	r3	r3	PROPN
ejde-610	220	19	|∇φun	|∇φun	NUM
ejde-610	220	20	|4	|4	X
ejde-610	220	21	dx−	dx−	NUM
ejde-610	220	22	∫	∫	PROPN
ejde-610	220	23	r3	r3	PROPN
ejde-610	220	24	h(x	h(x	PROPN
ejde-610	220	25	,	,	PUNCT
ejde-610	220	26	un	un	PROPN
ejde-610	220	27	)	)	PUNCT
ejde-610	220	28	‖un‖θ	‖un‖θ	PROPN
ejde-610	220	29	dx	dx	PROPN
ejde-610	220	30	)	)	PUNCT
ejde-610	220	31	→	→	SYM
ejde-610	220	32	−∞	−∞	NOUN
ejde-610	220	33	,	,	PUNCT
ejde-610	220	34	which	which	PRON
ejde-610	220	35	is	be	AUX
ejde-610	220	36	contradicts	contradict	VERB
ejde-610	220	37	i(un	i(un	NUM
ejde-610	220	38	)	)	PUNCT
ejde-610	220	39	≥	≥	NUM
ejde-610	220	40	ξ	ξ	X
ejde-610	220	41	.	.	PUNCT
ejde-610	221	1	the	the	DET
ejde-610	221	2	proof	proof	NOUN
ejde-610	221	3	is	be	AUX
ejde-610	221	4	complete	complete	ADJ
ejde-610	221	5	.	.	PUNCT
ejde-610	222	1	�	�	PROPN
ejde-610	222	2	proof	proof	NOUN
ejde-610	222	3	of	of	ADP
ejde-610	222	4	theorem	theorem	ADJ
ejde-610	222	5	1.1	1.1	NUM
ejde-610	222	6	.	.	PUNCT
ejde-610	223	1	we	we	PRON
ejde-610	223	2	will	will	AUX
ejde-610	223	3	find	find	VERB
ejde-610	223	4	a	a	DET
ejde-610	223	5	sequence	sequence	NOUN
ejde-610	223	6	of	of	ADP
ejde-610	223	7	critical	critical	ADJ
ejde-610	223	8	points	point	NOUN
ejde-610	223	9	{	{	PUNCT
ejde-610	223	10	un	un	PROPN
ejde-610	223	11	}	}	PUNCT
ejde-610	223	12	of	of	ADP
ejde-610	223	13	i	i	PRON
ejde-610	223	14	such	such	ADJ
ejde-610	223	15	that	that	SCONJ
ejde-610	223	16	i(un	i(un	NUM
ejde-610	223	17	)	)	PUNCT
ejde-610	223	18	→	→	PUNCT
ejde-610	224	1	+	+	X
ejde-610	224	2	∞.	∞.	PROPN
ejde-610	224	3	since	since	SCONJ
ejde-610	224	4	f(x	f(x	PROPN
ejde-610	224	5	,	,	PUNCT
ejde-610	224	6	t	t	PROPN
ejde-610	224	7	)	)	PUNCT
ejde-610	224	8	is	be	AUX
ejde-610	224	9	odd	odd	ADJ
ejde-610	224	10	in	in	ADP
ejde-610	224	11	t	t	PROPN
ejde-610	224	12	,	,	PUNCT
ejde-610	224	13	i	i	PRON
ejde-610	224	14	is	be	AUX
ejde-610	224	15	an	an	DET
ejde-610	224	16	even	even	ADJ
ejde-610	224	17	function	function	NOUN
ejde-610	224	18	.	.	PUNCT
ejde-610	225	1	from	from	ADP
ejde-610	225	2	lemma	lemma	PROPN
ejde-610	225	3	2.4	2.4	NUM
ejde-610	225	4	it	it	PRON
ejde-610	225	5	follows	follow	VERB
ejde-610	225	6	that	that	SCONJ
ejde-610	225	7	i	i	PRON
ejde-610	225	8	satisfies	satisfy	VERB
ejde-610	225	9	the	the	DET
ejde-610	225	10	(	(	PUNCT
ejde-610	225	11	ps	ps	NOUN
ejde-610	225	12	)	)	PUNCT
ejde-610	225	13	condition	condition	NOUN
ejde-610	225	14	.	.	PUNCT
ejde-610	226	1	therefore	therefore	ADV
ejde-610	226	2	,	,	PUNCT
ejde-610	226	3	it	it	PRON
ejde-610	226	4	suffices	suffice	VERB
ejde-610	226	5	to	to	PART
ejde-610	226	6	verify	verify	VERB
ejde-610	226	7	(	(	PUNCT
ejde-610	226	8	i	i	NOUN
ejde-610	226	9	)	)	PUNCT
ejde-610	226	10	and	and	CCONJ
ejde-610	226	11	(	(	PUNCT
ejde-610	226	12	ii	ii	NOUN
ejde-610	226	13	)	)	PUNCT
ejde-610	226	14	of	of	ADP
ejde-610	226	15	proposition	proposition	NOUN
ejde-610	226	16	2.2	2.2	NUM
ejde-610	226	17	.	.	PUNCT
ejde-610	227	1	since	since	SCONJ
ejde-610	227	2	dimyk	dimyk	PROPN
ejde-610	227	3	<	<	X
ejde-610	227	4	∞	∞	PROPN
ejde-610	227	5	,	,	PUNCT
ejde-610	227	6	by	by	ADP
ejde-610	227	7	lemma	lemma	PROPN
ejde-610	227	8	2.5	2.5	NUM
ejde-610	227	9	,	,	PUNCT
ejde-610	227	10	we	we	PRON
ejde-610	227	11	obtain	obtain	VERB
ejde-610	227	12	the	the	DET
ejde-610	227	13	conclusion	conclusion	NOUN
ejde-610	227	14	of	of	ADP
ejde-610	227	15	(	(	PUNCT
ejde-610	227	16	i	i	NOUN
ejde-610	227	17	)	)	PUNCT
ejde-610	227	18	.	.	PUNCT
ejde-610	228	1	by	by	ADP
ejde-610	228	2	(	(	PUNCT
ejde-610	228	3	a2	a2	PROPN
ejde-610	228	4	)	)	PUNCT
ejde-610	228	5	and	and	CCONJ
ejde-610	228	6	(	(	PUNCT
ejde-610	228	7	a3	a3	NOUN
ejde-610	228	8	)	)	PUNCT
ejde-610	228	9	,	,	PUNCT
ejde-610	228	10	we	we	PRON
ejde-610	228	11	have	have	VERB
ejde-610	228	12	|f(x	|f(x	PROPN
ejde-610	228	13	,	,	PUNCT
ejde-610	228	14	t)|	t)|	ADJ
ejde-610	228	15	≤	≤	NUM
ejde-610	228	16	ε|t|+	ε|t|+	PROPN
ejde-610	228	17	cε|t|p−1	cε|t|p−1	PROPN
ejde-610	228	18	,	,	PUNCT
ejde-610	228	19	|f	|f	PROPN
ejde-610	228	20	(	(	PUNCT
ejde-610	228	21	x	x	X
ejde-610	228	22	,	,	PUNCT
ejde-610	228	23	t)|	t)|	ADJ
ejde-610	228	24	≤	≤	NUM
ejde-610	228	25	ε	ε	PROPN
ejde-610	228	26	2	2	NUM
ejde-610	228	27	t2	t2	NOUN
ejde-610	228	28	+	+	CCONJ
ejde-610	228	29	cε	cε	VERB
ejde-610	228	30	p	p	NOUN
ejde-610	228	31	|t|p	|t|p	PROPN
ejde-610	228	32	,	,	PUNCT
ejde-610	228	33	where	where	SCONJ
ejde-610	228	34	ε	ε	PROPN
ejde-610	228	35	>	>	X
ejde-610	228	36	0	0	NUM
ejde-610	228	37	is	be	AUX
ejde-610	228	38	very	very	ADV
ejde-610	228	39	small	small	ADJ
ejde-610	228	40	.	.	PUNCT
ejde-610	229	1	then	then	ADV
ejde-610	229	2	we	we	PRON
ejde-610	229	3	have	have	VERB
ejde-610	229	4	|f	|f	PROPN
ejde-610	229	5	(	(	PUNCT
ejde-610	229	6	x	x	X
ejde-610	229	7	,	,	PUNCT
ejde-610	229	8	t)|	t)|	ADJ
ejde-610	229	9	≤	≤	NUM
ejde-610	229	10	b	b	SYM
ejde-610	229	11	2d2	2d2	NUM
ejde-610	229	12	2	2	NUM
ejde-610	229	13	t2	t2	NOUN
ejde-610	229	14	+	+	CCONJ
ejde-610	229	15	cb	cb	PROPN
ejde-610	229	16	p	p	PROPN
ejde-610	229	17	|t|p	|t|p	PROPN
ejde-610	229	18	,	,	PUNCT
ejde-610	229	19	(	(	PUNCT
ejde-610	229	20	2.17	2.17	NUM
ejde-610	229	21	)	)	PUNCT
ejde-610	229	22	where	where	SCONJ
ejde-610	229	23	b	b	NOUN
ejde-610	229	24	is	be	AUX
ejde-610	229	25	defined	define	VERB
ejde-610	229	26	in	in	ADP
ejde-610	229	27	(	(	PUNCT
ejde-610	229	28	2.4	2.4	NUM
ejde-610	229	29	)	)	PUNCT
ejde-610	229	30	.	.	PUNCT
ejde-610	230	1	we	we	PRON
ejde-610	230	2	assume	assume	VERB
ejde-610	230	3	that	that	SCONJ
ejde-610	230	4	0	0	NUM
ejde-610	230	5	∈	∈	PROPN
ejde-610	231	1	[	[	X
ejde-610	231	2	λl	λl	X
ejde-610	231	3	,	,	PUNCT
ejde-610	231	4	λl+1	λl+1	PROPN
ejde-610	231	5	)	)	PUNCT
ejde-610	231	6	.	.	PUNCT
ejde-610	232	1	then	then	ADV
ejde-610	232	2	if	if	SCONJ
ejde-610	232	3	k	k	PROPN
ejde-610	232	4	>	>	X
ejde-610	232	5	l	l	NOUN
ejde-610	232	6	,	,	PUNCT
ejde-610	232	7	we	we	PRON
ejde-610	232	8	have	have	VERB
ejde-610	232	9	that	that	PRON
ejde-610	232	10	zk	zk	PROPN
ejde-610	232	11	⊂	⊂	PROPN
ejde-610	232	12	e+	e+	NOUN
ejde-610	232	13	,	,	PUNCT
ejde-610	232	14	where	where	SCONJ
ejde-610	232	15	e+	e+	PRON
ejde-610	232	16	is	be	AUX
ejde-610	232	17	defined	define	VERB
ejde-610	232	18	in	in	ADP
ejde-610	232	19	(	(	PUNCT
ejde-610	232	20	2.3	2.3	NUM
ejde-610	232	21	)	)	PUNCT
ejde-610	232	22	.	.	PUNCT
ejde-610	233	1	now	now	ADV
ejde-610	233	2	we	we	PRON
ejde-610	233	3	have	have	VERB
ejde-610	233	4	n(u	n(u	PROPN
ejde-610	233	5	)	)	PUNCT
ejde-610	233	6	≥	≥	NOUN
ejde-610	234	1	b‖u‖2	b‖u‖2	PROPN
ejde-610	234	2	,	,	PUNCT
ejde-610	234	3	u	u	PROPN
ejde-610	234	4	∈	∈	PROPN
ejde-610	234	5	zk	zk	PROPN
ejde-610	234	6	(	(	PUNCT
ejde-610	234	7	2.18	2.18	NUM
ejde-610	234	8	)	)	PUNCT
ejde-610	234	9	10	10	NUM
ejde-610	234	10	l.	l.	PROPN
ejde-610	234	11	wang	wang	PROPN
ejde-610	234	12	,	,	PUNCT
ejde-610	234	13	p.	p.	PROPN
ejde-610	234	14	zhao	zhao	PROPN
ejde-610	234	15	,	,	PUNCT
ejde-610	234	16	d.	d.	PROPN
ejde-610	234	17	zhang	zhang	PROPN
ejde-610	234	18	ejde-2024/18	ejde-2024/18	PROPN
