id	sid	tid	token	lemma	pos
ma-257	1	1	2025	2025	NUM
ma-257	1	2	ada	ada	PROPN
ma-257	1	3	academica	academica	PROPN
ma-257	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-257	1	5	.	.	PUNCT
ma-257	2	1	j.	j.	PROPN
ma-257	2	2	math	math	PROPN
ma-257	2	3	.	.	PUNCT
ma-257	3	1	anal	anal	ADJ
ma-257	3	2	.	.	PUNCT
ma-257	4	1	5	5	NUM
ma-257	4	2	(	(	PUNCT
ma-257	4	3	2025	2025	NUM
ma-257	4	4	)	)	PUNCT
ma-257	5	1	1doi	1doi	NUM
ma-257	5	2	:	:	PUNCT
ma-257	5	3	10.28924	10.28924	NUM
ma-257	5	4	/	/	SYM
ma-257	5	5	ada	ada	PROPN
ma-257	5	6	/	/	SYM
ma-257	5	7	ma.5.1	ma.5.1	PROPN
ma-257	5	8	stability	stability	NOUN
ma-257	5	9	results	result	NOUN
ma-257	5	10	of	of	ADP
ma-257	5	11	positive	positive	ADJ
ma-257	5	12	weak	weak	ADJ
ma-257	5	13	solution	solution	NOUN
ma-257	5	14	for	for	ADP
ma-257	5	15	a	a	DET
ma-257	5	16	class	class	NOUN
ma-257	5	17	of	of	ADP
ma-257	5	18	chemically	chemically	ADV
ma-257	5	19	reacting	react	VERB
ma-257	5	20	systems	system	NOUN
ma-257	5	21	salah	salah	PROPN
ma-257	5	22	a.	a.	PROPN
ma-257	5	23	khafagy1	khafagy1	PROPN
ma-257	5	24	,	,	PUNCT
ma-257	5	25	a.	a.	PROPN
ma-257	5	26	ezzat	ezzat	PROPN
ma-257	5	27	mohamed2∗	mohamed2∗	PROPN
ma-257	5	28	1department	1department	NUM
ma-257	5	29	of	of	ADP
ma-257	5	30	mathematics	mathematic	NOUN
ma-257	5	31	,	,	PUNCT
ma-257	5	32	faculty	faculty	NOUN
ma-257	5	33	of	of	ADP
ma-257	5	34	science	science	NOUN
ma-257	5	35	,	,	PUNCT
ma-257	5	36	al	al	PROPN
ma-257	5	37	-	-	PUNCT
ma-257	5	38	azhar	azhar	PROPN
ma-257	5	39	university	university	PROPN
ma-257	5	40	(	(	PUNCT
ma-257	5	41	11884	11884	NUM
ma-257	5	42	)	)	PUNCT
ma-257	5	43	,	,	PUNCT
ma-257	5	44	cairo	cairo	PROPN
ma-257	5	45	,	,	PUNCT
ma-257	5	46	egypt	egypt	PROPN
ma-257	5	47	salahabdelnaby.211@azhar.edu.eg	salahabdelnaby.211@azhar.edu.eg	X
ma-257	5	48	2department	2department	NUM
ma-257	5	49	of	of	ADP
ma-257	5	50	mathematics	mathematic	NOUN
ma-257	5	51	,	,	PUNCT
ma-257	5	52	faculty	faculty	NOUN
ma-257	5	53	of	of	ADP
ma-257	5	54	science	science	NOUN
ma-257	5	55	,	,	PUNCT
ma-257	5	56	fayoum	fayoum	PROPN
ma-257	5	57	university	university	PROPN
ma-257	5	58	(	(	PUNCT
ma-257	5	59	63514	63514	NUM
ma-257	5	60	)	)	PUNCT
ma-257	5	61	,	,	PUNCT
ma-257	5	62	fayoum	fayoum	PROPN
ma-257	5	63	,	,	PUNCT
ma-257	5	64	egypt	egypt	PROPN
ma-257	5	65	aam35@fayoum.edu.eg	aam35@fayoum.edu.eg	PROPN
ma-257	5	66	∗correspondence	∗correspondence	PROPN
ma-257	5	67	:	:	PUNCT
ma-257	5	68	aam35@fayoum.edu.eg	aam35@fayoum.edu.eg	NOUN
ma-257	5	69	abstract	abstract	NOUN
ma-257	5	70	.	.	PUNCT
ma-257	6	1	this	this	DET
ma-257	6	2	paper	paper	NOUN
ma-257	6	3	aims	aim	VERB
ma-257	6	4	to	to	PART
ma-257	6	5	study	study	VERB
ma-257	6	6	the	the	DET
ma-257	6	7	existence	existence	NOUN
ma-257	6	8	and	and	CCONJ
ma-257	6	9	non	non	ADJ
ma-257	6	10	-	-	ADJ
ma-257	6	11	existence	existence	ADJ
ma-257	6	12	results	result	NOUN
ma-257	6	13	of	of	ADP
ma-257	6	14	positive	positive	ADJ
ma-257	6	15	weak	weak	ADJ
ma-257	6	16	solutionto	solutionto	ADP
ma-257	6	17	the	the	DET
ma-257	6	18	quasilinear	quasilinear	PROPN
ma-257	6	19	elliptic	elliptic	ADJ
ma-257	6	20	system:	system:	NOUN
ma-257	6	21	−∆pu	−∆pu	X
ma-257	7	1	=	=	X
ma-257	8	1	λa(x)[f	λa(x)[f	ADJ
ma-257	8	2	(	(	PUNCT
ma-257	8	3	u	u	NOUN
ma-257	8	4	,	,	PUNCT
ma-257	8	5	v)−	v)−	PROPN
ma-257	8	6	1	1	NUM
ma-257	8	7	uα	uα	PROPN
ma-257	8	8	]	]	X
ma-257	8	9	,	,	PUNCT
ma-257	8	10	x	x	PUNCT
ma-257	8	11	∈	∈	PROPN
ma-257	8	12	ω	ω	PROPN
ma-257	8	13	,	,	PUNCT
ma-257	8	14	−∆qv	−∆qv	PROPN
ma-257	8	15	=	=	SYM
ma-257	8	16	λb(x)[g(u	λb(x)[g(u	PROPN
ma-257	8	17	,	,	PUNCT
ma-257	8	18	v)−	v)−	PROPN
ma-257	8	19	1	1	NUM
ma-257	8	20	vβ	vβ	NOUN
ma-257	8	21	]	]	X
ma-257	8	22	,	,	PUNCT
ma-257	8	23	x	x	PUNCT
ma-257	8	24	∈	∈	PROPN
ma-257	8	25	ω	ω	PROPN
ma-257	8	26	,	,	PUNCT
ma-257	8	27	u	u	NOUN
ma-257	8	28	=	=	NOUN
ma-257	8	29	0	0	PUNCT
ma-257	8	30	=	=	SYM
ma-257	8	31	v	v	NOUN
ma-257	8	32	,	,	PUNCT
ma-257	8	33	x	x	SYM
ma-257	8	34	∈	∈	PROPN
ma-257	8	35	∂ω	∂ω	PROPN
ma-257	8	36	,	,	PUNCT
ma-257	8	37	where	where	SCONJ
ma-257	8	38	∆rw	∆rw	X
ma-257	8	39	=	=	SYM
ma-257	8	40	div(|∇w	div(|∇w	PROPN
ma-257	8	41	|r−2∇w	|r−2∇w	NUM
ma-257	8	42	)	)	PUNCT
ma-257	8	43	is	be	AUX
ma-257	8	44	the	the	DET
ma-257	8	45	r	r	NOUN
ma-257	8	46	-	-	PUNCT
ma-257	8	47	laplacian	laplacian	ADJ
ma-257	8	48	(	(	PUNCT
ma-257	8	49	r	r	NOUN
ma-257	8	50	=	=	SYM
ma-257	8	51	p	p	NOUN
ma-257	8	52	,	,	PUNCT
ma-257	8	53	q	q	NOUN
ma-257	8	54	)	)	PUNCT
ma-257	8	55	,	,	PUNCT
ma-257	8	56	r	r	NOUN
ma-257	8	57	>	>	X
ma-257	8	58	1	1	NUM
ma-257	8	59	,	,	PUNCT
ma-257	8	60	α	α	X
ma-257	8	61	,	,	PUNCT
ma-257	8	62	β	β	X
ma-257	8	63	∈	∈	PROPN
ma-257	8	64	(	(	PUNCT
ma-257	8	65	0	0	NUM
ma-257	8	66	,	,	PUNCT
ma-257	8	67	1	1	NUM
ma-257	8	68	)	)	PUNCT
ma-257	8	69	,	,	PUNCT
ma-257	8	70	ω	ω	PROPN
ma-257	8	71	is	be	AUX
ma-257	8	72	a	a	DET
ma-257	8	73	boundeddomain	boundeddomain	NOUN
ma-257	8	74	in	in	ADP
ma-257	8	75	rn(n	rn(n	NOUN
ma-257	8	76	>	>	X
ma-257	8	77	1	1	NUM
ma-257	8	78	)	)	PUNCT
ma-257	8	79	with	with	ADP
ma-257	8	80	smooth	smooth	ADJ
ma-257	8	81	boundary	boundary	ADJ
ma-257	8	82	∂ω	∂ω	PROPN
ma-257	8	83	and	and	CCONJ
ma-257	8	84	λ	λ	PROPN
ma-257	8	85	is	be	AUX
ma-257	8	86	a	a	DET
ma-257	8	87	positive	positive	ADJ
ma-257	8	88	parameter	parameter	NOUN
ma-257	8	89	.	.	PUNCT
ma-257	9	1	here	here	ADV
ma-257	9	2	f	f	X
ma-257	9	3	,	,	PUNCT
ma-257	9	4	g	g	PROPN
ma-257	9	5	are	be	AUX
ma-257	9	6	c1	c1	NOUN
ma-257	9	7	increasing	increase	VERB
ma-257	9	8	functions	function	NOUN
ma-257	9	9	such	such	ADJ
ma-257	9	10	that	that	SCONJ
ma-257	9	11	f	f	NOUN
ma-257	9	12	,	,	PUNCT
ma-257	9	13	g	g	PROPN
ma-257	9	14	:	:	PUNCT
ma-257	9	15	r+	r+	NOUN
ma-257	9	16	×	×	NOUN
ma-257	9	17	r+	r+	NOUN
ma-257	9	18	→	→	SYM
ma-257	9	19	r+	r+	X
ma-257	9	20	;	;	PUNCT
ma-257	9	21	f	f	PROPN
ma-257	9	22	(	(	PUNCT
ma-257	9	23	υ1	υ1	PROPN
ma-257	9	24	,	,	PUNCT
ma-257	9	25	υ2	υ2	NOUN
ma-257	9	26	)	)	PUNCT
ma-257	9	27	>	>	X
ma-257	9	28	0	0	NUM
ma-257	9	29	,	,	PUNCT
ma-257	9	30	g(υ1	g(υ1	NOUN
ma-257	9	31	,	,	PUNCT
ma-257	9	32	υ2	υ2	NOUN
ma-257	9	33	)	)	PUNCT
ma-257	9	34	>	>	X
ma-257	9	35	0	0	PUNCT
ma-257	9	36	for	for	ADP
ma-257	9	37	υ1	υ1	PROPN
ma-257	9	38	,	,	PUNCT
ma-257	9	39	υ2	υ2	PROPN
ma-257	9	40	>	>	SYM
ma-257	9	41	0.with	0.with	NUM
ma-257	9	42	c1	c1	PROPN
ma-257	9	43	sign	sign	NOUN
ma-257	9	44	-	-	PUNCT
ma-257	9	45	changing	change	VERB
ma-257	9	46	functions	function	NOUN
ma-257	9	47	a(x	a(x	NOUN
ma-257	9	48	)	)	PUNCT
ma-257	9	49	,	,	PUNCT
ma-257	9	50	b(x	b(x	NOUN
ma-257	9	51	)	)	PUNCT
ma-257	9	52	that	that	PRON
ma-257	9	53	perhaps	perhaps	ADV
ma-257	9	54	have	have	VERB
ma-257	9	55	negative	negative	ADJ
ma-257	9	56	values	value	NOUN
ma-257	9	57	nearby	nearby	ADV
ma-257	9	58	the	the	DET
ma-257	9	59	boundary.we	boundary.we	NOUN
ma-257	9	60	establish	establish	VERB
ma-257	9	61	our	our	PRON
ma-257	9	62	results	result	NOUN
ma-257	9	63	via	via	ADP
ma-257	9	64	the	the	DET
ma-257	9	65	sub	sub	NOUN
ma-257	9	66	-	-	ADJ
ma-257	9	67	supersolution	supersolution	NOUN
ma-257	9	68	method	method	NOUN
ma-257	9	69	.	.	PUNCT
ma-257	10	1	in	in	ADP
ma-257	10	2	addition	addition	NOUN
ma-257	10	3	,	,	PUNCT
ma-257	10	4	we	we	PRON
ma-257	10	5	study	study	VERB
ma-257	10	6	the	the	DET
ma-257	10	7	stability	stability	NOUN
ma-257	10	8	andinstability	andinstability	NOUN
ma-257	10	9	results	result	NOUN
ma-257	10	10	of	of	ADP
ma-257	10	11	positive	positive	ADJ
ma-257	10	12	weak	weak	ADJ
ma-257	10	13	solution	solution	NOUN
ma-257	10	14	with	with	ADP
ma-257	10	15	different	different	ADJ
ma-257	10	16	choices	choice	NOUN
ma-257	10	17	of	of	ADP
ma-257	10	18	f	f	PROPN
ma-257	10	19	and	and	CCONJ
ma-257	10	20	g.	g.	PROPN
ma-257	10	21	1	1	NUM
ma-257	10	22	.	.	PUNCT
ma-257	11	1	introduction	introduction	NOUN
ma-257	11	2	this	this	DET
ma-257	11	3	paper	paper	NOUN
ma-257	11	4	aims	aim	VERB
ma-257	11	5	to	to	PART
ma-257	11	6	study	study	VERB
ma-257	11	7	the	the	DET
ma-257	11	8	existence	existence	NOUN
ma-257	11	9	and	and	CCONJ
ma-257	11	10	non	non	ADJ
ma-257	11	11	-	-	ADJ
ma-257	11	12	existence	existence	ADJ
ma-257	11	13	results	result	NOUN
ma-257	11	14	of	of	ADP
ma-257	11	15	positive	positive	ADJ
ma-257	11	16	weak	weak	ADJ
ma-257	11	17	solution	solution	NOUN
ma-257	11	18	tothe	tothe	VERB
ma-257	11	19	quasilinear	quasilinear	PROPN
ma-257	11	20	elliptic	elliptic	ADJ
ma-257	11	21	system:	system:	NOUN
ma-257	11	22	−∆pu	−∆pu	X
ma-257	12	1	=	=	X
ma-257	13	1	λa(x)[f	λa(x)[f	ADJ
ma-257	13	2	(	(	PUNCT