ejde-610	234	19	and	and	CCONJ
ejde-610	234	20	,	,	PUNCT
ejde-610	234	21	as	as	ADP
ejde-610	234	22	in	in	ADP
ejde-610	234	23	the	the	DET
ejde-610	234	24	proof	proof	NOUN
ejde-610	234	25	of	of	ADP
ejde-610	234	26	[	[	X
ejde-610	234	27	29	29	NUM
ejde-610	234	28	,	,	PUNCT
ejde-610	234	29	lemma3.8	lemma3.8	PROPN
ejde-610	234	30	]	]	PUNCT
ejde-610	234	31	,	,	PUNCT
ejde-610	234	32	βk	βk	ADP
ejde-610	234	33	=	=	PUNCT
ejde-610	234	34	sup	sup	NOUN
ejde-610	234	35	u∈zk,‖u‖=1	u∈zk,‖u‖=1	NOUN
ejde-610	234	36	|u|p	|u|p	NOUN
ejde-610	234	37	→	→	SYM
ejde-610	234	38	0	0	NUM
ejde-610	234	39	,	,	PUNCT
ejde-610	234	40	as	as	SCONJ
ejde-610	234	41	k	k	PROPN
ejde-610	234	42	→∞.	→∞.	PROPN
ejde-610	234	43	(	(	PUNCT
ejde-610	234	44	2.19	2.19	NUM
ejde-610	234	45	)	)	PUNCT
ejde-610	234	46	let	let	VERB
ejde-610	234	47	rk	rk	NOUN
ejde-610	234	48	=	=	SYM
ejde-610	234	49	(	(	PUNCT
ejde-610	234	50	cpβpk)1/(2−p	cpβpk)1/(2−p	PROPN
ejde-610	234	51	)	)	PUNCT
ejde-610	234	52	,	,	PUNCT
ejde-610	234	53	where	where	SCONJ
ejde-610	234	54	c	c	PROPN
ejde-610	234	55	is	be	AUX
ejde-610	234	56	chosen	choose	VERB
ejde-610	234	57	as	as	ADP
ejde-610	234	58	in	in	ADP
ejde-610	234	59	(	(	PUNCT
ejde-610	234	60	2.17	2.17	NUM
ejde-610	234	61	)	)	PUNCT
ejde-610	234	62	.	.	PUNCT
ejde-610	235	1	for	for	ADP
ejde-610	235	2	u	u	PROPN
ejde-610	235	3	∈	∈	PROPN
ejde-610	235	4	zk	zk	PROPN
ejde-610	235	5	⊂	⊂	PROPN
ejde-610	235	6	e+	e+	AUX
ejde-610	235	7	with	with	ADP
ejde-610	235	8	‖u‖	‖u‖	PROPN
ejde-610	235	9	=	=	SYM
ejde-610	235	10	rk	rk	PROPN
ejde-610	235	11	,	,	PUNCT
ejde-610	235	12	φu	φu	ADJ
ejde-610	235	13	≤	≤	NOUN
ejde-610	235	14	0	0	NUM
ejde-610	235	15	,	,	PUNCT
ejde-610	235	16	by	by	ADP
ejde-610	235	17	(	(	PUNCT
ejde-610	235	18	2.18	2.18	NUM
ejde-610	235	19	)	)	PUNCT
ejde-610	235	20	we	we	PRON
ejde-610	235	21	deduce	deduce	VERB
ejde-610	235	22	that	that	SCONJ
ejde-610	235	23	i(u	i(u	PROPN
ejde-610	235	24	)	)	PUNCT
ejde-610	236	1	=	=	PUNCT
ejde-610	236	2	n(u)−	n(u)−	DET
ejde-610	236	3	1	1	NUM
ejde-610	236	4	2	2	NUM
ejde-610	236	5	ω	ω	NUM
ejde-610	236	6	∫	∫	PROPN
ejde-610	236	7	r3	r3	PROPN
ejde-610	236	8	φuu	φuu	VERB
ejde-610	236	9	2	2	NUM
ejde-610	236	10	dx+	dx+	NOUN
ejde-610	236	11	β	β	X
ejde-610	236	12	16π	16π	PROPN
ejde-610	236	13	∫	∫	PROPN
ejde-610	236	14	r3	r3	PROPN
ejde-610	236	15	|∇φu|4	|∇φu|4	NOUN
ejde-610	237	1	dx−	dx−	NUM
ejde-610	237	2	∫	∫	PROPN
ejde-610	237	3	r3	r3	PROPN
ejde-610	237	4	f	f	PROPN
ejde-610	237	5	(	(	PUNCT
ejde-610	237	6	x	x	X
ejde-610	237	7	,	,	PUNCT
ejde-610	237	8	u	u	NOUN
ejde-610	237	9	)	)	PUNCT
ejde-610	237	10	dx	dx	PROPN
ejde-610	237	11	≥	≥	NOUN
ejde-610	237	12	b‖u‖2	b‖u‖2	ADV
ejde-610	237	13	−	−	PROPN
ejde-610	237	14	b	b	NOUN
ejde-610	237	15	2d2	2d2	NUM
ejde-610	237	16	2	2	NUM
ejde-610	237	17	|u|22	|u|22	PROPN
ejde-610	237	18	−	−	PROPN
ejde-610	237	19	cb	cb	PROPN
ejde-610	237	20	p	p	PROPN
ejde-610	237	21	|u|pp	|u|pp	PROPN
ejde-610	237	22	≥	≥	PROPN
ejde-610	237	23	b	b	PROPN
ejde-610	237	24	(	(	PUNCT
ejde-610	237	25	1	1	NUM
ejde-610	237	26	2	2	NUM
ejde-610	237	27	‖u‖2	‖u‖2	ADJ
ejde-610	237	28	−	−	PROPN
ejde-610	237	29	cβpk	cβpk	NOUN
ejde-610	237	30	p	p	NOUN
ejde-610	237	31	‖u‖p	‖u‖p	NOUN
ejde-610	237	32	)	)	PUNCT
ejde-610	238	1	=	=	SYM
ejde-610	238	2	b	b	X
ejde-610	238	3	(	(	PUNCT
ejde-610	238	4	1	1	NUM
ejde-610	238	5	2	2	NUM
ejde-610	238	6	−	−	NUM
ejde-610	238	7	1	1	NUM
ejde-610	238	8	p2	p2	PROPN
ejde-610	238	9	)	)	PUNCT
ejde-610	238	10	(	(	PUNCT
ejde-610	238	11	cpβpk)2/(2−p	cpβpk)2/(2−p	NOUN
ejde-610	238	12	)	)	PUNCT
ejde-610	238	13	.	.	PUNCT
ejde-610	239	1	since	since	SCONJ
ejde-610	239	2	βk	βk	ADP
ejde-610	239	3	→	→	SYM
ejde-610	239	4	0	0	NUM
ejde-610	239	5	and	and	CCONJ
ejde-610	239	6	p	p	X
ejde-610	239	7	>	>	X
ejde-610	239	8	2	2	NUM
ejde-610	239	9	,	,	PUNCT
ejde-610	239	10	it	it	PRON
ejde-610	239	11	follows	follow	VERB
ejde-610	239	12	that	that	SCONJ
ejde-610	239	13	bk	bk	PROPN
ejde-610	239	14	=	=	SYM
ejde-610	239	15	inf	inf	PROPN
ejde-610	239	16	u∈zk,‖u‖=rk	u∈zk,‖u‖=rk	PROPN
ejde-610	239	17	i(u)→	i(u)→	NOUN
ejde-610	240	1	+	+	SYM
ejde-610	240	2	∞.	∞.	PROPN
ejde-610	240	3	we	we	PRON
ejde-610	240	4	obtain	obtain	VERB
ejde-610	240	5	the	the	DET
ejde-610	240	6	conclusion	conclusion	NOUN
ejde-610	240	7	of	of	ADP
ejde-610	240	8	(	(	PUNCT
ejde-610	240	9	ii	ii	NOUN
ejde-610	240	10	)	)	PUNCT
ejde-610	240	11	.	.	PUNCT
ejde-610	241	1	the	the	DET
ejde-610	241	2	proof	proof	NOUN
ejde-610	241	3	is	be	AUX
ejde-610	241	4	complete	complete	ADJ
ejde-610	241	5	.	.	PUNCT
ejde-610	242	1	�	�	PROPN
ejde-610	242	2	acknowledgments	acknowledgment	NOUN
ejde-610	242	3	.	.	PUNCT
ejde-610	243	1	this	this	DET
ejde-610	243	2	research	research	NOUN
ejde-610	243	3	was	be	AUX
ejde-610	243	4	partially	partially	ADV
ejde-610	243	5	supported	support	VERB
ejde-610	243	6	by	by	ADP
ejde-610	243	7	the	the	DET
ejde-610	243	8	scientific	scientific	ADJ
ejde-610	243	9	research	research	NOUN
ejde-610	243	10	program	program	NOUN
ejde-610	243	11	of	of	ADP
ejde-610	243	12	tianjin	tianjin	PROPN
ejde-610	243	13	education	education	PROPN
ejde-610	243	14	commission	commission	PROPN
ejde-610	243	15	(	(	PUNCT
ejde-610	243	16	2023kj216	2023kj216	NUM
ejde-610	243	17	)	)	PUNCT
ejde-610	243	18	.	.	PUNCT
ejde-610	244	1	references	reference	NOUN
ejde-610	244	2	[	[	X
ejde-610	244	3	1	1	NUM
ejde-610	244	4	]	]	PUNCT
ejde-610	244	5	f.	f.	PROPN
ejde-610	244	6	s.	s.	PROPN
ejde-610	244	7	b.	b.	PROPN
ejde-610	244	8	albuquerque	albuquerque	PROPN
ejde-610	244	9	,	,	PUNCT
ejde-610	244	10	s.	s.	PROPN
ejde-610	244	11	j.	j.	PROPN
ejde-610	244	12	chen	chen	PROPN
ejde-610	244	13	,	,	PUNCT
ejde-610	244	14	l.	l.	PROPN
ejde-610	244	15	li	li	PROPN
ejde-610	244	16	;	;	PUNCT
ejde-610	244	17	solitary	solitary	ADJ
ejde-610	244	18	wave	wave	NOUN
ejde-610	244	19	of	of	ADP
ejde-610	244	20	ground	ground	NOUN
ejde-610	244	21	state	state	NOUN
ejde-610	244	22	type	type	NOUN
ejde-610	244	23	for	for	ADP
ejde-610	244	24	a	a	DET
ejde-610	244	25	nonlinear	nonlinear	ADJ
ejde-610	244	26	klein	klein	PROPN
ejde-610	244	27	-	-	PUNCT
ejde-610	244	28	gordon	gordon	PROPN
ejde-610	244	29	equation	equation	NOUN
ejde-610	244	30	coupled	couple	VERB
ejde-610	244	31	with	with	ADP
ejde-610	244	32	born	bear	VERB
ejde-610	244	33	-	-	PUNCT
ejde-610	244	34	infeld	infeld	NOUN
ejde-610	244	35	theory	theory	NOUN
ejde-610	244	36	in	in	ADP
ejde-610	244	37	r2	r2	PROPN
ejde-610	244	38	,	,	PUNCT
ejde-610	244	39	electronic	electronic	ADJ
ejde-610	244	40	journal	journal	NOUN
ejde-610	244	41	of	of	ADP
ejde-610	244	42	qualitative	qualitative	ADJ
ejde-610	244	43	theory	theory	NOUN
ejde-610	244	44	of	of	ADP
ejde-610	244	45	differential	differential	ADJ
ejde-610	244	46	equations	equation	NOUN
ejde-610	244	47	,	,	PUNCT
ejde-610	244	48	12	12	NUM
ejde-610	244	49	,	,	PUNCT
ejde-610	244	50	2020	2020	NUM
ejde-610	244	51	,	,	PUNCT
ejde-610	244	52	1	1	NUM
ejde-610	244	53	-	-	SYM
ejde-610	244	54	18	18	NUM
ejde-610	244	55	.	.	PUNCT
ejde-610	245	1	[	[	X
ejde-610	245	2	2	2	NUM
ejde-610	245	3	]	]	PUNCT
ejde-610	245	4	c.	c.	PROPN
ejde-610	245	5	o.	o.	PROPN
ejde-610	245	6	alves	alves	PROPN
ejde-610	245	7	,	,	PUNCT
ejde-610	245	8	m.	m.	NOUN
ejde-610	245	9	a.	a.	PROPN
ejde-610	245	10	s.	s.	PROPN
ejde-610	245	11	souto	souto	PROPN
ejde-610	245	12	,	,	PUNCT
ejde-610	245	13	s.	s.	PROPN
ejde-610	245	14	h.	h.	PROPN
ejde-610	245	15	m.	m.	PROPN
ejde-610	245	16	soares	soares	PROPN
ejde-610	245	17	;	;	PUNCT
ejde-610	245	18	schrödinger	schrödinger	NOUN
ejde-610	245	19	-	-	PUNCT
ejde-610	245	20	poisson	poisson	NOUN
ejde-610	245	21	equations	equation	NOUN
ejde-610	245	22	without	without	ADP
ejde-610	245	23	ambrosetti	ambrosetti	NOUN
ejde-610	245	24	-	-	PUNCT
ejde-610	245	25	rabinowitz	rabinowitz	NOUN
ejde-610	245	26	condition	condition	NOUN
ejde-610	245	27	,	,	PUNCT
ejde-610	245	28	journal	journal	NOUN
ejde-610	245	29	of	of	ADP
ejde-610	245	30	mathematical	mathematical	ADJ
ejde-610	245	31	analysis	analysis	NOUN
ejde-610	245	32	and	and	CCONJ
ejde-610	245	33	applications	application	NOUN
ejde-610	245	34	,	,	PUNCT
ejde-610	245	35	377(2	377(2	NUM
ejde-610	245	36	)	)	PUNCT
ejde-610	245	37	,	,	PUNCT
ejde-610	245	38	2011	2011	NUM
ejde-610	245	39	,	,	PUNCT
ejde-610	245	40	584	584	NUM
ejde-610	245	41	-	-	SYM
ejde-610	245	42	592	592	NUM
ejde-610	245	43	.	.	PUNCT
ejde-610	246	1	[	[	X
ejde-610	246	2	3	3	X
ejde-610	246	3	]	]	X
ejde-610	246	4	t.	t.	PROPN
ejde-610	246	5	bartsch	bartsch	PROPN
ejde-610	246	6	,	,	PUNCT
ejde-610	246	7	z	z	PROPN
ejde-610	246	8	-	-	PUNCT
ejde-610	246	9	q.	q.	PROPN
ejde-610	246	10	wang	wang	PROPN
ejde-610	246	11	;	;	PUNCT
ejde-610	246	12	existence	existence	NOUN
ejde-610	246	13	and	and	CCONJ
ejde-610	246	14	multiplicity	multiplicity	NOUN
ejde-610	246	15	results	result	NOUN
ejde-610	246	16	for	for	ADP
ejde-610	246	17	some	some	DET
ejde-610	246	18	superlinear	superlinear	ADJ
ejde-610	246	19	elliptic	elliptic	ADJ
ejde-610	246	20	problem	problem	NOUN
ejde-610	246	21	on	on	ADP
ejde-610	246	22	rn	rn	PROPN
ejde-610	246	23	,	,	PUNCT
ejde-610	246	24	communications	communication	NOUN
ejde-610	246	25	in	in	ADP
ejde-610	246	26	partial	partial	ADJ
ejde-610	246	27	differential	differential	NOUN
ejde-610	246	28	equations	equation	NOUN
ejde-610	246	29	,	,	PUNCT
ejde-610	246	30	20	20	NUM
ejde-610	246	31	,	,	PUNCT
ejde-610	246	32	1995	1995	NUM
ejde-610	246	33	,	,	PUNCT
ejde-610	246	34	1725	1725	NUM
ejde-610	246	35	-	-	SYM
ejde-610	246	36	1741	1741	NUM
ejde-610	246	37	.	.	PUNCT
ejde-610	247	1	[	[	X
ejde-610	247	2	4	4	X
ejde-610	247	3	]	]	PUNCT
ejde-610	247	4	t.	t.	PROPN
ejde-610	247	5	bartsch	bartsch	X
ejde-610	247	6	;	;	PUNCT
ejde-610	247	7	infinitely	infinitely	ADV
ejde-610	247	8	many	many	ADJ
ejde-610	247	9	solutions	solution	NOUN
ejde-610	247	10	of	of	ADP
ejde-610	247	11	a	a	DET
ejde-610	247	12	symmetric	symmetric	ADJ
ejde-610	247	13	dirichlet	dirichlet	PROPN
ejde-610	247	14	problem	problem	NOUN
ejde-610	247	15	,	,	PUNCT
ejde-610	247	16	nonlinear	nonlinear	ADJ
ejde-610	247	17	analysis	analysis	NOUN
ejde-610	247	18	:	:	PUNCT
ejde-610	247	19	theory	theory	NOUN
ejde-610	247	20	,	,	PUNCT
ejde-610	247	21	methods	method	NOUN
ejde-610	247	22	and	and	CCONJ
ejde-610	247	23	applications	application	NOUN
ejde-610	247	24	,	,	PUNCT
ejde-610	247	25	20	20	NUM
ejde-610	247	26	,	,	PUNCT
ejde-610	247	27	1993	1993	NUM
ejde-610	247	28	,	,	PUNCT
ejde-610	247	29	1205	1205	NUM
ejde-610	247	30	-	-	SYM
ejde-610	247	31	1216	1216	NUM
ejde-610	247	32	.	.	PUNCT
ejde-610	248	1	[	[	X
ejde-610	248	2	5	5	NUM
ejde-610	248	3	]	]	PUNCT
ejde-610	248	4	v.	v.	PROPN
ejde-610	248	5	benci	benci	PROPN
ejde-610	248	6	,	,	PUNCT
ejde-610	248	7	d.	d.	PROPN
ejde-610	248	8	fortunato	fortunato	PROPN
ejde-610	248	9	,	,	PUNCT
ejde-610	248	10	a.	a.	PROPN
ejde-610	248	11	masiello	masiello	PROPN
ejde-610	248	12	,	,	PUNCT
ejde-610	248	13	l.	l.	PROPN
ejde-610	248	14	pisani	pisani	PROPN
ejde-610	248	15	;	;	PUNCT
ejde-610	248	16	solitons	soliton	NOUN
ejde-610	248	17	and	and	CCONJ
ejde-610	248	18	the	the	DET
ejde-610	248	19	electromagnetic	electromagnetic	ADJ
ejde-610	248	20	field	field	NOUN
ejde-610	248	21	,	,	PUNCT
ejde-610	248	22	mathematische	mathematische	NOUN
ejde-610	248	23	zeitschrift	zeitschrift	NOUN
ejde-610	248	24	,	,	PUNCT
ejde-610	248	25	3	3	NUM
ejde-610	248	26	,	,	PUNCT
ejde-610	248	27	2012	2012	NUM
ejde-610	248	28	,	,	PUNCT
ejde-610	248	29	299	299	NUM
ejde-610	248	30	-	-	SYM
ejde-610	248	31	301	301	NUM
ejde-610	248	32	.	.	PUNCT
ejde-610	249	1	[	[	X
ejde-610	249	2	6	6	NUM
ejde-610	249	3	]	]	PUNCT
ejde-610	249	4	v.	v.	PROPN
ejde-610	249	5	benci	benci	PROPN
ejde-610	249	6	,	,	PUNCT
ejde-610	249	7	d.	d.	PROPN
ejde-610	249	8	fortunato	fortunato	PROPN
ejde-610	249	9	;	;	PUNCT
ejde-610	249	10	solitary	solitary	ADJ
ejde-610	249	11	waves	wave	NOUN
ejde-610	249	12	of	of	ADP
ejde-610	249	13	the	the	DET
ejde-610	249	14	nonlinear	nonlinear	PROPN
ejde-610	249	15	klein	klein	PROPN
ejde-610	249	16	-	-	PUNCT
ejde-610	249	17	gordon	gordon	PROPN
ejde-610	249	18	equation	equation	NOUN
ejde-610	249	19	coupled	couple	VERB
ejde-610	249	20	with	with	ADP
ejde-610	249	21	the	the	DET
ejde-610	249	22	maxwell	maxwell	PROPN
ejde-610	249	23	equations	equation	NOUN
ejde-610	249	24	,	,	PUNCT
ejde-610	249	25	reviews	review	NOUN
ejde-610	249	26	in	in	ADP
ejde-610	249	27	mathematical	mathematical	ADJ
ejde-610	249	28	physics	physics	NOUN
ejde-610	249	29	,	,	PUNCT
ejde-610	249	30	14(4	14(4	NUM
ejde-610	249	31	)	)	PUNCT
ejde-610	249	32	,	,	PUNCT
ejde-610	249	33	2002	2002	NUM
ejde-610	249	34	,	,	PUNCT
ejde-610	249	35	409	409	NUM
ejde-610	249	36	-	-	SYM
ejde-610	249	37	420	420	NUM
ejde-610	249	38	.	.	PUNCT
ejde-610	250	1	[	[	X
ejde-610	250	2	7	7	X
ejde-610	250	3	]	]	X
ejde-610	250	4	m.	m.	NOUN
ejde-610	250	5	born	bear	VERB
ejde-610	250	6	;	;	PUNCT
ejde-610	250	7	modified	modify	VERB
ejde-610	250	8	field	field	NOUN
ejde-610	250	9	equations	equation	NOUN
ejde-610	250	10	with	with	ADP
ejde-610	250	11	a	a	DET
ejde-610	250	12	finite	finite	ADJ
ejde-610	250	13	radius	radius	NOUN
ejde-610	250	14	of	of	ADP
ejde-610	250	15	the	the	DET
ejde-610	250	16	electron	electron	NOUN
ejde-610	250	17	,	,	PUNCT
ejde-610	250	18	nature	nature	NOUN
ejde-610	250	19	,	,	PUNCT
ejde-610	250	20	132	132	NUM
ejde-610	250	21	,	,	PUNCT
ejde-610	250	22	1933	1933	NUM
ejde-610	250	23	,	,	PUNCT
ejde-610	250	24	282	282	NUM
ejde-610	250	25	.	.	PUNCT
ejde-610	251	1	[	[	X
ejde-610	251	2	8	8	NUM
ejde-610	251	3	]	]	X
ejde-610	251	4	m.	m.	NOUN
ejde-610	251	5	born	bear	VERB
ejde-610	251	6	;	;	PUNCT
ejde-610	251	7	quantum	quantum	ADJ
ejde-610	251	8	theory	theory	NOUN
ejde-610	251	9	of	of	ADP
ejde-610	251	10	the	the	DET
ejde-610	251	11	electromagnetic	electromagnetic	ADJ
ejde-610	251	12	field	field	NOUN
ejde-610	251	13	,	,	PUNCT
ejde-610	251	14	proceedings	proceeding	NOUN
ejde-610	251	15	of	of	ADP
ejde-610	251	16	the	the	DET
ejde-610	251	17	royal	royal	ADJ
ejde-610	251	18	society	society	NOUN
ejde-610	251	19	of	of	ADP
ejde-610	251	20	london	london	PROPN
ejde-610	251	21	series	series	PROPN