ma-257	13	3	u	u	NOUN
ma-257	13	4	,	,	PUNCT
ma-257	13	5	v)−	v)−	PROPN
ma-257	13	6	1	1	NUM
ma-257	13	7	uα	uα	PROPN
ma-257	13	8	]	]	X
ma-257	13	9	,	,	PUNCT
ma-257	13	10	x	x	PUNCT
ma-257	13	11	∈	∈	PROPN
ma-257	13	12	ω	ω	PROPN
ma-257	13	13	,	,	PUNCT
ma-257	13	14	−∆qv	−∆qv	PROPN
ma-257	13	15	=	=	SYM
ma-257	13	16	λb(x)[g(u	λb(x)[g(u	PROPN
ma-257	13	17	,	,	PUNCT
ma-257	13	18	v)−	v)−	PROPN
ma-257	13	19	1	1	NUM
ma-257	13	20	vβ	vβ	NOUN
ma-257	13	21	]	]	X
ma-257	13	22	,	,	PUNCT
ma-257	13	23	x	x	PUNCT
ma-257	13	24	∈	∈	PROPN
ma-257	13	25	ω	ω	PROPN
ma-257	13	26	,	,	PUNCT
ma-257	13	27	u	u	NOUN
ma-257	13	28	=	=	NOUN
ma-257	13	29	0	0	PUNCT
ma-257	13	30	=	=	SYM
ma-257	13	31	v	v	NOUN
ma-257	13	32	,	,	PUNCT
ma-257	13	33	x	x	X
ma-257	13	34	∈	∈	PROPN
ma-257	13	35	∂ω	∂ω	PROPN
ma-257	13	36	,	,	PUNCT
ma-257	13	37	(	(	PUNCT
ma-257	13	38	1	1	X
ma-257	13	39	)	)	PUNCT
ma-257	13	40	where	where	SCONJ
ma-257	13	41	∆rw	∆rw	X
ma-257	13	42	=	=	SYM
ma-257	13	43	div(|∇w	div(|∇w	PROPN
ma-257	13	44	|r−2∇w	|r−2∇w	NUM
ma-257	13	45	)	)	PUNCT
ma-257	13	46	is	be	AUX
ma-257	13	47	the	the	DET
ma-257	13	48	r	r	NOUN
ma-257	13	49	-	-	PUNCT
ma-257	13	50	laplacian	laplacian	ADJ
ma-257	13	51	(	(	PUNCT
ma-257	13	52	r	r	NOUN
ma-257	13	53	=	=	SYM
ma-257	13	54	p	p	NOUN
ma-257	13	55	,	,	PUNCT
ma-257	13	56	q	q	NOUN
ma-257	13	57	)	)	PUNCT
ma-257	13	58	,	,	PUNCT
ma-257	13	59	r	r	NOUN
ma-257	13	60	>	>	X
ma-257	13	61	1	1	NUM
ma-257	13	62	,	,	PUNCT
ma-257	13	63	α	α	X
ma-257	13	64	,	,	PUNCT
ma-257	13	65	β	β	X
ma-257	13	66	∈	∈	PROPN
ma-257	13	67	(	(	PUNCT
ma-257	13	68	0	0	NUM
ma-257	13	69	,	,	PUNCT
ma-257	13	70	1	1	NUM
ma-257	13	71	)	)	PUNCT
ma-257	13	72	,	,	PUNCT
ma-257	13	73	ω	ω	PROPN
ma-257	13	74	is	be	AUX
ma-257	13	75	a	a	DET
ma-257	13	76	boundeddomain	boundeddomain	NOUN
ma-257	13	77	in	in	ADP
ma-257	13	78	rn(n	rn(n	NOUN
ma-257	13	79	>	>	X
ma-257	13	80	1	1	NUM
ma-257	13	81	)	)	PUNCT
ma-257	13	82	with	with	ADP
ma-257	13	83	smooth	smooth	ADJ
ma-257	13	84	boundary	boundary	ADJ
ma-257	13	85	∂ω	∂ω	PROPN
ma-257	13	86	and	and	CCONJ
ma-257	13	87	λ	λ	PROPN
ma-257	13	88	is	be	AUX
ma-257	13	89	a	a	DET
ma-257	13	90	positive	positive	ADJ
ma-257	13	91	parameter	parameter	NOUN
ma-257	13	92	.	.	PUNCT
ma-257	14	1	here	here	ADV
ma-257	14	2	f	f	X
ma-257	14	3	,	,	PUNCT
ma-257	14	4	g	g	PROPN
ma-257	14	5	are	be	AUX
ma-257	14	6	c1increasing	c1increase	VERB
ma-257	14	7	functions	function	NOUN
ma-257	14	8	such	such	ADJ
ma-257	14	9	that	that	SCONJ
ma-257	14	10	f	f	NOUN
ma-257	14	11	,	,	PUNCT
ma-257	14	12	g	g	PROPN
ma-257	14	13	:	:	PUNCT
ma-257	14	14	r+	r+	NOUN
ma-257	14	15	×	×	NOUN
ma-257	14	16	r+	r+	NOUN
ma-257	14	17	→	→	SYM
ma-257	14	18	r+	r+	X
ma-257	14	19	;	;	PUNCT
ma-257	14	20	f	f	PROPN
ma-257	14	21	(	(	PUNCT
ma-257	14	22	υ1	υ1	PROPN
ma-257	14	23	,	,	PUNCT
ma-257	14	24	υ2	υ2	NOUN
ma-257	14	25	)	)	PUNCT
ma-257	14	26	>	>	X
ma-257	14	27	0	0	NUM
ma-257	14	28	,	,	PUNCT
ma-257	14	29	g(υ1	g(υ1	NOUN
ma-257	14	30	,	,	PUNCT
ma-257	14	31	υ2	υ2	NOUN
ma-257	14	32	)	)	PUNCT
ma-257	14	33	>	>	X
ma-257	14	34	0	0	PUNCT
ma-257	14	35	for	for	ADP
ma-257	14	36	υ1	υ1	PROPN
ma-257	14	37	,	,	PUNCT
ma-257	14	38	υ2	υ2	PROPN
ma-257	14	39	>	>	SYM
ma-257	14	40	0.with	0.with	NUM
ma-257	14	41	c1	c1	PROPN
ma-257	14	42	sign	sign	NOUN
ma-257	14	43	-	-	PUNCT
ma-257	14	44	changing	change	VERB
ma-257	14	45	functions	function	NOUN
ma-257	14	46	a(x	a(x	NOUN
ma-257	14	47	)	)	PUNCT
ma-257	14	48	,	,	PUNCT
ma-257	14	49	b(x	b(x	NOUN
ma-257	14	50	)	)	PUNCT
ma-257	14	51	that	that	PRON
ma-257	14	52	perhaps	perhaps	ADV
ma-257	14	53	have	have	VERB
ma-257	14	54	negative	negative	ADJ
ma-257	14	55	values	value	NOUN
ma-257	14	56	nearby	nearby	ADV
ma-257	14	57	the	the	DET
ma-257	14	58	boundary	boundary	NOUN
ma-257	14	59	.	.	PUNCT
ma-257	15	1	received	receive	VERB
ma-257	15	2	:	:	PUNCT
ma-257	15	3	17	17	NUM
ma-257	15	4	jul	jul	PROPN
ma-257	15	5	2024	2024	NUM
ma-257	15	6	.	.	PUNCT
ma-257	16	1	key	key	ADJ
ma-257	16	2	words	word	NOUN
ma-257	16	3	and	and	CCONJ
ma-257	16	4	phrases	phrase	NOUN
ma-257	16	5	.	.	PUNCT
ma-257	17	1	positive	positive	ADJ
ma-257	17	2	weak	weak	ADJ
ma-257	17	3	solution	solution	NOUN
ma-257	17	4	;	;	PUNCT
ma-257	17	5	sub	sub	ADJ
ma-257	17	6	-	-	ADJ
ma-257	17	7	supersolution	supersolution	NOUN
ma-257	17	8	method	method	NOUN
ma-257	17	9	;	;	PUNCT
ma-257	17	10	stability.1	stability.1	PROPN
ma-257	17	11	https://adac.ee	https://adac.ee	PROPN
ma-257	17	12	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	17	13	eur	eur	PROPN
ma-257	17	14	.	.	PUNCT
ma-257	18	1	j.	j.	PROPN
ma-257	18	2	math	math	PROPN
ma-257	18	3	.	.	PUNCT
ma-257	19	1	anal	anal	PROPN
ma-257	19	2	.	.	PUNCT
ma-257	20	1	10.28924	10.28924	NUM
ma-257	20	2	/	/	SYM
ma-257	20	3	ada	ada	PROPN
ma-257	20	4	/	/	SYM
ma-257	20	5	ma.5.1	ma.5.1	PROPN
ma-257	20	6	2general	2general	NUM
ma-257	20	7	evolutionary	evolutionary	ADJ
ma-257	20	8	problems	problem	NOUN
ma-257	20	9	are	be	AUX
ma-257	20	10	defined	define	VERB
ma-257	20	11	as	as	SCONJ
ma-257	20	12	follows	follow	VERB
ma-257	20	13	:	:	PUNCT
ma-257	21	1			PROPN
ma-257	21	2	ut	ut	PUNCT
ma-257	21	3	=	=	PRON
ma-257	21	4	η∆pu	η∆pu	PROPN
ma-257	21	5	+	+	CCONJ
ma-257	21	6	λa(x)[f	λa(x)[f	ADJ
ma-257	21	7	(	(	PUNCT
ma-257	21	8	u	u	NOUN
ma-257	21	9	,	,	PUNCT
ma-257	21	10	v)−	v)−	PROPN
ma-257	21	11	1	1	NUM
ma-257	21	12	uα	uα	PROPN
ma-257	21	13	]	]	X
ma-257	21	14	,	,	PUNCT
ma-257	21	15	x	x	PUNCT
ma-257	21	16	∈	∈	PROPN
ma-257	21	17	ω	ω	PROPN
ma-257	21	18	,	,	PUNCT
ma-257	21	19	vt	vt	PROPN
ma-257	21	20	=	=	PROPN
ma-257	21	21	δ∆qv	δ∆qv	PROPN
ma-257	21	22	+	+	CCONJ
ma-257	21	23	λb(x)[g(u	λb(x)[g(u	PROPN
ma-257	21	24	,	,	PUNCT
ma-257	21	25	v)−	v)−	PROPN
ma-257	21	26	1	1	NUM
ma-257	21	27	vβ	vβ	NOUN
ma-257	21	28	]	]	X
ma-257	21	29	,	,	PUNCT
ma-257	21	30	x	x	PUNCT
ma-257	21	31	∈	∈	PROPN
ma-257	21	32	ω	ω	PROPN
ma-257	21	33	,	,	PUNCT
ma-257	21	34	u	u	NOUN
ma-257	21	35	=	=	NOUN
ma-257	21	36	0	0	PUNCT
ma-257	21	37	=	=	SYM
ma-257	21	38	v	v	NOUN
ma-257	21	39	,	,	PUNCT
ma-257	21	40	x	x	X
ma-257	21	41	∈	∈	PROPN
ma-257	21	42	∂ω	∂ω	PROPN
ma-257	21	43	,	,	PUNCT
ma-257	21	44	(	(	PUNCT
ma-257	21	45	2	2	X
ma-257	21	46	)	)	PUNCT
ma-257	21	47	have	have	VERB
ma-257	21	48	stationary	stationary	ADJ
ma-257	21	49	counterpart	counterpart	NOUN
ma-257	21	50	of	of	ADP
ma-257	21	51	systems	system	NOUN
ma-257	21	52	of	of	ADP
ma-257	21	53	singular	singular	ADJ
ma-257	21	54	equations	equation	NOUN
ma-257	21	55	like	like	ADP
ma-257	21	56	(	(	PUNCT
ma-257	21	57	1	1	NUM
ma-257	21	58	)	)	PUNCT
ma-257	21	59	,	,	PUNCT
ma-257	21	60	such	such	ADJ
ma-257	21	61	that	that	SCONJ
ma-257	21	62	η	η	PROPN
ma-257	21	63	and	and	CCONJ
ma-257	21	64	δ	δ	PROPN
ma-257	21	65	are	be	AUX
ma-257	21	66	positiveparameters	positiveparameter	NOUN
ma-257	21	67	.	.	PUNCT
ma-257	22	1	system	system	NOUN
ma-257	22	2	(	(	PUNCT
ma-257	22	3	2	2	X
ma-257	22	4	)	)	PUNCT
ma-257	22	5	is	be	AUX
ma-257	22	6	an	an	DET
ma-257	22	7	inspiration	inspiration	NOUN
ma-257	22	8	from	from	ADP
ma-257	22	9	major	major	ADJ
ma-257	22	10	applications	application	NOUN
ma-257	22	11	in	in	ADP
ma-257	22	12	chemically	chemically	ADV
ma-257	22	13	reacting	react	VERB
ma-257	22	14	systems	system	NOUN
ma-257	22	15	,	,	PUNCT
ma-257	22	16	where	where	SCONJ
ma-257	22	17	the	the	DET
ma-257	22	18	activator	activator	NOUN
ma-257	22	19	chemical	chemical	NOUN
ma-257	22	20	substance	substance	NOUN
ma-257	22	21	’s	’s	PART
ma-257	22	22	density	density	NOUN
ma-257	22	23	is	be	AUX
ma-257	22	24	denoted	denote	VERB
ma-257	22	25	by	by	ADP
ma-257	22	26	u	u	NOUN
ma-257	22	27	,	,	PUNCT
ma-257	22	28	while	while	SCONJ
ma-257	22	29	an	an	DET
ma-257	22	30	inhibitor	inhibitor	NOUN
ma-257	22	31	is	be	AUX
ma-257	22	32	denotedby	denotedby	ADJ
ma-257	22	33	v	v	NOUN
ma-257	22	34	.	.	PUNCT
ma-257	23	1	the	the	DET
ma-257	23	2	slow	slow	ADJ
ma-257	23	3	and	and	CCONJ
ma-257	23	4	fast	fast	ADJ
ma-257	23	5	diffusion	diffusion	NOUN
ma-257	23	6	of	of	ADP
ma-257	23	7	u	u	NOUN
ma-257	23	8	and	and	CCONJ
ma-257	23	9	v	v	NOUN
ma-257	23	10	,	,	PUNCT
ma-257	23	11	respectively	respectively	ADV
ma-257	23	12	,	,	PUNCT
ma-257	23	13	are	be	AUX
ma-257	23	14	turned	turn	VERB
ma-257	23	15	into	into	ADP
ma-257	23	16	a	a	DET
ma-257	23	17	small	small	ADJ
ma-257	23	18	η	η	PROPN
ma-257	23	19	and	and	CCONJ
ma-257	23	20	large	large	ADJ
ma-257	23	21	δ	δ	PROPN
ma-257	23	22	(	(	PUNCT
ma-257	23	23	see	see	VERB
ma-257	23	24	[	[	X
ma-257	23	25	1	1	NUM
ma-257	23	26	]	]	NUM
ma-257	23	27	)	)	PUNCT