ejde-610	251	22	a	a	PRON
ejde-610	251	23	,	,	PUNCT
ejde-610	251	24	143(849	143(849	NUM
ejde-610	251	25	)	)	PUNCT
ejde-610	251	26	,	,	PUNCT
ejde-610	251	27	1934	1934	NUM
ejde-610	251	28	,	,	PUNCT
ejde-610	251	29	410	410	NUM
ejde-610	251	30	-	-	SYM
ejde-610	251	31	437	437	NUM
ejde-610	251	32	.	.	PUNCT
ejde-610	252	1	[	[	X
ejde-610	252	2	9	9	NUM
ejde-610	252	3	]	]	PUNCT
ejde-610	252	4	m.	m.	NOUN
ejde-610	252	5	born	bear	VERB
ejde-610	252	6	,	,	PUNCT
ejde-610	252	7	l.	l.	PROPN
ejde-610	252	8	infeld	infeld	PROPN
ejde-610	252	9	;	;	PUNCT
ejde-610	252	10	foundations	foundation	NOUN
ejde-610	252	11	of	of	ADP
ejde-610	252	12	the	the	DET
ejde-610	252	13	new	new	ADJ
ejde-610	252	14	field	field	NOUN
ejde-610	252	15	theory	theory	NOUN
ejde-610	252	16	,	,	PUNCT
ejde-610	252	17	nature	nature	NOUN
ejde-610	252	18	,	,	PUNCT
ejde-610	252	19	144(852	144(852	NUM
ejde-610	252	20	)	)	PUNCT
ejde-610	252	21	,	,	PUNCT
ejde-610	252	22	1934	1934	NUM
ejde-610	252	23	,	,	PUNCT
ejde-610	252	24	425	425	NUM
ejde-610	252	25	-	-	SYM
ejde-610	252	26	451	451	NUM
ejde-610	252	27	.	.	PUNCT
ejde-610	253	1	[	[	X
ejde-610	253	2	10	10	NUM
ejde-610	253	3	]	]	PUNCT
ejde-610	253	4	m.	m.	NOUN
ejde-610	253	5	carmeli	carmeli	NOUN
ejde-610	253	6	;	;	PUNCT
ejde-610	253	7	field	field	NOUN
ejde-610	253	8	theory	theory	NOUN
ejde-610	253	9	on	on	ADP
ejde-610	253	10	r×s	r×s	PROPN
ejde-610	253	11	3	3	NUM
ejde-610	253	12	topology	topology	NOUN
ejde-610	254	1	i	i	PRON
ejde-610	254	2	:	:	PUNCT
ejde-610	254	3	the	the	DET
ejde-610	254	4	klein	klein	PROPN
ejde-610	254	5	-	-	PUNCT
ejde-610	254	6	gordon	gordon	PROPN
ejde-610	254	7	and	and	CCONJ
ejde-610	254	8	schrödinger	schrödinger	NOUN
ejde-610	254	9	equations	equation	NOUN
ejde-610	254	10	,	,	PUNCT
ejde-610	254	11	foundations	foundation	NOUN
ejde-610	254	12	of	of	ADP
ejde-610	254	13	physics	physics	NOUN
ejde-610	254	14	,	,	PUNCT
ejde-610	254	15	15	15	NUM
ejde-610	254	16	,	,	PUNCT
ejde-610	254	17	1985	1985	NUM
ejde-610	254	18	,	,	PUNCT
ejde-610	254	19	175	175	NUM
ejde-610	254	20	-	-	SYM
ejde-610	254	21	184	184	NUM
ejde-610	254	22	.	.	PUNCT
ejde-610	255	1	[	[	X
ejde-610	255	2	11	11	NUM
ejde-610	255	3	]	]	X
ejde-610	255	4	g.	g.	PROPN
ejde-610	255	5	f.	f.	PROPN
ejde-610	255	6	che	che	PROPN
ejde-610	255	7	,	,	PUNCT
ejde-610	255	8	h.	h.	PROPN
ejde-610	255	9	b.	b.	PROPN
ejde-610	255	10	chen	chen	PROPN
ejde-610	255	11	;	;	PUNCT
ejde-610	255	12	infinitely	infinitely	ADV
ejde-610	255	13	many	many	ADJ
ejde-610	255	14	solutions	solution	NOUN
ejde-610	255	15	for	for	ADP
ejde-610	255	16	the	the	DET
ejde-610	255	17	klein	klein	PROPN
ejde-610	255	18	-	-	PUNCT
ejde-610	255	19	gordon	gordon	PROPN
ejde-610	255	20	equation	equation	NOUN
ejde-610	255	21	with	with	ADP
ejde-610	255	22	sublinear	sublinear	NOUN
ejde-610	255	23	nonlinearity	nonlinearity	NOUN
ejde-610	255	24	coupled	couple	VERB
ejde-610	255	25	with	with	ADP
ejde-610	255	26	born	bear	VERB
ejde-610	255	27	-	-	PUNCT
ejde-610	255	28	infeld	infeld	NOUN
ejde-610	255	29	theory	theory	NOUN
ejde-610	255	30	,	,	PUNCT
ejde-610	255	31	bulletin	bulletin	NOUN
ejde-610	255	32	of	of	ADP
ejde-610	255	33	the	the	DET
ejde-610	255	34	iranian	iranian	PROPN
ejde-610	255	35	mathematical	mathematical	PROPN
ejde-610	255	36	society	society	NOUN
ejde-610	255	37	,	,	PUNCT
ejde-610	255	38	46	46	NUM
ejde-610	255	39	,	,	PUNCT
ejde-610	255	40	2019	2019	NUM
ejde-610	255	41	,	,	PUNCT
ejde-610	255	42	1083	1083	NUM
ejde-610	255	43	-	-	SYM
ejde-610	255	44	1100	1100	NUM
ejde-610	255	45	.	.	PUNCT
ejde-610	256	1	[	[	X
ejde-610	256	2	12	12	NUM
ejde-610	256	3	]	]	PUNCT
ejde-610	256	4	s.	s.	PROPN
ejde-610	256	5	j.	j.	PROPN
ejde-610	256	6	chen	chen	PROPN
ejde-610	256	7	,	,	PUNCT
ejde-610	256	8	l.	l.	PROPN
ejde-610	256	9	li	li	PROPN
ejde-610	256	10	;	;	PUNCT
ejde-610	256	11	multiple	multiple	ADJ
ejde-610	256	12	solutions	solution	NOUN
ejde-610	256	13	for	for	ADP
ejde-610	256	14	the	the	DET
ejde-610	256	15	nonhomogeneous	nonhomogeneous	PROPN
ejde-610	256	16	klein	klein	PROPN
ejde-610	256	17	-	-	PUNCT
ejde-610	256	18	gordon	gordon	PROPN
ejde-610	256	19	equation	equation	NOUN
ejde-610	256	20	coupled	couple	VERB
ejde-610	256	21	with	with	ADP
ejde-610	256	22	born	bear	VERB
ejde-610	256	23	-	-	PUNCT
ejde-610	256	24	infeld	infeld	NOUN
ejde-610	256	25	theory	theory	NOUN
ejde-610	256	26	on	on	ADP
ejde-610	256	27	r3	r3	PROPN
ejde-610	256	28	,	,	PUNCT
ejde-610	256	29	journal	journal	NOUN
ejde-610	256	30	of	of	ADP
ejde-610	256	31	mathematical	mathematical	ADJ
ejde-610	256	32	analysis	analysis	NOUN
ejde-610	256	33	and	and	CCONJ
ejde-610	256	34	applications	application	NOUN
ejde-610	256	35	,	,	PUNCT
ejde-610	256	36	400(2	400(2	NUM
ejde-610	256	37	)	)	PUNCT
ejde-610	256	38	,	,	PUNCT
ejde-610	256	39	2013	2013	NUM
ejde-610	256	40	,	,	PUNCT
ejde-610	256	41	517	517	NUM
ejde-610	256	42	-	-	SYM
ejde-610	256	43	524	524	NUM
ejde-610	256	44	.	.	PUNCT
ejde-610	257	1	[	[	X
ejde-610	257	2	13	13	NUM
ejde-610	257	3	]	]	PUNCT
ejde-610	257	4	h.	h.	PROPN
ejde-610	257	5	y.	y.	PROPN
ejde-610	257	6	chen	chen	PROPN
ejde-610	257	7	,	,	PUNCT
ejde-610	257	8	s.	s.	PROPN
ejde-610	257	9	b.	b.	PROPN
ejde-610	257	10	liu	liu	PROPN
ejde-610	257	11	;	;	PUNCT
ejde-610	257	12	standing	stand	VERB
ejde-610	257	13	waves	wave	NOUN
ejde-610	257	14	with	with	ADP
ejde-610	257	15	large	large	ADJ
ejde-610	257	16	frequency	frequency	NOUN
ejde-610	257	17	for	for	ADP
ejde-610	257	18	4	4	NUM
ejde-610	257	19	-	-	PUNCT
ejde-610	257	20	superlinear	superlinear	ADJ
ejde-610	257	21	schrödingerpoisson	schrödingerpoisson	PROPN
ejde-610	257	22	systems	systems	PROPN
ejde-610	257	23	,	,	PUNCT
ejde-610	257	24	annali	annali	PROPN
ejde-610	257	25	di	di	PROPN
ejde-610	257	26	matematica	matematica	PROPN
ejde-610	257	27	,	,	PUNCT
ejde-610	257	28	194	194	NUM
ejde-610	257	29	,	,	PUNCT
ejde-610	257	30	2015	2015	NUM
ejde-610	257	31	,	,	PUNCT
ejde-610	257	32	43	43	NUM
ejde-610	257	33	-	-	SYM
ejde-610	257	34	53	53	NUM
ejde-610	257	35	.	.	PUNCT
ejde-610	257	36	ejde-2024/18	ejde-2024/18	NOUN
ejde-610	257	37	coupled	couple	VERB
ejde-610	257	38	klein	klein	PROPN
ejde-610	257	39	-	-	PUNCT
ejde-610	257	40	gordon	gordon	PROPN
ejde-610	257	41	and	and	CCONJ
ejde-610	257	42	born	bear	VERB
ejde-610	257	43	-	-	PUNCT
ejde-610	257	44	infeld	infeld	NOUN
ejde-610	257	45	equatons	equaton	NOUN
ejde-610	257	46	11	11	NUM
ejde-610	257	47	[	[	X
ejde-610	257	48	14	14	NUM
ejde-610	257	49	]	]	PUNCT
ejde-610	257	50	s.	s.	PROPN
ejde-610	257	51	j.	j.	PROPN
ejde-610	257	52	chen	chen	PROPN
ejde-610	257	53	,	,	PUNCT
ejde-610	257	54	s.	s.	PROPN
ejde-610	257	55	z.	z.	PROPN
ejde-610	257	56	song	song	PROPN
ejde-610	257	57	;	;	PUNCT
ejde-610	257	58	the	the	DET
ejde-610	257	59	existence	existence	NOUN
ejde-610	257	60	of	of	ADP
ejde-610	257	61	multiple	multiple	ADJ
ejde-610	257	62	solutions	solution	NOUN
ejde-610	257	63	for	for	ADP
ejde-610	257	64	the	the	DET
ejde-610	257	65	klein	klein	PROPN
ejde-610	257	66	-	-	PUNCT
ejde-610	257	67	gordon	gordon	PROPN
ejde-610	257	68	equation	equation	NOUN
ejde-610	257	69	with	with	ADP
ejde-610	257	70	concave	concave	ADJ
ejde-610	257	71	and	and	CCONJ
ejde-610	257	72	convex	convex	NOUN
ejde-610	257	73	nonlinearities	nonlinearitie	NOUN
ejde-610	257	74	coupled	couple	VERB
ejde-610	257	75	with	with	ADP
ejde-610	257	76	born	bear	VERB
ejde-610	257	77	-	-	PUNCT
ejde-610	257	78	infeld	infeld	NOUN
ejde-610	257	79	theory	theory	NOUN
ejde-610	257	80	on	on	ADP
ejde-610	257	81	r3	r3	PROPN
ejde-610	257	82	,	,	PUNCT
ejde-610	257	83	nonlinear	nonlinear	ADJ
ejde-610	257	84	analysis	analysis	NOUN
ejde-610	257	85	real	real	ADJ
ejde-610	257	86	world	world	NOUN
ejde-610	257	87	applications	application	NOUN
ejde-610	257	88	,	,	PUNCT
ejde-610	257	89	38	38	NUM
ejde-610	257	90	,	,	PUNCT
ejde-610	257	91	2017	2017	NUM
ejde-610	257	92	,	,	PUNCT
ejde-610	257	93	78	78	NUM
ejde-610	257	94	-	-	SYM
ejde-610	257	95	95	95	NUM
ejde-610	257	96	.	.	PUNCT
ejde-610	258	1	[	[	X
ejde-610	258	2	15	15	NUM
ejde-610	258	3	]	]	X
ejde-610	258	4	p.	p.	NOUN
ejde-610	258	5	d’avenia	d’avenia	PROPN
ejde-610	258	6	,	,	PUNCT
ejde-610	258	7	l.	l.	PROPN
ejde-610	258	8	pisani	pisani	PROPN
ejde-610	258	9	;	;	PUNCT
ejde-610	258	10	nonlinear	nonlinear	PROPN
ejde-610	258	11	klein	klein	PROPN
ejde-610	258	12	-	-	PUNCT
ejde-610	258	13	gordon	gordon	PROPN
ejde-610	258	14	equations	equation	NOUN
ejde-610	258	15	coupled	couple	VERB
ejde-610	258	16	with	with	ADP
ejde-610	258	17	born	bear	VERB
ejde-610	258	18	-	-	PUNCT
ejde-610	258	19	infeld	infeld	NOUN
ejde-610	258	20	type	type	NOUN
ejde-610	258	21	equations	equation	NOUN
ejde-610	258	22	,	,	PUNCT
ejde-610	258	23	electronic	electronic	ADJ
ejde-610	258	24	journal	journal	NOUN
ejde-610	258	25	of	of	ADP
ejde-610	258	26	differential	differential	ADJ
ejde-610	258	27	equations	equation	NOUN
ejde-610	258	28	,	,	PUNCT
ejde-610	258	29	vol	vol	NOUN
ejde-610	258	30	.	.	PROPN
ejde-610	258	31	2002(2002	2002(2002	NUM
ejde-610	258	32	)	)	PUNCT
ejde-610	258	33	,	,	PUNCT
ejde-610	258	34	no	no	INTJ
ejde-610	258	35	.	.	NOUN
ejde-610	258	36	26	26	NUM
ejde-610	258	37	,	,	PUNCT
ejde-610	258	38	pp	pp	ADJ
ejde-610	258	39	.	.	PUNCT
ejde-610	259	1	1	1	NUM
ejde-610	259	2	-	-	SYM
ejde-610	259	3	13	13	NUM
ejde-610	259	4	.	.	PUNCT
ejde-610	260	1	[	[	X
ejde-610	260	2	16	16	NUM
ejde-610	260	3	]	]	PUNCT
ejde-610	260	4	b.	b.	NOUN
ejde-610	260	5	felsager	felsager	NOUN
ejde-610	260	6	,	,	PUNCT
ejde-610	260	7	geometry	geometry	NOUN
ejde-610	260	8	;	;	PUNCT
ejde-610	260	9	particles	particle	NOUN
ejde-610	260	10	and	and	CCONJ
ejde-610	260	11	fields	field	NOUN
ejde-610	260	12	,	,	PUNCT
ejde-610	260	13	odense	odense	PROPN
ejde-610	260	14	university	university	PROPN
ejde-610	260	15	press	press	PROPN
ejde-610	260	16	,	,	PUNCT
ejde-610	260	17	odense	odense	PROPN
ejde-610	260	18	,	,	PUNCT
ejde-610	260	19	american	american	ADJ
ejde-610	260	20	journal	journal	PROPN
ejde-610	260	21	of	of	ADP
ejde-610	260	22	physics	physics	PROPN
ejde-610	260	23	,	,	PUNCT
ejde-610	260	24	52	52	NUM
ejde-610	260	25	,	,	PUNCT
ejde-610	260	26	573	573	NUM
ejde-610	260	27	,	,	PUNCT
ejde-610	260	28	1984	1984	NUM
ejde-610	260	29	.	.	PUNCT
ejde-610	261	1	[	[	X
ejde-610	261	2	17	17	NUM
ejde-610	261	3	]	]	X
ejde-610	261	4	d.	d.	PROPN
ejde-610	261	5	fortunato	fortunato	PROPN
ejde-610	261	6	,	,	PUNCT
ejde-610	261	7	l.	l.	PROPN
ejde-610	261	8	orsani	orsani	PROPN
ejde-610	261	9	,	,	PUNCT
ejde-610	261	10	l.	l.	PROPN
ejde-610	261	11	pisina	pisina	PROPN
ejde-610	261	12	;	;	PUNCT
ejde-610	261	13	born	bear	VERB
ejde-610	261	14	-	-	PUNCT
ejde-610	261	15	infeld	infeld	NOUN
ejde-610	261	16	type	type	NOUN
ejde-610	261	17	equations	equation	NOUN
ejde-610	261	18	for	for	ADP
ejde-610	261	19	electrostatic	electrostatic	ADJ
ejde-610	261	20	fields	field	NOUN
ejde-610	261	21	,	,	PUNCT
ejde-610	261	22	journal	journal	NOUN
ejde-610	261	23	of	of	ADP
ejde-610	261	24	mathematical	mathematical	ADJ
ejde-610	261	25	physics	physics	NOUN
ejde-610	261	26	,	,	PUNCT
ejde-610	261	27	43(11	43(11	NUM
ejde-610	261	28	)	)	PUNCT
ejde-610	261	29	,	,	PUNCT
ejde-610	261	30	2002	2002	NUM
ejde-610	261	31	,	,	PUNCT
ejde-610	261	32	5698	5698	NUM
ejde-610	261	33	-	-	SYM
ejde-610	261	34	5706	5706	NUM
ejde-610	261	35	.	.	PUNCT
ejde-610	262	1	[	[	X
ejde-610	262	2	18	18	NUM
ejde-610	262	3	]	]	X
ejde-610	262	4	l.	l.	PROPN
ejde-610	262	5	jeanjean	jeanjean	PROPN
ejde-610	262	6	;	;	PUNCT
ejde-610	262	7	on	on	ADP
ejde-610	262	8	the	the	DET
ejde-610	262	9	existence	existence	NOUN
ejde-610	262	10	of	of	ADP
ejde-610	262	11	bounded	bounded	ADJ
ejde-610	262	12	palais	palais	PROPN
ejde-610	262	13	-	-	PUNCT
ejde-610	262	14	smale	smale	ADJ
ejde-610	262	15	sequences	sequence	NOUN
ejde-610	262	16	and	and	CCONJ
ejde-610	262	17	application	application	NOUN
ejde-610	262	18	to	to	ADP
ejde-610	262	19	a	a	DET
ejde-610	262	20	landesman	landesman	ADJ
ejde-610	262	21	-	-	PUNCT
ejde-610	262	22	lazer	lazer	NOUN
ejde-610	262	23	-	-	PUNCT
ejde-610	262	24	type	type	NOUN
ejde-610	262	25	problem	problem	NOUN
ejde-610	262	26	set	set	VERB
ejde-610	262	27	on	on	ADP
ejde-610	262	28	rn	rn	PROPN
ejde-610	262	29	,	,	PUNCT
ejde-610	262	30	proceedings	proceeding	NOUN
ejde-610	262	31	of	of	ADP
ejde-610	262	32	the	the	DET
ejde-610	262	33	royal	royal	ADJ
ejde-610	262	34	society	society	NOUN
ejde-610	262	35	of	of	ADP
ejde-610	262	36	edinburgh	edinburgh	PROPN
ejde-610	262	37	,	,	PUNCT
ejde-610	262	38	129	129	NUM
ejde-610	262	39	,	,	PUNCT
ejde-610	262	40	1999	1999	NUM
ejde-610	262	41	,	,	PUNCT
ejde-610	262	42	787	787	NUM
ejde-610	262	43	-	-	SYM
ejde-610	262	44	809	809	NUM
ejde-610	262	45	.	.	PUNCT
ejde-610	263	1	[	[	X
ejde-610	263	2	19	19	NUM
ejde-610	263	3	]	]	X
ejde-610	263	4	c.	c.	PROPN
ejde-610	263	5	m.	m.	NOUN
ejde-610	263	6	he	he	PRON
ejde-610	263	7	,	,	PUNCT
ejde-610	263	8	l.	l.	PROPN
ejde-610	263	9	li	li	PROPN
ejde-610	263	10	,	,	PUNCT
ejde-610	263	11	s.	s.	PROPN
ejde-610	263	12	j.	j.	PROPN
ejde-610	263	13	chen	chen	PROPN
ejde-610	263	14	,	,	PUNCT
ejde-610	263	15	d.	d.	PROPN
ejde-610	263	16	o’regan	o’regan	PROPN
ejde-610	263	17	;	;	PUNCT
ejde-610	263	18	ground	ground	NOUN
ejde-610	263	19	state	state	NOUN
ejde-610	263	20	solution	solution	NOUN
ejde-610	263	21	for	for	ADP
ejde-610	263	22	the	the	DET
ejde-610	263	23	nonlinear	nonlinear	ADJ
ejde-610	263	24	kleingordon	kleingordon	NOUN
ejde-610	263	25	equation	equation	NOUN
ejde-610	263	26	coupled	couple	VERB
ejde-610	263	27	with	with	ADP
ejde-610	263	28	born	bear	VERB
ejde-610	263	29	-	-	PUNCT
ejde-610	263	30	infeld	infeld	NOUN
ejde-610	263	31	theory	theory	NOUN
ejde-610	263	32	with	with	ADP
ejde-610	263	33	critical	critical	ADJ
ejde-610	263	34	exponents	exponent	NOUN
ejde-610	263	35	,	,	PUNCT
ejde-610	263	36	analysis	analysis	NOUN
ejde-610	263	37	and	and	CCONJ
ejde-610	263	38	mathematical	mathematical	ADJ
ejde-610	263	39	physics	physics	NOUN
ejde-610	263	40	,	,	PUNCT
ejde-610	263	41	12	12	NUM
ejde-610	263	42	,	,	PUNCT
ejde-610	263	43	2022	2022	NUM
ejde-610	263	44	,	,	PUNCT