ma-257	23	28	.	.	PUNCT
ma-257	24	1	furthermore	furthermore	ADV
ma-257	24	2	,	,	PUNCT
ma-257	24	3	systems	system	NOUN
ma-257	24	4	like	like	ADP
ma-257	24	5	(	(	PUNCT
ma-257	24	6	1	1	X
ma-257	24	7	)	)	PUNCT
ma-257	24	8	appear	appear	VERB
ma-257	24	9	in	in	ADP
ma-257	24	10	many	many	ADJ
ma-257	24	11	contexts	context	NOUN
ma-257	24	12	in	in	ADP
ma-257	24	13	engineering	engineering	NOUN
ma-257	24	14	and	and	CCONJ
ma-257	24	15	biology.it	biology.it	NOUN
ma-257	24	16	presents	present	VERB
ma-257	24	17	a	a	DET
ma-257	24	18	simple	simple	ADJ
ma-257	24	19	model	model	NOUN
ma-257	24	20	where	where	SCONJ
ma-257	24	21	u	u	NOUN
ma-257	24	22	,	,	PUNCT
ma-257	24	23	v	v	NUM
ma-257	24	24	denote	denote	VERB
ma-257	24	25	the	the	DET
ma-257	24	26	density	density	NOUN
ma-257	24	27	of	of	ADP
ma-257	24	28	two	two	NUM
ma-257	24	29	diffusing	diffuse	VERB
ma-257	24	30	biological	biological	ADJ
ma-257	24	31	species	specie	NOUN
ma-257	24	32	fordescribing	fordescribe	VERB
ma-257	24	33	the	the	DET
ma-257	24	34	interaction	interaction	NOUN
ma-257	24	35	between	between	ADP
ma-257	24	36	these	these	DET
ma-257	24	37	two	two	NUM
ma-257	24	38	species.recently	species.recently	ADV
ma-257	24	39	,	,	PUNCT
ma-257	24	40	similar	similar	ADJ
ma-257	24	41	problems	problem	NOUN
ma-257	24	42	have	have	AUX
ma-257	24	43	been	be	AUX
ma-257	24	44	discussed	discuss	VERB
ma-257	24	45	in	in	ADP
ma-257	24	46	[	[	X
ma-257	24	47	2–5	2–5	NOUN
ma-257	24	48	]	]	PUNCT
ma-257	24	49	.	.	PUNCT
ma-257	25	1	the	the	DET
ma-257	25	2	authors	author	NOUN
ma-257	25	3	in	in	ADP
ma-257	25	4	[	[	X
ma-257	25	5	6	6	NUM
ma-257	25	6	]	]	PUNCT
ma-257	25	7	investigated	investigate	VERB
ma-257	25	8	thepositive	thepositive	ADJ
ma-257	25	9	weak	weak	ADJ
ma-257	25	10	solution	solution	NOUN
ma-257	25	11	of	of	ADP
ma-257	25	12	the	the	DET
ma-257	25	13	system:−∆u	system:−∆u	PROPN
ma-257	25	14	=	=	SYM
ma-257	25	15	λ[f	λ[f	X
ma-257	25	16	(	(	PUNCT
ma-257	25	17	u)−	u)−	PROPN
ma-257	25	18	1	1	NUM
ma-257	25	19	uα	uα	NOUN
ma-257	25	20	]	]	X
ma-257	25	21	,	,	PUNCT
ma-257	25	22	x	x	PUNCT
ma-257	25	23	∈	∈	PROPN
ma-257	25	24	ω	ω	PROPN
ma-257	25	25	,	,	PUNCT
ma-257	25	26	u	u	NOUN
ma-257	25	27	=	=	PROPN
ma-257	25	28	0	0	PROPN
ma-257	25	29	,	,	PUNCT
ma-257	25	30	x	x	X
ma-257	25	31	∈	∈	PROPN
ma-257	25	32	∂ω	∂ω	PROPN
ma-257	25	33	,	,	PUNCT
ma-257	25	34	(	(	PUNCT
ma-257	25	35	3	3	X
ma-257	25	36	)	)	PUNCT
ma-257	26	1	where	where	SCONJ
ma-257	26	2	f	f	PROPN
ma-257	26	3	∈	∈	PROPN
ma-257	26	4	c2(r+	c2(r+	PROPN
ma-257	26	5	)	)	PUNCT
ma-257	26	6	,	,	PUNCT
ma-257	26	7	f	f	PROPN
ma-257	26	8	′	′	NUM
ma-257	26	9	>	>	X
ma-257	26	10	0	0	NUM
ma-257	26	11	,	,	PUNCT
ma-257	26	12	f	f	PROPN
ma-257	26	13	(	(	PUNCT
ma-257	26	14	0	0	NUM
ma-257	26	15	)	)	PUNCT
ma-257	26	16	≥	≥	NOUN
ma-257	26	17	0	0	NUM
ma-257	26	18	,	,	PUNCT
ma-257	26	19	lim	lim	PROPN
ma-257	26	20	ε→∞	ε→∞	NUM
ma-257	26	21	f	f	PROPN
ma-257	26	22	(	(	PUNCT
ma-257	26	23	ε	ε	PROPN
ma-257	26	24	)	)	PUNCT
ma-257	26	25	ε	ε	PROPN
ma-257	26	26	=	=	SYM
ma-257	26	27	∞	∞	PROPN
ma-257	26	28	and	and	CCONJ
ma-257	26	29	ω	ω	NUM
ma-257	26	30	⊂	⊂	PROPN
ma-257	26	31	rn(n	rn(n	NUM
ma-257	26	32	≥	≥	NOUN
ma-257	26	33	1	1	NUM
ma-257	26	34	)	)	PUNCT
ma-257	26	35	.	.	PUNCT
ma-257	27	1	when	when	SCONJ
ma-257	27	2	n	n	X
ma-257	27	3	=	=	SYM
ma-257	27	4	1	1	NUM
ma-257	27	5	,	,	PUNCT
ma-257	27	6	theyused	theyuse	VERB
ma-257	27	7	the	the	DET
ma-257	27	8	quadrature	quadrature	NOUN
ma-257	27	9	method	method	NOUN
ma-257	27	10	to	to	PART
ma-257	27	11	discuss	discuss	VERB
ma-257	27	12	the	the	DET
ma-257	27	13	multiplicity	multiplicity	NOUN
ma-257	27	14	and	and	CCONJ
ma-257	27	15	uniqueness	uniqueness	NOUN
ma-257	27	16	results	result	NOUN
ma-257	27	17	,	,	PUNCT
ma-257	27	18	while	while	SCONJ
ma-257	27	19	for	for	ADP
ma-257	27	20	n	n	PROPN
ma-257	27	21	>	>	X
ma-257	27	22	1they	1they	PRON
ma-257	27	23	established	establish	VERB
ma-257	27	24	their	their	PRON
ma-257	27	25	existence	existence	NOUN
ma-257	27	26	results	result	NOUN
ma-257	27	27	using	use	VERB
ma-257	27	28	the	the	DET
ma-257	27	29	sub	sub	ADJ
ma-257	27	30	-	-	ADJ
ma-257	27	31	supersolution	supersolution	NOUN
ma-257	27	32	method	method	NOUN
ma-257	27	33	.	.	PUNCT
ma-257	28	1	in	in	ADP
ma-257	28	2	[	[	X
ma-257	28	3	7	7	NUM
ma-257	28	4	]	]	PUNCT
ma-257	28	5	,	,	PUNCT
ma-257	28	6	it	it	PRON
ma-257	28	7	was	be	AUX
ma-257	28	8	discussedthe	discussedthe	ADJ
ma-257	28	9	existence	existence	NOUN
ma-257	28	10	of	of	ADP
ma-257	28	11	positive	positive	ADJ
ma-257	28	12	weak	weak	ADJ
ma-257	28	13	solution	solution	NOUN
ma-257	28	14	to	to	ADP
ma-257	28	15	the	the	DET
ma-257	28	16	non	non	ADJ
ma-257	28	17	-	-	ADJ
ma-257	28	18	linear	linear	ADJ
ma-257	28	19	system	system	NOUN
ma-257	28	20	:	:	PUNCT
ma-257	29	1			PROPN
ma-257	29	2	−∆pu	−∆pu	NOUN
ma-257	30	1	=	=	X
ma-257	31	1	λa(x)[f	λa(x)[f	ADJ
ma-257	31	2	(	(	PUNCT
ma-257	31	3	v)−	v)−	PROPN
ma-257	31	4	1	1	NUM
ma-257	31	5	uα	uα	PROPN
ma-257	31	6	]	]	X
ma-257	31	7	,	,	PUNCT
ma-257	31	8	x	x	PUNCT
ma-257	31	9	∈	∈	PROPN
ma-257	31	10	ω	ω	PROPN
ma-257	31	11	,	,	PUNCT
ma-257	31	12	−∆qv	−∆qv	PROPN
ma-257	31	13	=	=	PUNCT
ma-257	32	1	λb(x)[g(u)−	λb(x)[g(u)−	NOUN
ma-257	32	2	1	1	NUM
ma-257	32	3	vβ	vβ	NOUN
ma-257	32	4	]	]	X
ma-257	32	5	,	,	PUNCT
ma-257	32	6	x	x	PUNCT
ma-257	32	7	∈	∈	PROPN
ma-257	32	8	ω	ω	PROPN
ma-257	32	9	,	,	PUNCT
ma-257	32	10	u	u	NOUN
ma-257	32	11	=	=	NOUN
ma-257	32	12	0	0	PUNCT
ma-257	32	13	=	=	SYM
ma-257	32	14	v	v	NOUN
ma-257	32	15	,	,	PUNCT
ma-257	32	16	x	x	X
ma-257	32	17	∈	∈	PROPN
ma-257	32	18	∂ω	∂ω	PROPN
ma-257	32	19	,	,	PUNCT
ma-257	32	20	(	(	PUNCT
ma-257	32	21	4	4	X
ma-257	32	22	)	)	PUNCT
ma-257	32	23	where	where	SCONJ
ma-257	32	24	∆rw	∆rw	X
ma-257	32	25	=	=	SYM
ma-257	32	26	div(|∇w	div(|∇w	PROPN
ma-257	32	27	|r−2∇w	|r−2∇w	NUM
ma-257	32	28	)	)	PUNCT
ma-257	32	29	is	be	AUX
ma-257	32	30	the	the	DET
ma-257	32	31	r	r	NOUN
ma-257	32	32	-	-	PUNCT
ma-257	32	33	laplacian	laplacian	ADJ
ma-257	32	34	(	(	PUNCT
ma-257	32	35	r	r	NOUN
ma-257	32	36	=	=	SYM
ma-257	32	37	p	p	NOUN
ma-257	32	38	,	,	PUNCT
ma-257	32	39	q	q	NOUN
ma-257	32	40	)	)	PUNCT
ma-257	32	41	,	,	PUNCT
ma-257	32	42	r	r	NOUN
ma-257	32	43	>	>	X
ma-257	32	44	1	1	NUM
ma-257	32	45	.	.	PUNCT
ma-257	33	1	here	here	ADV
ma-257	33	2	f	f	X
ma-257	33	3	,	,	PUNCT
ma-257	33	4	g	g	PROPN
ma-257	33	5	are	be	AUX
ma-257	33	6	c1	c1	PROPN
ma-257	33	7	increasingfunctions	increasingfunction	NOUN
ma-257	33	8	such	such	ADJ
ma-257	33	9	that	that	SCONJ
ma-257	33	10	f	f	NOUN
ma-257	33	11	,	,	PUNCT
ma-257	33	12	g	g	PROPN
ma-257	33	13	:	:	PUNCT
ma-257	33	14	r+	r+	X
ma-257	33	15	→	→	SYM
ma-257	33	16	r+	r+	X
ma-257	33	17	;	;	PUNCT
ma-257	33	18	f	f	PROPN
ma-257	33	19	(	(	PUNCT
ma-257	33	20	ω	ω	PROPN
ma-257	33	21	)	)	PUNCT
ma-257	33	22	>	>	X
ma-257	33	23	0	0	NUM
ma-257	33	24	,	,	PUNCT
ma-257	33	25	g(ω	g(ω	PROPN
ma-257	33	26	)	)	PUNCT
ma-257	33	27	>	>	X
ma-257	33	28	0	0	PUNCT
ma-257	34	1	for	for	ADP
ma-257	34	2	ω	ω	PROPN
ma-257	34	3	>	>	X
ma-257	34	4	0	0	PROPN
ma-257	35	1	and	and	CCONJ
ma-257	35	2	lim	lim	PROPN
ma-257	35	3	ω→∞	ω→∞	NUM
ma-257	35	4	f	f	X
ma-257	35	5	(	(	PUNCT
ma-257	35	6	mg(ω	mg(ω	X
ma-257	35	7	)	)	PUNCT
ma-257	35	8	1	1	NUM
ma-257	35	9	q−1	q−1	PROPN
ma-257	35	10	)	)	PUNCT
ma-257	35	11	ωp−1	ωp−1	NOUN
ma-257	35	12	=	=	SYM
ma-257	35	13	0	0	NUM
ma-257	35	14	∀	∀	NOUN
ma-257	35	15	m	m	VERB
ma-257	35	16	>	>	X
ma-257	35	17	0	0	X
ma-257	35	18	.	.	PUNCT
ma-257	36	1	with	with	ADP
ma-257	36	2	c1	c1	PROPN
ma-257	36	3	sign	sign	NOUN
ma-257	36	4	-	-	PUNCT
ma-257	36	5	changing	change	VERB
ma-257	36	6	functions	function	NOUN
ma-257	36	7	a(x	a(x	NOUN
ma-257	36	8	)	)	PUNCT
ma-257	36	9	,	,	PUNCT
ma-257	36	10	b(x	b(x	NOUN
ma-257	36	11	)	)	PUNCT
ma-257	36	12	that	that	PRON
ma-257	36	13	perhaps	perhaps	ADV
ma-257	36	14	have	have	AUX
ma-257	36	15	negative	negative	ADJ
ma-257	36	16	valuesnearby	valuesnearby	ADP
ma-257	36	17	the	the	DET
ma-257	36	18	boundary	boundary	NOUN
ma-257	36	19	.	.	PUNCT
ma-257	37	1	see	see	VERB
ma-257	37	2	[	[	X
ma-257	37	3	8	8	NUM
ma-257	37	4	]	]	PUNCT
ma-257	37	5	,	,	PUNCT
ma-257	37	6	where	where	SCONJ
ma-257	37	7	system	system	NOUN
ma-257	37	8	(	(	PUNCT
ma-257	37	9	4	4	X
ma-257	37	10	)	)	PUNCT
ma-257	37	11	studied	study	VERB
ma-257	37	12	by	by	ADP
ma-257	37	13	some	some	DET
ma-257	37	14	authors	author	NOUN
ma-257	37	15	when	when	SCONJ
ma-257	37	16	p	p	NOUN
ma-257	37	17	=	=	X
ma-257	37	18	q	q	NOUN
ma-257	37	19	=	=	PUNCT
ma-257	37	20	2.also	2.also	NUM