ejde-610	263	45	48	48	NUM
ejde-610	263	46	.	.	PUNCT
ejde-610	264	1	[	[	X
ejde-610	264	2	20	20	NUM
ejde-610	264	3	]	]	PUNCT
ejde-610	264	4	x.	x.	NOUN
ejde-610	264	5	m.	m.	PROPN
ejde-610	264	6	he	he	PRON
ejde-610	264	7	;	;	PUNCT
ejde-610	264	8	multiplicity	multiplicity	NOUN
ejde-610	264	9	of	of	ADP
ejde-610	264	10	solutions	solution	NOUN
ejde-610	264	11	for	for	ADP
ejde-610	264	12	a	a	DET
ejde-610	264	13	nonlinear	nonlinear	ADJ
ejde-610	264	14	klein	klein	PROPN
ejde-610	264	15	-	-	PUNCT
ejde-610	264	16	gordon	gordon	PROPN
ejde-610	264	17	-	-	PUNCT
ejde-610	264	18	maxwell	maxwell	PROPN
ejde-610	264	19	system	system	NOUN
ejde-610	264	20	,	,	PUNCT
ejde-610	264	21	acta	acta	PROPN
ejde-610	264	22	applicandae	applicandae	PROPN
ejde-610	264	23	mathematicae	mathematicae	PROPN
ejde-610	264	24	,	,	PUNCT
ejde-610	264	25	130	130	NUM
ejde-610	264	26	,	,	PUNCT
ejde-610	264	27	2014	2014	NUM
ejde-610	264	28	,	,	PUNCT
ejde-610	264	29	237	237	NUM
ejde-610	264	30	-	-	SYM
ejde-610	264	31	250	250	NUM
ejde-610	264	32	.	.	PUNCT
ejde-610	265	1	[	[	X
ejde-610	265	2	21	21	NUM
ejde-610	265	3	]	]	X
ejde-610	265	4	d.	d.	PROPN
ejde-610	265	5	mugnai	mugnai	PROPN
ejde-610	265	6	;	;	PUNCT
ejde-610	265	7	coupled	couple	VERB
ejde-610	265	8	klein	klein	PROPN
ejde-610	265	9	-	-	PUNCT
ejde-610	265	10	gorndon	gorndon	PROPN
ejde-610	265	11	and	and	CCONJ
ejde-610	265	12	born	bear	VERB
ejde-610	265	13	-	-	PUNCT
ejde-610	265	14	infeld	infeld	NOUN
ejde-610	265	15	type	type	NOUN
ejde-610	265	16	equations	equation	NOUN
ejde-610	265	17	:	:	PUNCT
ejde-610	265	18	looking	look	VERB
ejde-610	265	19	for	for	ADP
ejde-610	265	20	solitary	solitary	ADJ
ejde-610	265	21	waves	wave	NOUN
ejde-610	265	22	,	,	PUNCT
ejde-610	265	23	proceedings	proceeding	NOUN
ejde-610	265	24	of	of	ADP
ejde-610	265	25	the	the	DET
ejde-610	265	26	royal	royal	ADJ
ejde-610	265	27	society	society	NOUN
ejde-610	265	28	of	of	ADP
ejde-610	265	29	london	london	PROPN
ejde-610	265	30	a	a	DET
ejde-610	265	31	mathematical	mathematical	ADJ
ejde-610	265	32	physical	physical	ADJ
ejde-610	265	33	and	and	CCONJ
ejde-610	265	34	engineering	engineering	NOUN
ejde-610	265	35	sciences	science	NOUN
ejde-610	265	36	,	,	PUNCT
ejde-610	265	37	460(2045	460(2045	NUM
ejde-610	265	38	)	)	PUNCT
ejde-610	265	39	,	,	PUNCT
ejde-610	265	40	2004	2004	NUM
ejde-610	265	41	,	,	PUNCT
ejde-610	265	42	1519	1519	NUM
ejde-610	265	43	-	-	SYM
ejde-610	265	44	1527	1527	NUM
ejde-610	265	45	.	.	PUNCT
ejde-610	266	1	[	[	X
ejde-610	266	2	22	22	NUM
ejde-610	266	3	]	]	PUNCT
ejde-610	266	4	p.	p.	NOUN
ejde-610	266	5	h.	h.	PROPN
ejde-610	266	6	rabinowitz	rabinowitz	PROPN
ejde-610	266	7	;	;	PUNCT
ejde-610	266	8	minimax	minimax	NOUN
ejde-610	266	9	methods	method	NOUN
ejde-610	266	10	in	in	ADP
ejde-610	266	11	critical	critical	ADJ
ejde-610	266	12	point	point	NOUN
ejde-610	266	13	theory	theory	NOUN
ejde-610	266	14	with	with	ADP
ejde-610	266	15	applications	application	NOUN
ejde-610	266	16	to	to	PART
ejde-610	266	17	differential	differential	VERB
ejde-610	266	18	equations	equation	NOUN
ejde-610	266	19	,	,	PUNCT
ejde-610	266	20	cbms	cbms	PROPN
ejde-610	266	21	reg	reg	PROPN
ejde-610	266	22	.	.	PUNCT
ejde-610	266	23	conf	conf	PROPN
ejde-610	266	24	.	.	PUNCT
ejde-610	267	1	ser	ser	PROPN
ejde-610	267	2	.	.	PROPN
ejde-610	267	3	math	math	PROPN
ejde-610	267	4	.	.	PUNCT
ejde-610	268	1	65	65	NUM
ejde-610	268	2	,	,	PUNCT
ejde-610	268	3	american	american	PROPN
ejde-610	268	4	mathematical	mathematical	ADJ
ejde-610	268	5	society	society	NOUN
ejde-610	268	6	,	,	PUNCT
ejde-610	268	7	providence	providence	NOUN
ejde-610	268	8	,	,	PUNCT
ejde-610	268	9	1986	1986	NUM
ejde-610	268	10	.	.	PUNCT
ejde-610	269	1	[	[	X
ejde-610	269	2	23	23	NUM
ejde-610	269	3	]	]	PUNCT
ejde-610	269	4	k.	k.	PROPN
ejde-610	270	1	m.	m.	PROPN
ejde-610	270	2	teng	teng	PROPN
ejde-610	270	3	,	,	PUNCT
ejde-610	270	4	k.	k.	PROPN
ejde-610	270	5	zhang	zhang	PROPN
ejde-610	270	6	;	;	PUNCT
ejde-610	270	7	existence	existence	NOUN
ejde-610	270	8	of	of	ADP
ejde-610	270	9	solitary	solitary	ADJ
ejde-610	270	10	wave	wave	NOUN
ejde-610	270	11	solutions	solution	NOUN
ejde-610	270	12	for	for	ADP
ejde-610	270	13	the	the	DET
ejde-610	270	14	nonlinear	nonlinear	PROPN
ejde-610	270	15	klein	klein	PROPN
ejde-610	270	16	-	-	PUNCT
ejde-610	270	17	gordon	gordon	PROPN
ejde-610	270	18	equation	equation	NOUN
ejde-610	270	19	coupled	couple	VERB
ejde-610	270	20	with	with	ADP
ejde-610	270	21	born	bear	VERB
ejde-610	270	22	-	-	PUNCT
ejde-610	270	23	infeld	infeld	NOUN
ejde-610	270	24	theory	theory	NOUN
ejde-610	270	25	with	with	ADP
ejde-610	270	26	critical	critical	ADJ
ejde-610	270	27	sobolev	sobolev	NOUN
ejde-610	270	28	exponent	exponent	NOUN
ejde-610	270	29	,	,	PUNCT
ejde-610	270	30	nonlinear	nonlinear	ADJ
ejde-610	270	31	analysis	analysis	NOUN
ejde-610	270	32	:	:	PUNCT
ejde-610	270	33	an	an	DET
ejde-610	270	34	international	international	ADJ
ejde-610	270	35	multidisciplinary	multidisciplinary	ADJ
ejde-610	270	36	journal	journal	NOUN
ejde-610	270	37	,	,	PUNCT
ejde-610	270	38	74(12	74(12	NUM
ejde-610	270	39	)	)	PUNCT
ejde-610	270	40	,	,	PUNCT
ejde-610	270	41	2011	2011	NUM
ejde-610	270	42	,	,	PUNCT
ejde-610	270	43	4241	4241	NUM
ejde-610	270	44	-	-	SYM
ejde-610	270	45	4251	4251	NUM
ejde-610	270	46	.	.	PUNCT
ejde-610	271	1	[	[	X
ejde-610	271	2	24	24	NUM
ejde-610	271	3	]	]	PUNCT
ejde-610	271	4	k.	k.	PROPN
ejde-610	272	1	m.	m.	PROPN
ejde-610	272	2	teng	teng	PROPN
ejde-610	272	3	;	;	PUNCT
ejde-610	272	4	existence	existence	NOUN
ejde-610	272	5	and	and	CCONJ
ejde-610	272	6	multiple	multiple	NOUN
ejde-610	272	7	of	of	ADP
ejde-610	272	8	the	the	DET
ejde-610	272	9	solutions	solution	NOUN
ejde-610	272	10	for	for	ADP
ejde-610	272	11	the	the	DET
ejde-610	272	12	nonlinear	nonlinear	PROPN
ejde-610	272	13	klein	klein	PROPN
ejde-610	272	14	-	-	PUNCT
ejde-610	272	15	gordon	gordon	PROPN
ejde-610	272	16	equation	equation	NOUN
ejde-610	272	17	coupled	couple	VERB
ejde-610	272	18	with	with	ADP
ejde-610	272	19	born	bear	VERB
ejde-610	272	20	-	-	PUNCT
ejde-610	272	21	infeld	infeld	NOUN
ejde-610	272	22	theory	theory	NOUN
ejde-610	272	23	on	on	ADP
ejde-610	272	24	boundary	boundary	ADJ
ejde-610	272	25	domain	domain	NOUN
ejde-610	272	26	,	,	PUNCT
ejde-610	272	27	differential	differential	ADJ
ejde-610	272	28	equations	equation	NOUN
ejde-610	272	29	and	and	CCONJ
ejde-610	272	30	applications	application	NOUN
ejde-610	272	31	,	,	PUNCT
ejde-610	272	32	4(3	4(3	NUM
ejde-610	272	33	)	)	PUNCT
ejde-610	272	34	,	,	PUNCT
ejde-610	272	35	2012	2012	NUM
ejde-610	272	36	,	,	PUNCT
ejde-610	272	37	445	445	NUM
ejde-610	272	38	-	-	SYM
ejde-610	272	39	457	457	NUM
ejde-610	272	40	.	.	PUNCT
ejde-610	273	1	[	[	X
ejde-610	273	2	25	25	NUM
ejde-610	273	3	]	]	PUNCT
ejde-610	273	4	f.	f.	PROPN
ejde-610	273	5	z.	z.	PROPN
ejde-610	273	6	wang	wang	PROPN
ejde-610	273	7	;	;	PUNCT
ejde-610	273	8	solitary	solitary	ADJ
ejde-610	273	9	waves	wave	NOUN
ejde-610	273	10	for	for	ADP
ejde-610	273	11	the	the	DET
ejde-610	273	12	coupled	couple	VERB