ma-257	37	21	,	,	PUNCT
ma-257	37	22	we	we	PRON
ma-257	37	23	studied	study	VERB
ma-257	37	24	in	in	ADP
ma-257	37	25	[	[	X
ma-257	37	26	9	9	NUM
ma-257	37	27	]	]	PUNCT
ma-257	37	28	the	the	DET
ma-257	37	29	existence	existence	NOUN
ma-257	37	30	and	and	CCONJ
ma-257	37	31	non	non	ADJ
ma-257	37	32	-	-	ADJ
ma-257	37	33	existence	existence	ADJ
ma-257	37	34	results	result	NOUN
ma-257	37	35	of	of	ADP
ma-257	37	36	positive	positive	ADJ
ma-257	37	37	weak	weak	ADJ
ma-257	37	38	solution	solution	NOUN
ma-257	37	39	of	of	ADP
ma-257	37	40	(	(	PUNCT
ma-257	37	41	1	1	X
ma-257	37	42	)	)	PUNCT
ma-257	37	43	incase	incase	VERB
ma-257	37	44	p	p	NOUN
ma-257	37	45	=	=	PUNCT
ma-257	37	46	q	q	NOUN
ma-257	37	47	=	=	SYM
ma-257	37	48	2	2	NUM
ma-257	37	49	,	,	PUNCT
ma-257	37	50	where	where	SCONJ
ma-257	37	51	f	f	PROPN
ma-257	37	52	,	,	PUNCT
ma-257	37	53	g	g	PROPN
ma-257	37	54	are	be	AUX
ma-257	37	55	c1	c1	NOUN
ma-257	37	56	increasing	increase	VERB
ma-257	37	57	functions	function	NOUN
ma-257	37	58	,	,	PUNCT
ma-257	37	59	lim	lim	PROPN
ma-257	37	60	ω→∞	ω→∞	NUM
ma-257	37	61	f	f	PROPN
ma-257	37	62	(	(	PUNCT
ma-257	37	63	ω	ω	PROPN
ma-257	37	64	,	,	PUNCT
ma-257	37	65	mg(ω	mg(ω	PROPN
ma-257	37	66	,	,	PUNCT
ma-257	37	67	ω	ω	NOUN
ma-257	37	68	)	)	PUNCT
ma-257	37	69	)	)	PUNCT
ma-257	38	1	ω	ω	X
ma-257	38	2	=	=	SYM
ma-257	38	3	0	0	NUM
ma-257	38	4	∀	∀	NOUN
ma-257	38	5	m	m	VERB
ma-257	38	6	>	>	X
ma-257	38	7	0	0	PUNCT
ma-257	39	1	and	and	CCONJ
ma-257	39	2	lim	lim	PROPN
ma-257	39	3	ω→∞	ω→∞	NUM
ma-257	39	4	g(ω	g(ω	PROPN
ma-257	39	5	,	,	PUNCT
ma-257	39	6	ω	ω	NUM
ma-257	39	7	)	)	PUNCT
ma-257	39	8	ω	ω	NOUN
ma-257	39	9	=	=	SYM
ma-257	39	10	0	0	PROPN
ma-257	39	11	.	.	PUNCT
ma-257	40	1	with	with	ADP
ma-257	40	2	c1	c1	PROPN
ma-257	40	3	sign	sign	NOUN
ma-257	40	4	-	-	PUNCT
ma-257	40	5	changing	change	VERB
ma-257	40	6	functions	function	NOUN
ma-257	40	7	a(x	a(x	NOUN
ma-257	40	8	)	)	PUNCT
ma-257	40	9	,	,	PUNCT
ma-257	40	10	b(x	b(x	NOUN
ma-257	40	11	)	)	PUNCT
ma-257	40	12	that	that	PRON
ma-257	40	13	perhaps	perhaps	ADV
ma-257	40	14	have	have	AUX
ma-257	40	15	negative	negative	ADJ
ma-257	40	16	valuesnearby	valuesnearby	ADP
ma-257	40	17	the	the	DET
ma-257	40	18	boundary	boundary	NOUN
ma-257	40	19	.	.	PUNCT
ma-257	41	1	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	41	2	eur	eur	PROPN
ma-257	41	3	.	.	PUNCT
ma-257	42	1	j.	j.	PROPN
ma-257	42	2	math	math	PROPN
ma-257	42	3	.	.	PUNCT
ma-257	43	1	anal	anal	PROPN
ma-257	43	2	.	.	PUNCT
ma-257	44	1	10.28924	10.28924	NUM
ma-257	44	2	/	/	SYM
ma-257	44	3	ada	ada	PROPN
ma-257	44	4	/	/	SYM
ma-257	44	5	ma.5.1	ma.5.1	PROPN
ma-257	44	6	3our	3our	NUM
ma-257	44	7	first	first	ADJ
ma-257	44	8	aim	aim	NOUN
ma-257	44	9	of	of	ADP
ma-257	44	10	this	this	DET
ma-257	44	11	paper	paper	NOUN
ma-257	44	12	is	be	AUX
ma-257	44	13	to	to	PART
ma-257	44	14	study	study	VERB
ma-257	44	15	system	system	NOUN
ma-257	44	16	(	(	PUNCT
ma-257	44	17	1	1	NUM
ma-257	44	18	)	)	PUNCT
ma-257	44	19	as	as	ADP
ma-257	44	20	an	an	DET
ma-257	44	21	extension	extension	NOUN
ma-257	44	22	of	of	ADP
ma-257	44	23	system	system	NOUN
ma-257	44	24	(	(	PUNCT
ma-257	44	25	4	4	NUM
ma-257	44	26	)	)	PUNCT
ma-257	44	27	with	with	ADP
ma-257	44	28	c1	c1	PROPN
ma-257	44	29	increasingfunctions	increasingfunctions	PROPN
ma-257	44	30	f	f	PROPN
ma-257	44	31	,	,	PUNCT
ma-257	44	32	g	g	PROPN
ma-257	44	33	satisfying	satisfy	VERB
ma-257	44	34	lim	lim	PROPN
ma-257	44	35	ξ→∞	ξ→∞	PROPN
ma-257	44	36	f	f	X
ma-257	44	37	(	(	PUNCT
ma-257	44	38	ξ	ξ	PROPN
ma-257	44	39	,	,	PUNCT
ma-257	44	40	m[g(ξ	m[g(ξ	NOUN
ma-257	44	41	,	,	PUNCT
ma-257	44	42	ξ	ξ	NOUN
ma-257	44	43	)	)	PUNCT
ma-257	44	44	]	]	PUNCT
ma-257	44	45	1	1	NUM
ma-257	44	46	q−1	q−1	PROPN
ma-257	44	47	)	)	PUNCT
ma-257	44	48	ξp−1	ξp−1	PROPN
ma-257	44	49	=	=	PUNCT
ma-257	44	50	0	0	NUM
ma-257	44	51	∀m	∀m	PROPN
ma-257	44	52	>	>	X
ma-257	44	53	0	0	PROPN
ma-257	44	54	,	,	PUNCT
ma-257	44	55	lim	lim	PROPN
ma-257	44	56	ξ→∞	ξ→∞	NUM
ma-257	44	57	g(ξ	g(ξ	PROPN
ma-257	44	58	,	,	PUNCT
ma-257	44	59	ξ	ξ	NOUN
ma-257	44	60	)	)	PUNCT
ma-257	44	61	ξq−1	ξq−1	NOUN
ma-257	44	62	=	=	PUNCT
ma-257	44	63	0	0	X
ma-257	44	64	.	.	PUNCT
ma-257	45	1	on	on	ADP
ma-257	45	2	the	the	DET
ma-257	45	3	other	other	ADJ
ma-257	45	4	side	side	NOUN
ma-257	45	5	,	,	PUNCT
ma-257	45	6	many	many	ADJ
ma-257	45	7	authors	author	NOUN
ma-257	45	8	have	have	VERB
ma-257	45	9	an	an	DET
ma-257	45	10	interest	interest	NOUN
ma-257	45	11	in	in	ADP
ma-257	45	12	studying	study	VERB
ma-257	45	13	the	the	DET
ma-257	45	14	stability	stability	NOUN
ma-257	45	15	and	and	CCONJ
ma-257	45	16	instability	instability	NOUN
ma-257	45	17	ofpositive	ofpositive	ADJ
ma-257	45	18	solution	solution	NOUN
ma-257	45	19	to	to	ADP
ma-257	45	20	semiposiotne	semiposiotne	NOUN
ma-257	45	21	[	[	X
ma-257	45	22	10–12	10–12	NUM
ma-257	45	23	]	]	PUNCT
ma-257	45	24	,	,	PUNCT
ma-257	45	25	linear	linear	PROPN
ma-257	46	1	[	[	X
ma-257	46	2	13	13	NUM
ma-257	46	3	]	]	PUNCT
ma-257	46	4	,	,	PUNCT
ma-257	46	5	semilinear	semilinear	NOUN
ma-257	47	1	[	[	X
ma-257	47	2	14–17	14–17	NUM
ma-257	47	3	]	]	PUNCT
ma-257	47	4	and	and	CCONJ
ma-257	47	5	fractional	fractional	ADJ
ma-257	47	6	[	[	X
ma-257	47	7	19,20	19,20	X
ma-257	47	8	]	]	PUNCT
ma-257	47	9	sys	sys	PROPN
ma-257	47	10	-	-	NOUN
ma-257	47	11	tems	tem	NOUN
ma-257	47	12	,	,	PUNCT
ma-257	47	13	they	they	PRON
ma-257	47	14	are	be	AUX
ma-257	47	15	used	use	VERB
ma-257	47	16	in	in	ADP
ma-257	47	17	several	several	ADJ
ma-257	47	18	applications	application	NOUN
ma-257	47	19	such	such	ADJ
ma-257	47	20	as	as	ADP
ma-257	47	21	fluid	fluid	ADJ
ma-257	47	22	mechanics	mechanic	NOUN
ma-257	47	23	,	,	PUNCT
ma-257	47	24	newtonian	newtonian	ADJ
ma-257	47	25	fluids	fluid	NOUN
ma-257	47	26	,	,	PUNCT
ma-257	47	27	populationdynamics	populationdynamic	NOUN
ma-257	47	28	,	,	PUNCT
ma-257	47	29	reaction	reaction	NOUN
ma-257	47	30	-	-	PUNCT
ma-257	47	31	diffusion	diffusion	NOUN
ma-257	47	32	problems	problem	NOUN
ma-257	47	33	,	,	PUNCT
ma-257	47	34	glaciology	glaciology	NOUN
ma-257	47	35	,	,	PUNCT
ma-257	47	36	etc	etc	X
ma-257	47	37	.	.	X
ma-257	47	38	;	;	PUNCT
ma-257	47	39	see	see	VERB
ma-257	47	40	[	[	X
ma-257	47	41	20–22	20–22	NUM
ma-257	47	42	]	]	SYM
ma-257	47	43	.	.	PUNCT
ma-257	48	1	shivaji	shivaji	PROPN
ma-257	48	2	and	and	CCONJ
ma-257	48	3	brown	brown	ADJ
ma-257	48	4	in	in	ADP
ma-257	48	5	[	[	X
ma-257	48	6	11	11	NUM
ma-257	48	7	]	]	PUNCT
ma-257	48	8	discussed	discuss	VERB
ma-257	48	9	the	the	DET
ma-257	48	10	stability	stability	NOUN
ma-257	48	11	properties	property	NOUN
ma-257	48	12	of	of	ADP
ma-257	48	13	positive	positive	ADJ
ma-257	48	14	solution	solution	NOUN
ma-257	48	15	for	for	ADP
ma-257	48	16	the	the	DET
ma-257	48	17	system:−∆u	system:−∆u	NOUN
ma-257	48	18	=	=	PRON
ma-257	48	19	λf	λf	ADJ
ma-257	48	20	(	(	PUNCT
ma-257	48	21	u	u	NOUN
ma-257	48	22	)	)	PUNCT
ma-257	48	23	,	,	PUNCT
ma-257	48	24	x	x	PUNCT
ma-257	48	25	∈	∈	PROPN
ma-257	48	26	ω	ω	PROPN
ma-257	48	27	,	,	PUNCT
ma-257	48	28	u	u	NOUN
ma-257	48	29	=	=	PROPN
ma-257	48	30	0	0	PROPN
ma-257	48	31	,	,	PUNCT
ma-257	48	32	x	x	X
ma-257	48	33	∈	∈	PROPN
ma-257	48	34	∂ω	∂ω	PROPN
ma-257	48	35	,	,	PUNCT
ma-257	48	36	(	(	PUNCT
ma-257	48	37	5	5	X
ma-257	48	38	)	)	PUNCT
ma-257	48	39	they	they	PRON
ma-257	48	40	proved	prove	VERB
ma-257	48	41	that	that	SCONJ
ma-257	48	42	every	every	DET
ma-257	48	43	positive	positive	ADJ
ma-257	48	44	solution	solution	NOUN
ma-257	48	45	of	of	ADP
ma-257	48	46	(	(	PUNCT
ma-257	48	47	5	5	NUM
ma-257	48	48	)	)	PUNCT
ma-257	48	49	is	be	AUX
ma-257	48	50	unstable	unstable	ADJ
ma-257	48	51	when	when	SCONJ
ma-257	48	52	f	f	X
ma-257	48	53	(	(	PUNCT
ma-257	48	54	0	0	NUM
ma-257	48	55	)	)	PUNCT
ma-257	48	56	≤	≤	NOUN
ma-257	48	57	0	0	NUM
ma-257	49	1	and	and	CCONJ
ma-257	49	2	f	f	X
ma-257	49	3	′′	′′	PROPN
ma-257	49	4	≥	≥	NOUN
ma-257	49	5	0	0	NUM
ma-257	49	6	.	.	PUNCT
ma-257	49	7	see	see	VERB
ma-257	49	8	[	[	X
ma-257	49	9	12	12	NUM
ma-257	49	10	]	]	PUNCT
ma-257	49	11	,	,	PUNCT
ma-257	49	12	wheretertikas	wheretertikas	PROPN
ma-257	49	13	proved	prove	VERB
ma-257	49	14	the	the	DET
ma-257	49	15	non	non	ADJ
ma-257	49	16	-	-	ADJ
ma-257	49	17	monotone	monotone	ADJ
ma-257	49	18	case	case	NOUN
ma-257	49	19	.	.	PUNCT
ma-257	50	1	maya	maya	PROPN
ma-257	50	2	and	and	CCONJ
ma-257	50	3	shivaji	shivaji	PROPN
ma-257	50	4	in	in	ADP
ma-257	50	5	[	[	X
ma-257	50	6	16	16	NUM
ma-257	50	7	]	]	PUNCT
ma-257	50	8	overcame	overcome	VERB
ma-257	50	9	the	the	DET
ma-257	50	10	non	non	ADJ
ma-257	50	11	-	-	ADJ
ma-257	50	12	monotone	monotone	ADJ
ma-257	50	13	casethrough	casethrough	NOUN
ma-257	50	14	re	re	NOUN
ma-257	50	15	-	-	VERB
ma-257	50	16	formulating	formulate	VERB
ma-257	50	17	f	f	NOUN
ma-257	50	18	as	as	ADP
ma-257	50	19	a	a	DET
ma-257	50	20	combination	combination	NOUN
ma-257	50	21	of	of	ADP
ma-257	50	22	a	a	DET
ma-257	50	23	linear	linear	ADJ
ma-257	50	24	and	and	CCONJ
ma-257	50	25	monotone	monotone	ADJ
ma-257	50	26	function	function	NOUN
ma-257	50	27	.	.	PUNCT
ma-257	51	1	simon	simon	PROPN
ma-257	51	2	and	and	CCONJ
ma-257	51	3	karatsongave	karatsongave	VERB
ma-257	51	4	a	a	DET
ma-257	51	5	direct	direct	ADJ
ma-257	51	6	proof	proof	NOUN
ma-257	51	7	of	of	ADP
ma-257	51	8	the	the	DET
ma-257	51	9	result	result	NOUN
ma-257	51	10	(	(	PUNCT
ma-257	51	11	see	see	VERB
ma-257	51	12	[	[	X
ma-257	51	13	14	14	NUM
ma-257	51	14	]	]	SYM
ma-257	51	15	)	)	PUNCT
ma-257	51	16	.	.	PUNCT
ma-257	52	1	in	in	ADP
ma-257	52	2	summary	summary	NOUN
ma-257	52	3	,	,	PUNCT
ma-257	52	4	if	if	SCONJ
ma-257	52	5	f	f	PROPN
ma-257	52	6	(	(	PUNCT
ma-257	52	7	0	0	NUM
ma-257	52	8	)	)	PUNCT
ma-257	52	9	≥	≥	NOUN
ma-257	52	10	0	0	NUM
ma-257	52	11	(	(	PUNCT
ma-257	52	12	≤	≤	NOUN
ma-257	52	13	0	0	NUM
ma-257	52	14	)	)	PUNCT
ma-257	52	15	and	and	CCONJ
ma-257	52	16	f	f	X
ma-257	52	17	′′	′′	PROPN
ma-257	52	18	<	<	X
ma-257	52	19	0	0	PUNCT
ma-257	53	1	(	(	PUNCT
ma-257	53	2	>	>	X
ma-257	53	3	0	0	NUM
ma-257	53	4	)	)	PUNCT
ma-257	53	5	,	,	PUNCT
ma-257	53	6	thenevery	thenevery	NOUN
ma-257	53	7	positive	positive	ADJ
ma-257	53	8	solution	solution	NOUN
ma-257	53	9	of	of	ADP
ma-257	53	10	(	(	PUNCT
ma-257	53	11	5	5	NUM
ma-257	53	12	)	)	PUNCT
ma-257	53	13	is	be	AUX
ma-257	53	14	stable	stable	ADJ
ma-257	53	15	(	(	PUNCT
ma-257	53	16	unstable	unstable	ADJ
ma-257	53	17	)	)	PUNCT
ma-257	53	18	.	.	PUNCT
ma-257	54	1	also	also	ADV
ma-257	54	2	in	in	ADP
ma-257	54	3	[	[	X
ma-257	54	4	9	9	NUM
ma-257	54	5	]	]	PUNCT
ma-257	54	6	,	,	PUNCT
ma-257	54	7	we	we	PRON
ma-257	54	8	studied	study	VERB
ma-257	54	9	the	the	DET
ma-257	54	10	stability	stability	NOUN
ma-257	54	11	and	and	CCONJ
ma-257	54	12	instabilityproperties	instabilitypropertie	NOUN
ma-257	54	13	of	of	ADP
ma-257	54	14	system	system	NOUN
ma-257	54	15	(	(	PUNCT
ma-257	54	16	1	1	NUM
ma-257	54	17	)	)	PUNCT
ma-257	54	18	in	in	ADP
ma-257	54	19	case	case	NOUN
ma-257	54	20	p	p	X
ma-257	54	21	=	=	X
ma-257	54	22	q	q	NOUN
ma-257	54	23	=	=	NOUN
ma-257	54	24	2	2	NUM
ma-257	54	25	,	,	PUNCT
ma-257	54	26	under	under	ADP
ma-257	54	27	certain	certain	ADJ
ma-257	54	28	conditions	condition	NOUN
ma-257	54	29	such	such	ADJ
ma-257	54	30	that	that	SCONJ
ma-257	54	31	every	every	DET
ma-257	54	32	weak	weak	ADJ
ma-257	54	33	solution	solution	NOUN
ma-257	54	34	isstable	isstable	ADJ
ma-257	54	35	near	near	ADP
ma-257	54	36	the	the	DET
ma-257	54	37	boundary	boundary	NOUN
ma-257	54	38	;	;	PUNCT
ma-257	54	39	otherwise	otherwise	ADV
ma-257	54	40	,	,	PUNCT
ma-257	54	41	it	it	PRON
ma-257	54	42	is	be	AUX
ma-257	54	43	unstable	unstable	ADJ
ma-257	54	44	.	.	PUNCT
ma-257	55	1	in	in	ADP
ma-257	55	2	[	[	X
ma-257	55	3	23	23	NUM
ma-257	55	4	]	]	PUNCT
ma-257	55	5	,	,	PUNCT
ma-257	55	6	some	some	DET
ma-257	55	7	authors	author	NOUN
ma-257	55	8	investigated	investigate	VERB
ma-257	55	9	the	the	DET
ma-257	55	10	stabilityof	stabilityof	PROPN
ma-257	55	11	non	non	ADJ
ma-257	55	12	-	-	ADJ
ma-257	55	13	negative	negative	ADJ
ma-257	55	14	weak	weak	ADJ
ma-257	55	15	solution	solution	NOUN
ma-257	55	16	for	for	ADP
ma-257	55	17	the	the	DET
ma-257	55	18	nonlinear	nonlinear	ADJ
ma-257	55	19	system:−∆pu	system:−∆pu	PROPN
ma-257	55	20	=	=	PRON
ma-257	56	1	λf	λf	X
ma-257	56	2	(	(	PUNCT
ma-257	56	3	x	x	NOUN
ma-257	56	4	,	,	PUNCT
ma-257	56	5	u	u	NOUN
ma-257	56	6	)	)	PUNCT
ma-257	56	7	,	,	PUNCT
ma-257	56	8	x	x	PUNCT
ma-257	56	9	∈	∈	PROPN
ma-257	56	10	ω	ω	PROPN
ma-257	56	11	,	,	PUNCT
ma-257	56	12	bu	bu	ADP
ma-257	56	13	=	=	NOUN
ma-257	56	14	0	0	PROPN
ma-257	56	15	,	,	PUNCT
ma-257	56	16	x	x	X
ma-257	56	17	∈	∈	PROPN
ma-257	56	18	∂ω	∂ω	PROPN
ma-257	56	19	,	,	PUNCT
ma-257	56	20	(	(	PUNCT
ma-257	56	21	6	6	NUM
ma-257	56	22	)	)	PUNCT
ma-257	56	23	where	where	SCONJ
ma-257	56	24	f	f	NOUN
ma-257	56	25	:	:	PUNCT
ma-257	56	26	ω×	ω×	PUNCT
ma-257	57	1	[	[	X
ma-257	57	2	0,∞)→	0,∞)→	NOUN
ma-257	57	3	r	r	NOUN
ma-257	57	4	be	be	VERB
ma-257	57	5	a	a	DET
ma-257	57	6	continuous	continuous	ADJ
ma-257	57	7	function	function	NOUN
ma-257	57	8	.	.	PUNCT
ma-257	58	1	they	they	PRON
ma-257	58	2	discussed	discuss	VERB
ma-257	58	3	(	(	PUNCT
ma-257	58	4	6	6	NUM
ma-257	58	5	)	)	PUNCT
ma-257	59	1	when	when	SCONJ
ma-257	59	2	f	f	PROPN
ma-257	59	3	(	(	PUNCT
ma-257	59	4	x	x	NOUN
ma-257	59	5	,	,	PUNCT
ma-257	59	6	u	u	NOUN
ma-257	59	7	)	)	PUNCT
ma-257	59	8	=	=	SYM
ma-257	60	1	w(x)f	w(x)f	PROPN
ma-257	60	2	(	(	PUNCT
ma-257	60	3	u),where	u),where	ADP
ma-257	60	4	w(x	w(x	NOUN
ma-257	60	5	)	)	PUNCT
ma-257	60	6	is	be	AUX
ma-257	60	7	a	a	DET
ma-257	60	8	continuous	continuous	ADJ
ma-257	60	9	weight	weight	NOUN
ma-257	60	10	function	function	NOUN
ma-257	60	11	.	.	PUNCT
ma-257	61	1	they	they	PRON
ma-257	61	2	showed	show	VERB
ma-257	61	3	that	that	SCONJ
ma-257	61	4	every	every	DET
ma-257	61	5	positive	positive	ADJ
ma-257	61	6	solution	solution	NOUN
ma-257	61	7	is	be	AUX
ma-257	61	8	unstable(stable	unstable(stable	ADJ
ma-257	61	9	)	)	PUNCT
ma-257	61	10	if	if	SCONJ
ma-257	61	11	f	f	PROPN
ma-257	61	12	(	(	PUNCT
ma-257	61	13	x	x	NOUN
ma-257	61	14	,	,	PUNCT
ma-257	61	15	u	u	NOUN
ma-257	61	16	)	)	PUNCT
ma-257	61	17	up−1	up−1	PROPN
ma-257	61	18	is	be	AUX
ma-257	61	19	strictly	strictly	ADV
ma-257	61	20	increasing	increase	VERB
ma-257	61	21	(	(	PUNCT
ma-257	61	22	decreasing	decrease	VERB
ma-257	61	23	)	)	PUNCT
ma-257	61	24	function	function	NOUN
ma-257	61	25	.	.	PUNCT
ma-257	62	1	our	our	PRON
ma-257	62	2	second	second	ADJ
ma-257	62	3	aim	aim	NOUN
ma-257	62	4	of	of	ADP
ma-257	62	5	this	this	DET
ma-257	62	6	paper	paper	NOUN
ma-257	62	7	is	be	AUX
ma-257	62	8	to	to	PART
ma-257	62	9	extend	extend	VERB
ma-257	62	10	these	these	DET
ma-257	62	11	results	result	NOUN
ma-257	62	12	to	to	ADP
ma-257	62	13	(	(	PUNCT
ma-257	62	14	1	1	X
ma-257	62	15	)	)	PUNCT
ma-257	62	16	with	with	ADP
ma-257	62	17	different	different	ADJ
ma-257	62	18	choices	choice	NOUN
ma-257	62	19	of	of	ADP
ma-257	62	20	f	f	PROPN
ma-257	62	21	,	,	PUNCT
ma-257	62	22	g.	g.	PROPN
ma-257	62	23	forfurther	forfurther	PROPN
ma-257	62	24	stability	stability	NOUN
ma-257	62	25	and	and	CCONJ
ma-257	62	26	instability	instability	NOUN
ma-257	62	27	results	result	NOUN
ma-257	62	28	on	on	ADP
ma-257	62	29	elliptic	elliptic	ADJ
ma-257	62	30	systems	system	NOUN
ma-257	62	31	(	(	PUNCT
ma-257	62	32	see	see	VERB
ma-257	62	33	[	[	X
ma-257	62	34	10,17,18,24–26]).let	10,17,18,24–26]).let	NUM
ma-257	62	35	λ1,r	λ1,r	PROPN
ma-257	62	36	>	>	X
ma-257	62	37	0	0	NUM
ma-257	62	38	,	,	PUNCT
ma-257	62	39	r	r	NOUN
ma-257	62	40	=	=	SYM
ma-257	62	41	p	p	NOUN
ma-257	62	42	,	,	PUNCT
ma-257	62	43	q	q	INTJ
ma-257	62	44	,	,	PUNCT
ma-257	62	45	be	be	AUX
ma-257	62	46	the	the	DET
ma-257	62	47	principal	principal	ADJ
ma-257	62	48	eigenvalue	eigenvalue	NOUN
ma-257	62	49	of	of	ADP
ma-257	62	50	the	the	DET
ma-257	62	51	following	follow	VERB
ma-257	62	52	eigenvalue	eigenvalue	PROPN
ma-257	62	53	problem	problem	NOUN
ma-257	62	54	toaccurately	toaccurately	ADV
ma-257	62	55	state	state	VERB
ma-257	62	56	our	our	PRON
ma-257	62	57	existence	existence	NOUN
ma-257	62	58	results:−∆rϕ	results:−∆rϕ	PROPN
ma-257	62	59	=	=	SYM
ma-257	62	60	λ|ϕ|r−2ϕ	λ|ϕ|r−2ϕ	PROPN
ma-257	62	61	,	,	PUNCT
ma-257	62	62	x	x	X
ma-257	62	63	∈	∈	PROPN
ma-257	62	64	ω	ω	PROPN
ma-257	62	65	,	,	PUNCT
ma-257	62	66	ϕ	ϕ	X
ma-257	62	67	=	=	SYM
ma-257	62	68	0	0	NUM
ma-257	62	69	,	,	PUNCT
ma-257	62	70	x	x	X
ma-257	62	71	∈	∈	PROPN
ma-257	62	72	∂ω	∂ω	PROPN
ma-257	62	73	,	,	PUNCT
ma-257	62	74	(	(	PUNCT
ma-257	62	75	7	7	X
ma-257	62	76	)	)	PUNCT
ma-257	62	77	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	62	78	eur	eur	PROPN
ma-257	62	79	.	.	PUNCT
ma-257	63	1	j.	j.	PROPN
ma-257	63	2	math	math	PROPN
ma-257	63	3	.	.	PUNCT
ma-257	64	1	anal	anal	PROPN
ma-257	64	2	.	.	PUNCT
ma-257	65	1	10.28924	10.28924	NUM
ma-257	65	2	/	/	SYM
ma-257	65	3	ada	ada	PROPN
ma-257	65	4	/	/	SYM
ma-257	65	5	ma.5.1	ma.5.1	PROPN
ma-257	65	6	4where	4where	NUM
ma-257	65	7	ϕ1,r	ϕ1,r	PROPN
ma-257	65	8	be	be	AUX
ma-257	65	9	the	the	DET
ma-257	65	10	corresponding	correspond	VERB
ma-257	65	11	eigenfunction	eigenfunction	NOUN