ejde-610	273	13	nonlinear	nonlinear	PROPN
ejde-610	273	14	klein	klein	PROPN
ejde-610	273	15	-	-	PUNCT
ejde-610	273	16	gordon	gordon	PROPN
ejde-610	273	17	and	and	CCONJ
ejde-610	273	18	born	bear	VERB
ejde-610	273	19	-	-	PUNCT
ejde-610	273	20	infeld	infeld	NOUN
ejde-610	273	21	type	type	NOUN
ejde-610	273	22	equations	equation	NOUN
ejde-610	273	23	,	,	PUNCT
ejde-610	273	24	electronic	electronic	ADJ
ejde-610	273	25	journal	journal	NOUN
ejde-610	273	26	of	of	ADP
ejde-610	273	27	differential	differential	ADJ
ejde-610	273	28	equations	equation	NOUN
ejde-610	273	29	,	,	PUNCT
ejde-610	273	30	vol	vol	NOUN
ejde-610	273	31	.	.	PROPN
ejde-610	273	32	2012	2012	NUM
ejde-610	273	33	(	(	PUNCT
ejde-610	273	34	2012	2012	NUM
ejde-610	273	35	)	)	PUNCT
ejde-610	273	36	,	,	PUNCT
ejde-610	273	37	no	no	INTJ
ejde-610	273	38	.	.	NOUN
ejde-610	273	39	82	82	NUM
ejde-610	273	40	,	,	PUNCT
ejde-610	273	41	pp	pp	ADJ
ejde-610	273	42	.	.	PUNCT
ejde-610	274	1	1	1	NUM
ejde-610	274	2	-	-	SYM
ejde-610	274	3	12	12	NUM
ejde-610	274	4	.	.	PUNCT
ejde-610	275	1	[	[	X
ejde-610	275	2	26	26	NUM
ejde-610	275	3	]	]	PUNCT
ejde-610	275	4	l.	l.	PROPN
ejde-610	275	5	x.	x.	PROPN
ejde-610	275	6	wang	wang	PROPN
ejde-610	275	7	,	,	PUNCT
ejde-610	275	8	c.	c.	PROPN
ejde-610	275	9	l.	l.	PROPN
ejde-610	275	10	xiong	xiong	PROPN
ejde-610	275	11	,	,	PUNCT
ejde-610	275	12	p.	p.	PROPN
ejde-610	276	1	p.	p.	PROPN
ejde-610	277	1	zhao	zhao	PROPN
ejde-610	277	2	;	;	PUNCT
ejde-610	277	3	two	two	NUM
ejde-610	277	4	solutions	solution	NOUN
ejde-610	277	5	for	for	ADP
ejde-610	277	6	the	the	DET
ejde-610	277	7	nonhomogeneous	nonhomogeneous	PROPN
ejde-610	277	8	klein	klein	PROPN
ejde-610	277	9	-	-	PUNCT
ejde-610	277	10	gordon	gordon	PROPN
ejde-610	277	11	equation	equation	NOUN
ejde-610	277	12	coupled	couple	VERB
ejde-610	277	13	with	with	ADP
ejde-610	277	14	born	bear	VERB
ejde-610	277	15	-	-	PUNCT
ejde-610	277	16	infeld	infeld	NOUN
ejde-610	277	17	theory	theory	NOUN
ejde-610	277	18	on	on	ADP
ejde-610	277	19	r3	r3	PROPN
ejde-610	277	20	,	,	PUNCT
ejde-610	277	21	electronic	electronic	ADJ
ejde-610	277	22	journal	journal	NOUN
ejde-610	277	23	of	of	ADP
ejde-610	277	24	differential	differential	ADJ
ejde-610	277	25	equations	equation	NOUN
ejde-610	277	26	,	,	PUNCT
ejde-610	277	27	vol	vol	NOUN
ejde-610	277	28	.	.	PUNCT
ejde-610	277	29	2022	2022	NUM
ejde-610	277	30	(	(	PUNCT
ejde-610	277	31	2022	2022	NUM
ejde-610	277	32	)	)	PUNCT
ejde-610	277	33	,	,	PUNCT
ejde-610	277	34	no	no	INTJ
ejde-610	277	35	.	.	NOUN
ejde-610	277	36	74	74	NUM
ejde-610	277	37	,	,	PUNCT
ejde-610	277	38	pp	pp	ADJ
ejde-610	277	39	.	.	PUNCT
ejde-610	278	1	1	1	NUM
ejde-610	278	2	-	-	SYM
ejde-610	278	3	11	11	NUM
ejde-610	278	4	.	.	PUNCT
ejde-610	279	1	[	[	X
ejde-610	279	2	27	27	NUM
ejde-610	279	3	]	]	X
ejde-610	279	4	l.	l.	PROPN
ejde-610	279	5	x.	x.	PROPN
ejde-610	279	6	wang	wang	PROPN
ejde-610	279	7	,	,	PUNCT
ejde-610	279	8	c.	c.	PROPN
ejde-610	279	9	l.	l.	PROPN
ejde-610	279	10	xiong	xiong	PROPN
ejde-610	279	11	,	,	PUNCT
ejde-610	279	12	d.	d.	PROPN
ejde-610	279	13	zhang	zhang	PROPN
ejde-610	279	14	;	;	PUNCT
ejde-610	279	15	multiple	multiple	ADJ
ejde-610	279	16	solutions	solution	NOUN
ejde-610	279	17	for	for	ADP
ejde-610	279	18	nonhomogeneous	nonhomogeneous	ADJ
ejde-610	279	19	klein	klein	PROPN
ejde-610	279	20	-	-	PUNCT
ejde-610	279	21	gordon	gordon	PROPN
ejde-610	279	22	equation	equation	NOUN
ejde-610	279	23	with	with	ADP
ejde-610	279	24	sign	sign	NOUN
ejde-610	279	25	-	-	PUNCT
ejde-610	279	26	changing	change	VERB
ejde-610	279	27	potential	potential	NOUN
ejde-610	279	28	coupled	couple	VERB
ejde-610	279	29	with	with	ADP
ejde-610	279	30	born	bear	VERB
ejde-610	279	31	-	-	PUNCT
ejde-610	279	32	infeld	infeld	NOUN
ejde-610	279	33	theory	theory	NOUN
ejde-610	279	34	,	,	PUNCT
ejde-610	279	35	journal	journal	NOUN
ejde-610	279	36	of	of	ADP
ejde-610	279	37	applied	apply	VERB
ejde-610	279	38	analysis	analysis	NOUN
ejde-610	279	39	and	and	CCONJ
ejde-610	279	40	computation	computation	NOUN
ejde-610	279	41	,	,	PUNCT
ejde-610	279	42	14	14	NUM
ejde-610	279	43	(	(	PUNCT
ejde-610	279	44	1	1	NUM
ejde-610	279	45	)	)	PUNCT
ejde-610	279	46	,	,	PUNCT
ejde-610	279	47	2024	2024	NUM
ejde-610	279	48	,	,	PUNCT
ejde-610	279	49	84	84	NUM
ejde-610	279	50	-	-	SYM
ejde-610	279	51	105	105	NUM
ejde-610	279	52	.	.	PUNCT
ejde-610	280	1	[	[	X
ejde-610	280	2	28	28	NUM
ejde-610	280	3	]	]	X
ejde-610	280	4	l.	l.	PROPN
ejde-610	280	5	x.	x.	PROPN
ejde-610	280	6	wen	wen	PROPN
ejde-610	280	7	,	,	PUNCT
ejde-610	280	8	x.	x.	PROPN
ejde-610	280	9	h.	h.	PROPN
ejde-610	280	10	tang	tang	PROPN
ejde-610	280	11	,	,	PUNCT
ejde-610	280	12	s.	s.	PROPN
ejde-610	280	13	t.	t.	PROPN
ejde-610	280	14	chen	chen	PROPN
ejde-610	280	15	;	;	PUNCT
ejde-610	280	16	infinitely	infinitely	ADV
ejde-610	280	17	many	many	ADJ
ejde-610	280	18	solutions	solution	NOUN
ejde-610	280	19	and	and	CCONJ
ejde-610	280	20	least	least	ADJ
ejde-610	280	21	energy	energy	NOUN
ejde-610	280	22	solutions	solution	NOUN
ejde-610	280	23	for	for	ADP
ejde-610	280	24	klein	klein	PROPN
ejde-610	280	25	-	-	PUNCT
ejde-610	280	26	gordon	gordon	PROPN
ejde-610	280	27	equation	equation	NOUN
ejde-610	280	28	coupled	couple	VERB
ejde-610	280	29	with	with	ADP
ejde-610	280	30	born	bear	VERB
ejde-610	280	31	-	-	PUNCT
ejde-610	280	32	infeld	infeld	NOUN
ejde-610	280	33	theory	theory	NOUN
ejde-610	280	34	,	,	PUNCT
ejde-610	280	35	complex	complex	ADJ
ejde-610	280	36	variables	variable	NOUN
ejde-610	280	37	and	and	CCONJ
ejde-610	280	38	elliptic	elliptic	ADJ
ejde-610	280	39	equations	equation	NOUN
ejde-610	280	40	,	,	PUNCT
ejde-610	280	41	2019	2019	NUM
ejde-610	280	42	,	,	PUNCT
ejde-610	280	43	1572124	1572124	NUM
ejde-610	280	44	[	[	X
ejde-610	280	45	29	29	NUM
ejde-610	280	46	]	]	X
ejde-610	280	47	m.	m.	NOUN
ejde-610	280	48	willem	willem	PROPN
ejde-610	280	49	;	;	PUNCT
ejde-610	280	50	minimax	minimax	NOUN
ejde-610	280	51	theorems	theorem	NOUN
ejde-610	280	52	,	,	PUNCT
ejde-610	280	53	progress	progress	NOUN
ejde-610	280	54	in	in	ADP
ejde-610	280	55	nonlineäuser	nonlineäuser	PROPN
ejde-610	280	56	boston	boston	PROPN
ejde-610	280	57	inc	inc	PROPN
ejde-610	280	58	.	.	PROPN
ejde-610	280	59	,	,	PUNCT
ejde-610	280	60	boston	boston	PROPN
ejde-610	280	61	,	,	PUNCT
ejde-610	280	62	ma	ma	PROPN
ejde-610	280	63	,	,	PUNCT
ejde-610	280	64	1996	1996	NUM
ejde-610	280	65	.	.	PUNCT
ejde-610	281	1	[	[	X
ejde-610	281	2	30	30	NUM
ejde-610	281	3	]	]	X
ejde-610	281	4	y.	y.	PROPN
ejde-610	281	5	yang	yang	PROPN
ejde-610	281	6	;	;	PUNCT
ejde-610	281	7	classical	classical	ADJ
ejde-610	281	8	solutions	solution	NOUN
ejde-610	281	9	in	in	ADP
ejde-610	281	10	the	the	DET
ejde-610	281	11	born	bear	VERB
ejde-610	281	12	-	-	PUNCT
ejde-610	281	13	infeld	infeld	NOUN
ejde-610	281	14	theory	theory	NOUN
ejde-610	281	15	,	,	PUNCT
ejde-610	281	16	proceedings	proceeding	NOUN
ejde-610	281	17	of	of	ADP
ejde-610	281	18	the	the	DET
ejde-610	281	19	royal	royal	ADJ
ejde-610	281	20	society	society	NOUN
ejde-610	281	21	a	a	DET