ma-257	65	12	satisfying	satisfy	VERB
ma-257	65	13	ϕ1,r	ϕ1,r	PROPN
ma-257	65	14	(	(	PUNCT
ma-257	65	15	x	x	X
ma-257	65	16	)	)	PUNCT
ma-257	65	17	>	>	X
ma-257	65	18	0	0	PUNCT
ma-257	66	1	in	in	ADP
ma-257	66	2	ω	ω	NUM
ma-257	66	3	with	with	ADP
ma-257	66	4	‖ϕ1,r‖∞	‖ϕ1,r‖∞	PUNCT
ma-257	66	5	=	=	SYM
ma-257	66	6	1.suppose	1.suppose	NUM
ma-257	66	7	µ	µ	X
ma-257	66	8	,	,	PUNCT
ma-257	66	9	δ	δ	PROPN
ma-257	66	10	,	,	PUNCT
ma-257	66	11	m	m	VERB
ma-257	66	12	>	>	X
ma-257	66	13	0	0	PUNCT
ma-257	66	14	be	be	AUX
ma-257	66	15	such	such	ADJ
ma-257	66	16	that	that	SCONJ
ma-257	66	17	r	r	NOUN
ma-257	66	18	sr	sr	PROPN
ma-257	66	19	(	(	PUNCT
ma-257	66	20	1−	1−	NUM
ma-257	66	21	r	r	NOUN
ma-257	66	22	s	s	NOUN
ma-257	66	23	s	s	X
ma-257	66	24	+	+	NOUN
ma-257	66	25	1	1	NUM
ma-257	66	26	)	)	PUNCT
ma-257	66	27	|∇ϕ1,r	|∇ϕ1,r	NOUN
ma-257	66	28	|r	|r	PROPN
ma-257	66	29	≥	≥	PROPN
ma-257	66	30	m	m	PROPN
ma-257	66	31	,	,	PUNCT
ma-257	66	32	x	x	SYM
ma-257	66	33	∈	∈	PROPN
ma-257	66	34	ω̄δ	ω̄δ	PROPN
ma-257	66	35	,	,	PUNCT
ma-257	66	36	(	(	PUNCT
ma-257	66	37	8)	8)	NUM
ma-257	66	38	µ	µ	PRON
ma-257	66	39	≤	≤	NOUN
ma-257	66	40	ϕ1,r	ϕ1,r	PROPN
ma-257	66	41	≤	≤	NOUN
ma-257	66	42	1	1	NUM
ma-257	66	43	,	,	PUNCT
ma-257	66	44	x	x	SYM
ma-257	66	45	∈	∈	NOUN
ma-257	66	46	ω−	ω−	VERB
ma-257	66	47	ω̄δ	ω̄δ	PROPN
ma-257	66	48	,	,	PUNCT
ma-257	66	49	(	(	PUNCT
ma-257	66	50	9)for	9)for	PROPN
ma-257	66	51	s	s	NOUN
ma-257	66	52	=	=	SYM
ma-257	66	53	α	α	PROPN
ma-257	66	54	,	,	PUNCT
ma-257	66	55	β	β	X
ma-257	66	56	and	and	CCONJ
ma-257	66	57	sr	sr	PROPN
ma-257	66	58	=	=	PUNCT
ma-257	66	59	(	(	PUNCT
ma-257	66	60	s	s	X
ma-257	66	61	+	+	NOUN
ma-257	66	62	1)r−1	1)r−1	NUM
ma-257	66	63	,	,	PUNCT
ma-257	66	64	where	where	SCONJ
ma-257	66	65	ω̄δ	ω̄δ	PRON
ma-257	66	66	:	:	PUNCT
ma-257	66	67	=	=	SYM
ma-257	66	68	{	{	PUNCT
ma-257	66	69	x	x	PUNCT
ma-257	66	70	∈	∈	PROPN
ma-257	66	71	ω	ω	NUM
ma-257	66	72	|	|	CCONJ
ma-257	66	73	d(x	d(x	NOUN
ma-257	66	74	,	,	PUNCT
ma-257	66	75	∂ω	∂ω	ADJ
ma-257	66	76	)	)	PUNCT
ma-257	66	77	≤	≤	NUM
ma-257	66	78	δ	δ	PROPN
ma-257	66	79	}	}	PUNCT
ma-257	66	80	.	.	PUNCT
ma-257	67	1	by	by	ADP
ma-257	67	2	hopf	hopf	PROPN
ma-257	67	3	’s	’s	PART
ma-257	67	4	lemma	lemma	PROPN
ma-257	67	5	,	,	PUNCT
ma-257	67	6	we	we	PRON
ma-257	67	7	findthis	findthis	VERB
ma-257	67	8	available	available	ADJ
ma-257	67	9	since	since	SCONJ
ma-257	67	10	ϕ1,r	ϕ1,r	PROPN
ma-257	67	11	=	=	SYM
ma-257	67	12	0	0	PUNCT
ma-257	68	1	while	while	SCONJ
ma-257	68	2	|∇ϕ1,r	|∇ϕ1,r	NOUN
ma-257	68	3	|	|	ADV
ma-257	68	4	6=	6=	PROPN
ma-257	68	5	0	0	NUM
ma-257	68	6	on	on	ADP
ma-257	68	7	∂ω	∂ω	PROPN
ma-257	69	1	.	.	PUNCT
ma-257	70	1	furthermore	furthermore	ADV
ma-257	70	2	,	,	PUNCT
ma-257	70	3	we	we	PRON
ma-257	70	4	suppose	suppose	VERB
ma-257	70	5	er	er	INTJ
ma-257	70	6	∈	∈	PROPN
ma-257	70	7	w	w	PROPN
ma-257	70	8	1,r	1,r	NUM
ma-257	70	9	0	0	NUM
ma-257	70	10	(	(	PUNCT
ma-257	70	11	ω	ω	NOUN
ma-257	70	12	)	)	PUNCT
ma-257	70	13	bethe	bethe	ADJ
ma-257	70	14	unique	unique	ADJ
ma-257	70	15	solution	solution	NOUN
ma-257	70	16	of	of	ADP
ma-257	70	17	the	the	DET
ma-257	70	18	problem	problem	NOUN
ma-257	70	19	:	:	PUNCT
ma-257	70	20	−∆rer	−∆rer	NOUN
ma-257	70	21	=	=	SYM
ma-257	70	22	1	1	NUM
ma-257	70	23	,	,	PUNCT
ma-257	70	24	x	x	SYM
ma-257	70	25	∈	∈	PROPN
ma-257	70	26	ω	ω	PROPN
ma-257	70	27	,	,	PUNCT
ma-257	70	28	er	er	INTJ
ma-257	70	29	=	=	SYM
ma-257	70	30	0	0	NUM
ma-257	70	31	,	,	PUNCT
ma-257	70	32	x	x	X
ma-257	70	33	∈	∈	PROPN
ma-257	70	34	∂ω	∂ω	PROPN
ma-257	70	35	,	,	PUNCT
ma-257	70	36	(	(	PUNCT
ma-257	70	37	10	10	NUM
ma-257	70	38	)	)	PUNCT
ma-257	70	39	where	where	SCONJ
ma-257	70	40	∂	∂	ADJ
ma-257	70	41	∂n	∂n	PROPN
ma-257	70	42	is	be	AUX
ma-257	70	43	the	the	DET
ma-257	70	44	outer	outer	ADJ
ma-257	70	45	normal	normal	ADJ
ma-257	70	46	derivative	derivative	NOUN
ma-257	70	47	,	,	PUNCT
ma-257	70	48	er	er	INTJ
ma-257	70	49	>	>	X
ma-257	70	50	0	0	PUNCT
ma-257	70	51	in	in	ADP
ma-257	70	52	ω	ω	PROPN
ma-257	70	53	and	and	CCONJ
ma-257	70	54	∂er	∂er	NOUN
ma-257	70	55	∂n	∂n	PROPN
ma-257	70	56	<	<	X
ma-257	70	57	0	0	PUNCT
ma-257	70	58	on	on	ADP
ma-257	70	59	∂ω	∂ω	PROPN
ma-257	70	60	(	(	PUNCT
ma-257	70	61	see	see	VERB
ma-257	70	62	[	[	X
ma-257	70	63	27	27	NUM
ma-257	70	64	]	]	NUM
ma-257	70	65	)	)	PUNCT
ma-257	70	66	.	.	PUNCT
ma-257	71	1	to	to	PART
ma-257	71	2	be	be	AUX
ma-257	71	3	morespecific	morespecific	ADJ
ma-257	71	4	,	,	PUNCT
ma-257	71	5	we	we	PRON
ma-257	71	6	will	will	AUX
ma-257	71	7	split	split	VERB
ma-257	71	8	our	our	PRON
ma-257	71	9	results	result	NOUN
ma-257	71	10	into	into	ADP
ma-257	71	11	two	two	NUM
ma-257	71	12	cases	case	NOUN
ma-257	71	13	:	:	PUNCT
ma-257	71	14	•	•	NUM
ma-257	71	15	case(i	case(i	PROPN
ma-257	71	16	):	):	PUNCT
ma-257	71	17	when	when	SCONJ
ma-257	71	18	x	x	SYM
ma-257	71	19	∈	∈	PROPN
ma-257	71	20	ω̄δ	ω̄δ	PROPN
ma-257	71	21	;	;	PUNCT
ma-257	71	22	assume	assume	VERB
ma-257	71	23	a(x	a(x	NOUN
ma-257	71	24	)	)	PUNCT
ma-257	71	25	,	,	PUNCT
ma-257	71	26	b(x	b(x	NOUN
ma-257	71	27	)	)	PUNCT
ma-257	71	28	<	<	X
ma-257	71	29	0	0	NUM
ma-257	71	30	with	with	ADP
ma-257	71	31	a0	a0	PROPN
ma-257	71	32	,	,	PUNCT
ma-257	71	33	a0	a0	PROPN
ma-257	71	34	,	,	PUNCT
ma-257	71	35	b0	b0	NOUN
ma-257	71	36	,	,	PUNCT
ma-257	71	37	b0	b0	NOUN
ma-257	71	38	>	>	X
ma-257	71	39	0	0	NUM
ma-257	72	1	:	:	PUNCT
ma-257	72	2	−a0	−a0	ADJ
ma-257	72	3	≤	≤	NUM
ma-257	72	4	a(x	a(x	PROPN
ma-257	72	5	)	)	PUNCT
ma-257	72	6	≤	≤	PUNCT
ma-257	73	1	−a0	−a0	PROPN
ma-257	73	2	,	,	PUNCT
ma-257	73	3	−b0	−b0	PROPN
ma-257	73	4	≤	≤	NUM
ma-257	73	5	b(x	b(x	NOUN
ma-257	73	6	)	)	PUNCT
ma-257	73	7	≤	≤	PUNCT
ma-257	74	1	−b0	−b0	PROPN
ma-257	74	2	.	.	PUNCT
ma-257	75	1	•	•	NUM
ma-257	75	2	case(ii	case(ii	ADJ
ma-257	75	3	):	):	PUNCT
ma-257	75	4	when	when	SCONJ
ma-257	75	5	x	x	SYM
ma-257	75	6	∈	∈	NOUN
ma-257	75	7	ω−	ω−	ADP
ma-257	75	8	ω̄δ	ω̄δ	NUM
ma-257	75	9	;	;	PUNCT
ma-257	75	10	assume	assume	VERB
ma-257	75	11	a(x	a(x	NOUN
ma-257	75	12	)	)	PUNCT
ma-257	75	13	,	,	PUNCT
ma-257	75	14	b(x	b(x	NOUN
ma-257	75	15	)	)	PUNCT
ma-257	75	16	>	>	X
ma-257	75	17	0	0	PUNCT
ma-257	75	18	with	with	ADP
ma-257	75	19	a1	a1	NOUN
ma-257	75	20	,	,	PUNCT
ma-257	75	21	a1	a1	NOUN
ma-257	75	22	,	,	PUNCT
ma-257	75	23	b1	b1	NOUN
ma-257	75	24	,	,	PUNCT
ma-257	75	25	b1	b1	NOUN
ma-257	75	26	>	>	X
ma-257	75	27	0	0	PUNCT
ma-257	75	28	:	:	PUNCT
ma-257	75	29	a1	a1	VERB
ma-257	75	30	≤	≤	NUM
ma-257	75	31	a(x	a(x	PROPN
ma-257	75	32	)	)	PUNCT
ma-257	75	33	≤	≤	NUM
ma-257	75	34	a1	a1	NOUN
ma-257	75	35	,	,	PUNCT
ma-257	75	36	b1	b1	NOUN
ma-257	75	37	≤	≤	NUM
ma-257	75	38	b(x	b(x	NOUN
ma-257	75	39	)	)	PUNCT
ma-257	75	40	≤	≤	NUM
ma-257	75	41	b1	b1	NOUN
ma-257	75	42	.	.	PUNCT
ma-257	76	1	2	2	X
ma-257	76	2	.	.	X
ma-257	76	3	existence	existence	NOUN
ma-257	76	4	and	and	CCONJ
ma-257	76	5	non	non	ADJ
ma-257	76	6	-	-	ADJ
ma-257	76	7	existence	existence	ADJ
ma-257	76	8	results	result	NOUN
ma-257	76	9	in	in	ADP
ma-257	76	10	this	this	DET
ma-257	76	11	section	section	NOUN
ma-257	76	12	,	,	PUNCT
ma-257	76	13	the	the	DET
ma-257	76	14	results	result	NOUN
ma-257	76	15	of	of	ADP
ma-257	76	16	the	the	DET
ma-257	76	17	existence	existence	NOUN
ma-257	76	18	and	and	CCONJ
ma-257	76	19	non	non	ADJ
ma-257	76	20	-	-	NOUN
ma-257	76	21	existence	existence	NOUN
ma-257	76	22	are	be	AUX
ma-257	76	23	established	establish	VERB
ma-257	76	24	by	by	ADP
ma-257	76	25	using	use	VERB
ma-257	76	26	thesub	thesub	NOUN
ma-257	76	27	-	-	PUNCT
ma-257	76	28	supersolution	supersolution	NOUN
ma-257	76	29	method	method	NOUN
ma-257	76	30	.	.	PUNCT
ma-257	77	1	definition	definition	NOUN
ma-257	77	2	2.1	2.1	NUM
ma-257	77	3	.	.	PUNCT
ma-257	78	1	a	a	DET
ma-257	78	2	pair	pair	NOUN
ma-257	78	3	of	of	ADP
ma-257	78	4	non	non	ADJ
ma-257	78	5	-	-	ADJ
ma-257	78	6	negative	negative	ADJ
ma-257	78	7	functions	function	NOUN
ma-257	78	8	(	(	PUNCT
ma-257	78	9	u	u	NOUN
ma-257	78	10	,	,	PUNCT
ma-257	78	11	v	v	NOUN
ma-257	78	12	)	)	PUNCT
ma-257	78	13	is	be	AUX
ma-257	78	14	called	call	VERB
ma-257	78	15	a	a	DET
ma-257	78	16	positive	positive	ADJ
ma-257	78	17	weak	weak	ADJ
ma-257	78	18	solution	solution	NOUN
ma-257	78	19	of	of	ADP
ma-257	78	20	(	(	PUNCT
ma-257	78	21	1	1	X
ma-257	78	22	)	)	PUNCT
ma-257	78	23	such	such	ADJ
ma-257	78	24	that	that	SCONJ
ma-257	78	25	(	(	PUNCT
ma-257	78	26	u	u	NOUN
ma-257	78	27	,	,	PUNCT
ma-257	78	28	v	v	NOUN
ma-257	78	29	)	)	PUNCT
ma-257	78	30	∈	∈	PROPN
ma-257	78	31	w	w	PROPN
ma-257	78	32	1,p	1,p	PROPN
ma-257	78	33	0	0	NUM
ma-257	78	34	(	(	PUNCT
ma-257	78	35	ω)×w	ω)×w	PROPN
ma-257	78	36	1,q	1,q	NUM
ma-257	78	37	0	0	NUM
ma-257	78	38	(	(	PUNCT
ma-257	78	39	ω	ω	NOUN
ma-257	78	40	)	)	PUNCT
ma-257	78	41	if	if	SCONJ
ma-257	78	42	they	they	PRON
ma-257	78	43	satisfy∫	satisfy∫	VERB
ma-257	78	44	ω	ω	NUM
ma-257	78	45	|∇u|p−2∇u	|∇u|p−2∇u	NUM
ma-257	78	46	·	·	PUNCT
ma-257	79	1	∇ζ	∇ζ	NOUN
ma-257	79	2	dx	dx	PROPN
ma-257	79	3	=	=	SYM
ma-257	79	4	λ	λ	PROPN
ma-257	79	5	∫	∫	PROPN
ma-257	79	6	ω	ω	NUM
ma-257	79	7	a(x)[f	a(x)[f	PROPN
ma-257	79	8	(	(	PUNCT
ma-257	79	9	u	u	NOUN
ma-257	79	10	,	,	PUNCT
ma-257	79	11	v)−	v)−	PROPN
ma-257	79	12	1	1	NUM
ma-257	79	13	uα	uα	NOUN
ma-257	79	14	]	]	X
ma-257	79	15	ζ	ζ	NOUN
ma-257	79	16	dx,∫	dx,∫	PROPN
ma-257	79	17	ω	ω	NUM
ma-257	79	18	|∇v	|∇v	NOUN
ma-257	79	19	|q−2∇v	|q−2∇v	NOUN
ma-257	79	20	·	·	PUNCT
ma-257	80	1	∇ζ	∇ζ	NOUN
ma-257	80	2	dx	dx	NOUN
ma-257	80	3	=	=	SYM
ma-257	80	4	λ	λ	PROPN
ma-257	80	5	∫	∫	PROPN
ma-257	80	6	ω	ω	PROPN
ma-257	80	7	b(x)[g(u	b(x)[g(u	PROPN
ma-257	80	8	,	,	PUNCT
ma-257	80	9	v)−	v)−	PROPN
ma-257	80	10	1	1	NUM
ma-257	80	11	vβ	vβ	NOUN
ma-257	80	12	]	]	X
ma-257	80	13	ζ	ζ	X
ma-257	80	14	dx	dx	PROPN
ma-257	80	15	,	,	PUNCT
ma-257	80	16	∀	∀	X
ma-257	80	17	ζ	ζ	NOUN
ma-257	80	18	∈	∈	NOUN
ma-257	80	19	w	w	NOUN
ma-257	80	20	:	:	PUNCT
ma-257	80	21	=	=	SYM
ma-257	80	22	{	{	PUNCT
ma-257	80	23	ζ	ζ	NOUN
ma-257	80	24	∈	∈	NOUN
ma-257	80	25	c∞0	c∞0	X
ma-257	80	26	(	(	PUNCT
ma-257	80	27	ω	ω	NOUN
ma-257	80	28	)	)	PUNCT
ma-257	81	1	|	|	ADV
ma-257	81	2	ζ	ζ	NOUN
ma-257	81	3	≥	≥	NOUN
ma-257	81	4	0	0	NUM
ma-257	81	5	,	,	PUNCT
ma-257	81	6	x	x	X
ma-257	81	7	∈	∈	PROPN
ma-257	81	8	ω	ω	PROPN
ma-257	81	9	}	}	PUNCT
ma-257	81	10	.	.	PUNCT
ma-257	82	1	definition	definition	NOUN
ma-257	82	2	2.2	2.2	NUM
ma-257	82	3	.	.	PUNCT
ma-257	83	1	a	a	DET
ma-257	83	2	pair	pair	NOUN
ma-257	83	3	of	of	ADP
ma-257	83	4	non	non	ADJ
ma-257	83	5	-	-	ADJ
ma-257	83	6	negative	negative	ADJ
ma-257	83	7	functions	function	NOUN
ma-257	83	8	(	(	PUNCT
ma-257	83	9	ψ1	ψ1	NOUN
ma-257	83	10	,	,	PUNCT
ma-257	83	11	ψ2	ψ2	NOUN
ma-257	83	12	)	)	PUNCT
ma-257	83	13	and	and	CCONJ
ma-257	83	14	(	(	PUNCT
ma-257	83	15	z1	z1	PROPN
ma-257	83	16	,	,	PUNCT
ma-257	83	17	z2	z2	PROPN
ma-257	83	18	)	)	PUNCT
ma-257	83	19	are	be	AUX
ma-257	83	20	called	call	VERB
ma-257	83	21	a	a	DET
ma-257	83	22	positive	positive	ADJ
ma-257	83	23	weak	weak	ADJ
ma-257	83	24	subsolution	subsolution	NOUN
ma-257	83	25	and	and	CCONJ
ma-257	83	26	supersolution	supersolution	NOUN
ma-257	83	27	of	of	ADP
ma-257	83	28	(	(	PUNCT
ma-257	83	29	1	1	NUM
ma-257	83	30	)	)	PUNCT
ma-257	83	31	,	,	PUNCT
ma-257	83	32	respectively	respectively	ADV
ma-257	83	33	,	,	PUNCT
ma-257	83	34	such	such	ADJ
ma-257	83	35	that	that	SCONJ
ma-257	83	36	(	(	PUNCT
ma-257	83	37	ψ1	ψ1	NOUN
ma-257	83	38	,	,	PUNCT
ma-257	83	39	ψ2	ψ2	NOUN
ma-257	83	40	)	)	PUNCT
ma-257	83	41	,	,	PUNCT
ma-257	83	42	(	(	PUNCT
ma-257	83	43	z1	z1	PROPN
ma-257	83	44	,	,	PUNCT
ma-257	83	45	z2	z2	PROPN
ma-257	83	46	)	)	PUNCT
ma-257	83	47	∈	∈	PROPN
ma-257	83	48	w	w	PROPN
ma-257	83	49	1,p	1,p	PROPN
ma-257	83	50	0	0	NUM
ma-257	83	51	(	(	PUNCT
ma-257	83	52	ω)×w	ω)×w	PROPN
ma-257	83	53	1,q	1,q	NUM
ma-257	83	54	0	0	NUM
ma-257	83	55	(	(	PUNCT
ma-257	83	56	ω	ω	NOUN
ma-257	83	57	)	)	PUNCT
ma-257	83	58	if	if	SCONJ
ma-257	83	59	they	they	PRON
ma-257	83	60	satisfy	satisfy	VERB
ma-257	83	61	∫	∫	PROPN
ma-257	83	62	ω	ω	NUM
ma-257	83	63	|∇ψ1|p−2∇ψ1	|∇ψ1|p−2∇ψ1	NOUN
ma-257	83	64	·	·	PUNCT
ma-257	84	1	∇ζ	∇ζ	NOUN
ma-257	84	2	dx	dx	PROPN
ma-257	84	3	≤	≤	PROPN
ma-257	85	1	λ	λ	PROPN
ma-257	85	2	∫	∫	PROPN
ma-257	85	3	ω	ω	NUM
ma-257	85	4	a(x)[f	a(x)[f	PROPN
ma-257	85	5	(	(	PUNCT
ma-257	85	6	ψ1	ψ1	ADJ
ma-257	85	7	,	,	PUNCT
ma-257	85	8	ψ2)−	ψ2)−	X
ma-257	85	9	1	1	NUM
ma-257	85	10	ψα1	ψα1	NOUN
ma-257	85	11	]	]	X
ma-257	85	12	ζ	ζ	X
ma-257	85	13	dx,∫	dx,∫	PROPN
ma-257	85	14	ω	ω	NUM
ma-257	85	15	|∇ψ2|q−2∇ψ2	|∇ψ2|q−2∇ψ2	PROPN
ma-257	85	16	·	·	PUNCT
ma-257	86	1	∇ζ	∇ζ	NOUN
ma-257	86	2	dx	dx	PROPN
ma-257	86	3	≤	≤	PROPN
ma-257	87	1	λ	λ	PROPN
ma-257	87	2	∫	∫	PROPN
ma-257	87	3	ω	ω	NUM
ma-257	87	4	b(x)[g(ψ1	b(x)[g(ψ1	NOUN
ma-257	87	5	,	,	PUNCT
ma-257	87	6	ψ2)−	ψ2)−	X
ma-257	87	7	1	1	NUM
ma-257	87	8	ψβ2	ψβ2	NOUN
ma-257	87	9	]	]	X
ma-257	87	10	ζ	ζ	X
ma-257	87	11	dx	dx	PROPN
ma-257	87	12	,	,	PUNCT
ma-257	87	13	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	87	14	eur	eur	PROPN
ma-257	87	15	.	.	PUNCT
ma-257	88	1	j.	j.	PROPN
ma-257	88	2	math	math	PROPN
ma-257	88	3	.	.	PUNCT
ma-257	89	1	anal	anal	PROPN
ma-257	89	2	.	.	PUNCT
ma-257	90	1	10.28924	10.28924	NUM
ma-257	90	2	/	/	SYM
ma-257	90	3	ada	ada	PROPN
ma-257	90	4	/	/	SYM
ma-257	90	5	ma.5.1	ma.5.1	PROPN
ma-257	90	6	5	5	NUM
ma-257	90	7	and	and	CCONJ
ma-257	90	8	∫	∫	PROPN
ma-257	90	9	ω	ω	PROPN
ma-257	90	10	|∇z1|p−2∇z1	|∇z1|p−2∇z1	PROPN
ma-257	90	11	·	·	PUNCT
ma-257	91	1	∇ζ	∇ζ	PROPN
ma-257	91	2	dx	dx	PROPN
ma-257	91	3	≥	≥	PROPN
ma-257	91	4	λ	λ	X
ma-257	91	5	∫	∫	PROPN
ma-257	91	6	ω	ω	NUM
ma-257	91	7	a(x)[f	a(x)[f	PROPN
ma-257	91	8	(	(	PUNCT
ma-257	91	9	z1	z1	PROPN
ma-257	91	10	,	,	PUNCT
ma-257	91	11	z2)−	z2)−	PROPN
ma-257	91	12	1	1	NUM
ma-257	91	13	zα1	zα1	NOUN
ma-257	91	14	]	]	X
ma-257	91	15	ζ	ζ	X
ma-257	91	16	dx,∫	dx,∫	PROPN
ma-257	91	17	ω	ω	NUM
ma-257	91	18	|∇z2|q−2∇z2	|∇z2|q−2∇z2	PROPN
ma-257	91	19	·	·	PUNCT
ma-257	92	1	∇ζ	∇ζ	PROPN
ma-257	92	2	dx	dx	PROPN
ma-257	92	3	≥	≥	PROPN
ma-257	92	4	λ	λ	X
ma-257	92	5	∫	∫	PROPN
ma-257	92	6	ω	ω	PROPN
ma-257	92	7	b(x)[g(z1	b(x)[g(z1	NOUN
ma-257	92	8	,	,	PUNCT
ma-257	92	9	z2)−	z2)−	PROPN
ma-257	92	10	1	1	NUM
ma-257	92	11	zβ2	zβ2	NOUN
ma-257	92	12	]	]	X
ma-257	92	13	ζ	ζ	X
ma-257	92	14	dx	dx	PROPN
ma-257	92	15	,	,	PUNCT
ma-257	92	16	∀	∀	X
ma-257	92	17	ζ	ζ	NOUN
ma-257	92	18	∈	∈	NOUN
ma-257	92	19	w	w	NOUN
ma-257	92	20	:	:	PUNCT
ma-257	92	21	=	=	SYM
ma-257	92	22	{	{	PUNCT
ma-257	92	23	ζ	ζ	NOUN
ma-257	92	24	∈	∈	NOUN
ma-257	92	25	c∞0	c∞0	X
ma-257	92	26	(	(	PUNCT
ma-257	92	27	ω	ω	NOUN
ma-257	92	28	)	)	PUNCT
ma-257	92	29	|	|	ADV
ma-257	92	30	ζ	ζ	NOUN
ma-257	92	31	≥	≥	NOUN
ma-257	92	32	0	0	NUM
ma-257	92	33	,	,	PUNCT
ma-257	92	34	x	x	X
ma-257	92	35	∈	∈	PROPN
ma-257	92	36	ω	ω	PROPN
ma-257	92	37	}	}	PUNCT
ma-257	92	38	.	.	PUNCT
ma-257	93	1	now	now	ADV
ma-257	93	2	,	,	PUNCT
ma-257	93	3	we	we	PRON
ma-257	93	4	state	state	VERB
ma-257	93	5	our	our	PRON
ma-257	93	6	results	result	NOUN
ma-257	93	7	as	as	SCONJ
ma-257	93	8	follows	follow	VERB
ma-257	93	9	:	:	PUNCT
ma-257	93	10	lemma	lemma	PROPN
ma-257	93	11	2.1	2.1	NUM
ma-257	93	12	.	.	PUNCT
ma-257	94	1	(	(	PUNCT
ma-257	94	2	see	see	VERB
ma-257	94	3	[	[	X
ma-257	94	4	2	2	NUM
ma-257	94	5	]	]	PUNCT
ma-257	94	6	):	):	PUNCT
ma-257	94	7	let	let	VERB
ma-257	94	8	(	(	PUNCT
ma-257	94	9	ψ1	ψ1	ADJ
ma-257	94	10	,	,	PUNCT
ma-257	94	11	ψ2	ψ2	NOUN
ma-257	94	12	)	)	PUNCT
ma-257	94	13	and	and	CCONJ
ma-257	94	14	(	(	PUNCT
ma-257	94	15	z1	z1	PROPN
ma-257	94	16	,	,	PUNCT
ma-257	94	17	z2	z2	PROPN
ma-257	94	18	)	)	PUNCT
ma-257	94	19	be	be	VERB
ma-257	94	20	a	a	DET
ma-257	94	21	subsolution	subsolution	NOUN
ma-257	94	22	and	and	CCONJ
ma-257	94	23	supersolution	supersolution	NOUN
ma-257	94	24	of	of	ADP
ma-257	94	25	(	(	PUNCT
ma-257	94	26	1	1	NUM
ma-257	94	27	)	)	PUNCT
ma-257	94	28	,	,	PUNCT
ma-257	94	29	respectively	respectively	ADV
ma-257	94	30	,	,	PUNCT
ma-257	94	31	with	with	ADP
ma-257	94	32	ψ1	ψ1	ADJ
ma-257	94	33	≤	≤	ADJ
ma-257	94	34	z1	z1	NOUN
ma-257	94	35	and	and	CCONJ
ma-257	94	36	ψ2	ψ2	NOUN
ma-257	94	37	≤	≤	NOUN
ma-257	94	38	z2	z2	NOUN
ma-257	94	39	.	.	PUNCT
ma-257	95	1	therefore	therefore	ADV
ma-257	95	2	,	,	PUNCT
ma-257	95	3	system	system	NOUN
ma-257	95	4	(	(	PUNCT
ma-257	95	5	1	1	X
ma-257	95	6	)	)	PUNCT
ma-257	95	7	has	have	VERB
ma-257	95	8	a	a	DET
ma-257	95	9	solution	solution	NOUN
ma-257	95	10	(	(	PUNCT
ma-257	95	11	u	u	NOUN
ma-257	95	12	,	,	PUNCT
ma-257	95	13	v	v	NOUN
ma-257	95	14	)	)	PUNCT
ma-257	95	15	with	with	ADP
ma-257	95	16	ψ1	ψ1	ADJ
ma-257	95	17	≤	≤	NUM
ma-257	95	18	u	u	NOUN
ma-257	95	19	≤	≤	NOUN
ma-257	95	20	z1	z1	NOUN
ma-257	95	21	and	and	CCONJ
ma-257	95	22	ψ2	ψ2	NOUN
ma-257	95	23	≤	≤	X
ma-257	95	24	v	v	PRON
ma-257	95	25	≤	≤	NUM
ma-257	95	26	z2	z2	NOUN
ma-257	95	27	.	.	PUNCT
ma-257	96	1	our	our	PRON
ma-257	96	2	assumptions	assumption	NOUN
ma-257	96	3	are	be	AUX
ma-257	96	4	as	as	SCONJ
ma-257	96	5	follows	follow	VERB
ma-257	96	6	:	:	PUNCT
ma-257	96	7	(	(	PUNCT
ma-257	96	8	s1	s1	NOUN
ma-257	96	9	)	)	PUNCT
ma-257	96	10	f	f	NOUN
ma-257	96	11	,	,	PUNCT
ma-257	96	12	g	g	PROPN