ejde-610	281	22	:	:	PUNCT
ejde-610	281	23	mathematical	mathematical	ADJ
ejde-610	281	24	,	,	PUNCT
ejde-610	281	25	physical	physical	ADJ
ejde-610	281	26	and	and	CCONJ
ejde-610	281	27	engineering	engineering	NOUN
ejde-610	281	28	sciences	science	NOUN
ejde-610	281	29	,	,	PUNCT
ejde-610	281	30	456	456	NUM
ejde-610	281	31	,	,	PUNCT
ejde-610	281	32	1995	1995	NUM
ejde-610	281	33	,	,	PUNCT
ejde-610	281	34	615	615	NUM
ejde-610	281	35	-	-	SYM
ejde-610	281	36	640	640	NUM
ejde-610	281	37	.	.	PUNCT
ejde-610	282	1	[	[	X
ejde-610	282	2	31	31	NUM
ejde-610	282	3	]	]	X
ejde-610	282	4	y.	y.	PROPN
ejde-610	282	5	yu	yu	PROPN
ejde-610	282	6	;	;	PUNCT
ejde-610	282	7	solitary	solitary	ADJ
ejde-610	282	8	waves	wave	NOUN
ejde-610	282	9	for	for	ADP
ejde-610	282	10	nonlinear	nonlinear	PROPN
ejde-610	282	11	klein	klein	PROPN
ejde-610	282	12	-	-	PUNCT
ejde-610	282	13	gordon	gordon	PROPN
ejde-610	282	14	equations	equation	NOUN
ejde-610	282	15	coupled	couple	VERB
ejde-610	282	16	with	with	ADP
ejde-610	282	17	born	bear	VERB
ejde-610	282	18	-	-	PUNCT
ejde-610	282	19	infeld	infeld	NOUN
ejde-610	282	20	theory	theory	NOUN
ejde-610	282	21	,	,	PUNCT
ejde-610	282	22	annales	annales	PROPN
ejde-610	282	23	de	de	X
ejde-610	282	24	l’institut	l’institut	PROPN
ejde-610	282	25	henri	henri	PROPN
ejde-610	282	26	poincare	poincare	PROPN
ejde-610	282	27	(	(	PUNCT
ejde-610	282	28	c	c	X
ejde-610	282	29	)	)	PUNCT
ejde-610	282	30	non	non	NOUN
ejde-610	282	31	linear	linear	ADJ
ejde-610	282	32	analysis	analysis	NOUN
ejde-610	282	33	,	,	PUNCT
ejde-610	282	34	27(1	27(1	NUM
ejde-610	282	35	)	)	PUNCT
ejde-610	282	36	,	,	PUNCT
ejde-610	282	37	2010	2010	NUM
ejde-610	282	38	,	,	PUNCT
ejde-610	282	39	351	351	NUM
ejde-610	282	40	-	-	SYM
ejde-610	282	41	376	376	NUM
ejde-610	282	42	.	.	PUNCT
ejde-610	283	1	[	[	X
ejde-610	283	2	32	32	NUM
ejde-610	283	3	]	]	PUNCT
ejde-610	283	4	z.	z.	PROPN
ejde-610	283	5	h.	h.	PROPN
ejde-610	283	6	zhang	zhang	PROPN
ejde-610	283	7	,	,	PUNCT
ejde-610	283	8	j.	j.	PROPN
ejde-610	283	9	l.	l.	PROPN
ejde-610	283	10	liu	liu	PROPN
ejde-610	283	11	;	;	PUNCT
ejde-610	283	12	existence	existence	NOUN
ejde-610	283	13	and	and	CCONJ
ejde-610	283	14	multiplicity	multiplicity	NOUN
ejde-610	283	15	of	of	ADP
ejde-610	283	16	sign	sign	NOUN
ejde-610	283	17	-	-	PUNCT
ejde-610	283	18	changing	change	VERB
ejde-610	283	19	solutions	solution	NOUN
ejde-610	283	20	for	for	ADP
ejde-610	283	21	klein	klein	PROPN
ejde-610	283	22	-	-	PUNCT
ejde-610	283	23	gordon	gordon	PROPN
ejde-610	283	24	equation	equation	NOUN
ejde-610	283	25	coupled	couple	VERB
ejde-610	283	26	with	with	ADP
ejde-610	283	27	born	bear	VERB
ejde-610	283	28	-	-	PUNCT
ejde-610	283	29	infeld	infeld	NOUN
ejde-610	283	30	theory	theory	NOUN
ejde-610	283	31	with	with	ADP
ejde-610	283	32	subcritical	subcritical	ADJ
ejde-610	283	33	exponent	exponent	NOUN
ejde-610	283	34	,	,	PUNCT
ejde-610	283	35	qualitative	qualitative	ADJ
ejde-610	283	36	theory	theory	NOUN
ejde-610	283	37	of	of	ADP
ejde-610	283	38	dynamical	dynamical	ADJ
ejde-610	283	39	systems	system	NOUN
ejde-610	283	40	,	,	PUNCT
ejde-610	283	41	22(1	22(1	NUM
ejde-610	283	42	)	)	PUNCT
ejde-610	283	43	,	,	PUNCT
ejde-610	283	44	2023	2023	NUM
ejde-610	283	45	,	,	PUNCT
ejde-610	283	46	1	1	NUM
ejde-610	283	47	-	-	SYM
ejde-610	283	48	9	9	NUM
ejde-610	283	49	.	.	PUNCT
ejde-610	284	1	[	[	X
ejde-610	284	2	33	33	NUM
ejde-610	284	3	]	]	PUNCT
ejde-610	284	4	w.	w.	PROPN
ejde-610	284	5	m.	m.	PROPN
ejde-610	284	6	zou	zou	PROPN
ejde-610	284	7	,	,	PUNCT
ejde-610	284	8	m.	m.	NOUN
ejde-610	284	9	schechter	schechter	NOUN
ejde-610	284	10	;	;	PUNCT
ejde-610	284	11	critical	critical	ADJ
ejde-610	284	12	point	point	NOUN
ejde-610	284	13	theory	theory	NOUN
ejde-610	284	14	and	and	CCONJ
ejde-610	284	15	its	its	PRON
ejde-610	284	16	applications	application	NOUN
ejde-610	284	17	,	,	PUNCT
ejde-610	284	18	springer	springer	NOUN
ejde-610	284	19	,	,	PUNCT
ejde-610	284	20	new	new	PROPN
ejde-610	284	21	york	york	PROPN
ejde-610	284	22	,	,	PUNCT
ejde-610	284	23	2006	2006	NUM
ejde-610	284	24	.	.	PUNCT
ejde-610	285	1	lixia	lixia	PROPN
ejde-610	285	2	wang	wang	PROPN
ejde-610	285	3	school	school	PROPN
ejde-610	285	4	of	of	ADP
ejde-610	285	5	sciences	sciences	PROPN
ejde-610	285	6	,	,	PUNCT
ejde-610	285	7	tianjin	tianjin	PROPN
ejde-610	285	8	chengjian	chengjian	PROPN
ejde-610	285	9	university	university	PROPN
ejde-610	285	10	,	,	PUNCT
ejde-610	285	11	tianjin	tianjin	PROPN
ejde-610	285	12	300384	300384	NUM
ejde-610	285	13	,	,	PUNCT
ejde-610	285	14	china	china	PROPN
ejde-610	285	15	email	email	NOUN
ejde-610	285	16	address	address	NOUN
ejde-610	285	17	:	:	PUNCT
ejde-610	285	18	wanglixia0311@126.com	wanglixia0311@126.com	X
ejde-610	286	1	pingping	pingpe	VERB
ejde-610	286	2	zhao	zhao	PROPN
ejde-610	286	3	school	school	PROPN
ejde-610	286	4	of	of	ADP
ejde-610	286	5	sciences	sciences	PROPN
ejde-610	286	6	,	,	PUNCT
ejde-610	286	7	tianjin	tianjin	PROPN
ejde-610	286	8	chengjian	chengjian	PROPN
ejde-610	286	9	university	university	PROPN
ejde-610	286	10	,	,	PUNCT
ejde-610	286	11	tianjin	tianjin	PROPN
ejde-610	286	12	300384	300384	NUM
ejde-610	286	13	,	,	PUNCT
ejde-610	286	14	china	china	PROPN
ejde-610	286	15	email	email	NOUN
ejde-610	286	16	address	address	NOUN
ejde-610	286	17	:	:	PUNCT
ejde-610	286	18	ppzhao	ppzhao	PROPN
ejde-610	286	19	math@126.com	math@126.com	PROPN
ejde-610	286	20	12	12	NUM
ejde-610	286	21	l.	l.	PROPN
ejde-610	286	22	wang	wang	PROPN
ejde-610	286	23	,	,	PUNCT
ejde-610	286	24	p.	p.	PROPN
ejde-610	286	25	zhao	zhao	PROPN
ejde-610	286	26	,	,	PUNCT
ejde-610	286	27	d.	d.	PROPN
ejde-610	286	28	zhang	zhang	PROPN
ejde-610	286	29	ejde-2024/18	ejde-2024/18	PROPN
ejde-610	286	30	dong	dong	PROPN
ejde-610	286	31	zhang	zhang	PROPN
ejde-610	286	32	(	(	PUNCT
ejde-610	286	33	corresponding	correspond	VERB
ejde-610	286	34	author	author	NOUN
ejde-610	286	35	)	)	PUNCT
ejde-610	286	36	school	school	NOUN
ejde-610	286	37	of	of	ADP
ejde-610	286	38	sciences	science	NOUN
ejde-610	286	39	,	,	PUNCT
ejde-610	286	40	tianjin	tianjin	PROPN
ejde-610	286	41	chengjian	chengjian	PROPN
ejde-610	286	42	university	university	PROPN
ejde-610	286	43	,	,	PUNCT
ejde-610	286	44	tianjin	tianjin	PROPN
ejde-610	286	45	300384	300384	NUM
ejde-610	286	46	,	,	PUNCT
ejde-610	286	47	china	china	PROPN
ejde-610	286	48	email	email	NOUN
ejde-610	286	49	address	address	NOUN
ejde-610	286	50	:	:	PUNCT
ejde-610	286	51	zhangdongtg@126.com	zhangdongtg@126.com	NUM
ejde-610	286	52	1	1	NUM
ejde-610	286	53	.	.	PUNCT
ejde-610	286	54	introduction	introduction	NOUN
ejde-610	286	55	and	and	CCONJ
ejde-610	286	56	main	main	ADJ
ejde-610	286	57	results	result	NOUN
ejde-610	286	58	2	2	NUM
ejde-610	286	59	.	.	X
ejde-610	286	60	variational	variational	ADJ
ejde-610	286	61	setting	setting	NOUN
ejde-610	286	62	and	and	CCONJ
ejde-610	286	63	compactness	compactness	NOUN
ejde-610	286	64	condition	condition	NOUN
ejde-610	286	65	acknowledgments	acknowledgment	NOUN
ejde-610	286	66	references	reference	NOUN