ma-257	96	13	:	:	PUNCT
ma-257	96	14	r+	r+	NOUN
ma-257	96	15	×	×	NOUN
ma-257	96	16	r+	r+	NOUN
ma-257	96	17	→	→	SYM
ma-257	96	18	r+	r+	PRON
ma-257	96	19	are	be	AUX
ma-257	96	20	c1	c1	NOUN
ma-257	96	21	increasing	increase	VERB
ma-257	96	22	functions	function	NOUN
ma-257	96	23	such	such	ADJ
ma-257	96	24	that	that	SCONJ
ma-257	96	25	f	f	PROPN
ma-257	96	26	(	(	PUNCT
ma-257	96	27	υ1	υ1	PROPN
ma-257	96	28	,	,	PUNCT
ma-257	96	29	υ2	υ2	NOUN
ma-257	96	30	)	)	PUNCT
ma-257	96	31	>	>	X
ma-257	96	32	0	0	NUM
ma-257	96	33	,	,	PUNCT
ma-257	96	34	g(υ1	g(υ1	NOUN
ma-257	96	35	,	,	PUNCT
ma-257	96	36	υ2	υ2	NOUN
ma-257	96	37	)	)	PUNCT
ma-257	96	38	>	>	X
ma-257	96	39	0	0	PUNCT
ma-257	96	40	for	for	ADP
ma-257	96	41	υ1	υ1	PROPN
ma-257	96	42	,	,	PUNCT
ma-257	96	43	υ2	υ2	NOUN
ma-257	96	44	>	>	X
ma-257	96	45	0	0	PUNCT
ma-257	97	1	and	and	CCONJ
ma-257	97	2	lim	lim	PROPN
ma-257	97	3	υ1,υ2→∞	υ1,υ2→∞	PROPN
ma-257	97	4	f	f	PROPN
ma-257	97	5	(	(	PUNCT
ma-257	97	6	υ1	υ1	PROPN
ma-257	97	7	,	,	PUNCT
ma-257	97	8	υ2	υ2	NOUN
ma-257	97	9	)	)	PUNCT
ma-257	97	10	=	=	PROPN
ma-257	97	11	lim	lim	PROPN
ma-257	97	12	υ1,υ2→∞	υ1,υ2→∞	PROPN
ma-257	97	13	g(υ1	g(υ1	NOUN
ma-257	97	14	,	,	PUNCT
ma-257	97	15	υ2	υ2	NOUN
ma-257	97	16	)	)	PUNCT
ma-257	97	17	=	=	NOUN
ma-257	97	18	∞	∞	PROPN
ma-257	97	19	,	,	PUNCT
ma-257	97	20	(	(	PUNCT
ma-257	97	21	s2	s2	PROPN
ma-257	97	22	)	)	PUNCT
ma-257	97	23	lim	lim	PROPN
ma-257	97	24	ξ→∞	ξ→∞	PROPN
ma-257	97	25	f	f	X
ma-257	97	26	(	(	PUNCT
ma-257	97	27	ξ	ξ	PROPN
ma-257	97	28	,	,	PUNCT
ma-257	97	29	m[g(ξ	m[g(ξ	NOUN
ma-257	97	30	,	,	PUNCT
ma-257	97	31	ξ	ξ	NOUN
ma-257	97	32	)	)	PUNCT
ma-257	97	33	]	]	PUNCT
ma-257	97	34	1	1	NUM
ma-257	97	35	q−1	q−1	PROPN
ma-257	97	36	)	)	PUNCT
ma-257	97	37	ξp−1	ξp−1	PROPN
ma-257	98	1	=	=	PUNCT
ma-257	98	2	0	0	NUM
ma-257	98	3	∀m	∀m	PROPN
ma-257	98	4	>	>	X
ma-257	98	5	0	0	PUNCT
ma-257	98	6	and	and	CCONJ
ma-257	98	7	lim	lim	PROPN
ma-257	98	8	ξ→∞	ξ→∞	PROPN
ma-257	98	9	g(ξ	g(ξ	PROPN
ma-257	98	10	,	,	PUNCT
ma-257	98	11	ξ	ξ	NOUN
ma-257	98	12	)	)	PUNCT
ma-257	98	13	ξq−1	ξq−1	NOUN
ma-257	98	14	=	=	PUNCT
ma-257	98	15	0	0	NUM
ma-257	98	16	,	,	PUNCT
ma-257	98	17	(	(	PUNCT
ma-257	98	18	s3	s3	PROPN
ma-257	98	19	)	)	PUNCT
ma-257	98	20	let	let	VERB
ma-257	98	21	εo	εo	ADP
ma-257	98	22	>	>	X
ma-257	98	23	0	0	NUM
ma-257	99	1	such	such	ADJ
ma-257	99	2	that	that	SCONJ
ma-257	99	3	:	:	PUNCT
ma-257	99	4	(	(	PUNCT
ma-257	99	5	i	i	NOUN
ma-257	99	6	)	)	PUNCT
ma-257	99	7	n	n	PROPN
ma-257	99	8	=	=	SYM
ma-257	99	9	f	f	PROPN
ma-257	99	10	(	(	PUNCT
ma-257	99	11	µε	µε	ADP
ma-257	99	12	1	1	NUM
ma-257	99	13	p−1	p−1	PROPN
ma-257	99	14	o	o	X
ma-257	99	15	po	po	NOUN
ma-257	99	16	,	,	PUNCT
ma-257	99	17	µε	µε	ADP
ma-257	99	18	1	1	NUM
ma-257	99	19	q−1	q−1	NOUN
ma-257	99	20	o	o	NOUN
ma-257	99	21	qo	qo	NOUN
ma-257	99	22	)	)	PUNCT
ma-257	99	23	−	−	PROPN
ma-257	99	24	(	(	PUNCT
ma-257	99	25	po	po	NOUN
ma-257	99	26	µε	µε	ADP
ma-257	99	27	1	1	NUM
ma-257	99	28	p−1	p−1	PROPN
ma-257	99	29	o	o	NOUN
ma-257	99	30	)	)	PUNCT
ma-257	99	31	α	α	X
ma-257	99	32	>	>	X
ma-257	99	33	0	0	NUM
ma-257	99	34	,	,	PUNCT
ma-257	99	35	and	and	CCONJ
ma-257	99	36	m	m	PROPN
ma-257	99	37	=	=	ADJ
ma-257	99	38	g	g	PROPN
ma-257	99	39	(	(	PUNCT
ma-257	99	40	µε	µε	ADP
ma-257	99	41	1	1	NUM
ma-257	99	42	p−1	p−1	PROPN
ma-257	99	43	o	o	X
ma-257	99	44	po	po	NOUN
ma-257	99	45	,	,	PUNCT
ma-257	99	46	µε	µε	ADP
ma-257	99	47	1	1	NUM
ma-257	99	48	q−1	q−1	NOUN
ma-257	99	49	o	o	NOUN
ma-257	99	50	qo	qo	NOUN
ma-257	99	51	)	)	PUNCT
ma-257	99	52	−	−	PROPN
ma-257	100	1	(	(	PUNCT
ma-257	100	2	qo	qo	NOUN
ma-257	100	3	µε	µε	ADP
ma-257	100	4	1	1	NUM
ma-257	100	5	q−1	q−1	NOUN
ma-257	100	6	o	o	NOUN
ma-257	100	7	)	)	PUNCT
ma-257	101	1	β	β	X
ma-257	101	2	>	>	X
ma-257	101	3	0	0	NUM
ma-257	101	4	,	,	PUNCT
ma-257	101	5	(	(	PUNCT
ma-257	101	6	ii	ii	NOUN
ma-257	101	7	)	)	PUNCT
ma-257	101	8	f	f	PROPN
ma-257	102	1	(	(	PUNCT
ma-257	102	2	ε	ε	PROPN
ma-257	102	3	1	1	NUM
ma-257	102	4	p−1	p−1	PROPN
ma-257	102	5	o	o	PROPN
ma-257	102	6	,	,	PUNCT
ma-257	102	7	ε	ε	PROPN
ma-257	102	8	1	1	NUM
ma-257	103	1	q−1	q−1	PROPN
ma-257	103	2	o	o	NOUN
ma-257	103	3	)	)	PUNCT
ma-257	104	1	m	m	VERB
ma-257	104	2	≤	≤	NUM
ma-257	104	3	min	min	NOUN
ma-257	104	4	{	{	PUNCT
ma-257	104	5	pαo	pαo	NOUN
ma-257	104	6	αp	αp	NOUN
ma-257	105	1	λ1,p	λ1,p	NOUN
ma-257	105	2	pε	pε	NOUN
ma-257	105	3	α	α	PRON
ma-257	105	4	p−1	p−1	PROPN
ma-257	105	5	o	o	PROPN
ma-257	105	6	,	,	PUNCT
ma-257	105	7	nαpa1	nαpa1	PROPN
ma-257	106	1	λ1,p	λ1,p	PROPN
ma-257	106	2	pa0	pa0	PROPN
ma-257	106	3	,	,	PUNCT
ma-257	106	4	qβo	qβo	PROPN
ma-257	107	1	βqb0	βqb0	PROPN
ma-257	107	2	λ1,q	λ1,q	PROPN
ma-257	107	3	ε	ε	PROPN
ma-257	107	4	β	β	X
ma-257	107	5	q−1	q−1	PROPN
ma-257	107	6	o	o	NOUN
ma-257	107	7	pa0	pa0	NOUN
ma-257	107	8	,	,	PUNCT
ma-257	107	9	mβqb1	mβqb1	PROPN
ma-257	108	1	λ1,q	λ1,q	PROPN
ma-257	108	2	pa0	pa0	NOUN
ma-257	108	3	}	}	PUNCT
ma-257	108	4	,	,	PUNCT
ma-257	108	5	(	(	PUNCT
ma-257	108	6	iii	iii	X
ma-257	108	7	)	)	PUNCT
ma-257	108	8	g(ε	g(ε	NOUN
ma-257	108	9	1	1	NUM
ma-257	108	10	p−1	p−1	PROPN
ma-257	108	11	o	o	PROPN
ma-257	108	12	,	,	PUNCT
ma-257	108	13	ε	ε	PROPN
ma-257	108	14	1	1	NUM
ma-257	109	1	q−1	q−1	PROPN
ma-257	109	2	o	o	NOUN
ma-257	109	3	)	)	PUNCT
ma-257	110	1	m	m	VERB
ma-257	110	2	≤	≤	NUM
ma-257	110	3	min	min	NOUN
ma-257	110	4	{	{	PUNCT
ma-257	110	5	qβo	qβo	PROPN
ma-257	110	6	βq	βq	ADJ
ma-257	110	7	λ1,q	λ1,q	PROPN
ma-257	110	8	qε	qε	ADV
ma-257	110	9	β	β	X
ma-257	110	10	q−1	q−1	PROPN
ma-257	110	11	o	o	INTJ
ma-257	110	12	,	,	PUNCT
ma-257	110	13	nαpa1	nαpa1	PROPN
ma-257	111	1	λ1,p	λ1,p	PROPN
ma-257	111	2	qb0	qb0	NOUN
ma-257	111	3	,	,	PUNCT
ma-257	112	1	pαo	pαo	NOUN
ma-257	112	2	αpa0	αpa0	NOUN
ma-257	112	3	λ1,p	λ1,p	PROPN
ma-257	112	4	ε	ε	PROPN
ma-257	112	5	α	α	NOUN
ma-257	112	6	p−1	p−1	PROPN
ma-257	112	7	o	o	PROPN
ma-257	112	8	qb0	qb0	NOUN
ma-257	112	9	,	,	PUNCT
ma-257	112	10	mβqb1	mβqb1	PROPN
ma-257	113	1	λ1,q	λ1,q	PUNCT
ma-257	113	2	qb0	qb0	NOUN
ma-257	113	3	}	}	PUNCT
ma-257	113	4	,	,	PUNCT
ma-257	113	5	with	with	ADP
ma-257	113	6	po	po	NOUN
ma-257	114	1	=	=	PUNCT
ma-257	114	2	p	p	PROPN
ma-257	114	3	p−1	p−1	PROPN
ma-257	114	4	,	,	PUNCT
ma-257	114	5	qo	qo	PROPN
ma-257	114	6	=	=	PUNCT
ma-257	114	7	q	q	PROPN
ma-257	115	1	q−1	q−1	PROPN
ma-257	115	2	,	,	PUNCT
ma-257	115	3	αp	αp	NOUN
ma-257	115	4	=	=	SYM
ma-257	115	5	(	(	PUNCT
ma-257	115	6	α+	α+	PROPN
ma-257	115	7	1)p−1	1)p−1	NUM
ma-257	115	8	and	and	CCONJ
ma-257	115	9	βq	βq	ADJ
ma-257	115	10	=	=	SYM
ma-257	115	11	(	(	PUNCT
ma-257	115	12	β	β	X
ma-257	115	13	+	+	X
ma-257	115	14	1)q−1	1)q−1	NUM
ma-257	115	15	.	.	PUNCT
ma-257	116	1	(	(	PUNCT
ma-257	116	2	s4	s4	PROPN
ma-257	116	3	)	)	PUNCT
ma-257	116	4	there	there	PRON
ma-257	116	5	exist	exist	VERB
ma-257	116	6	f0	f0	PROPN
ma-257	116	7	,	,	PUNCT
ma-257	116	8	g0	g0	NOUN
ma-257	116	9	>	>	X
ma-257	116	10	0	0	NUM
ma-257	117	1	where	where	SCONJ
ma-257	117	2	f	f	PROPN
ma-257	117	3	(	(	PUNCT
ma-257	117	4	υ1	υ1	PROPN
ma-257	117	5	,	,	PUNCT
ma-257	117	6	υ2	υ2	NOUN
ma-257	117	7	)	)	PUNCT
ma-257	117	8	≤	≤	NUM
ma-257	117	9	f0υ	f0υ	PROPN
ma-257	117	10	γ1	γ1	NOUN
ma-257	117	11	1	1	NUM
ma-257	117	12	υ	υ	PROPN
ma-257	117	13	κ1	κ1	PROPN
ma-257	117	14	2	2	NUM
ma-257	117	15	and	and	CCONJ
ma-257	117	16	g(υ1	g(υ1	NOUN
ma-257	117	17	,	,	PUNCT
ma-257	117	18	υ2	υ2	NOUN
ma-257	117	19	)	)	PUNCT
ma-257	117	20	≤	≤	NUM
ma-257	117	21	g0υ	g0υ	PROPN
ma-257	117	22	κ2	κ2	NOUN
ma-257	117	23	1	1	NUM
ma-257	117	24	υ	υ	NOUN
ma-257	117	25	γ2	γ2	PROPN
ma-257	117	26	2	2	NUM
ma-257	117	27	such	such	ADJ
ma-257	117	28	that	that	DET
ma-257	117	29	γ1	γ1	NOUN
ma-257	117	30	,	,	PUNCT
ma-257	117	31	γ2	γ2	PROPN
ma-257	117	32	,	,	PUNCT
ma-257	117	33	κ1	κ1	NOUN
ma-257	117	34	,	,	PUNCT
ma-257	117	35	κ2	κ2	NOUN
ma-257	117	36	are	be	AUX
ma-257	117	37	positive	positive	ADJ
ma-257	117	38	parameters	parameter	NOUN
ma-257	117	39	,	,	PUNCT
ma-257	117	40	γ1,γ2	γ1,γ2	PROPN
ma-257	117	41	∈	∈	PROPN
ma-257	117	42	(	(	PUNCT
ma-257	117	43	0	0	NUM
ma-257	117	44	,	,	PUNCT
ma-257	117	45	1	1	NUM
ma-257	117	46	)	)	PUNCT
ma-257	117	47	and	and	CCONJ
ma-257	117	48	κ2	κ2	NOUN
ma-257	117	49	+	+	CCONJ
ma-257	117	50	γ2	γ2	PROPN
ma-257	117	51	<	<	X
ma-257	117	52	min{1	min{1	PROPN
ma-257	117	53	,	,	PUNCT
ma-257	117	54	1	1	NUM
ma-257	117	55	κ1	κ1	NOUN
ma-257	117	56	}	}	PUNCT
ma-257	117	57	.to	.to	PUNCT
ma-257	118	1	be	be	AUX
ma-257	118	2	more	more	ADV
ma-257	118	3	specific	specific	ADJ
ma-257	118	4	we	we	PRON
ma-257	118	5	consider	consider	VERB
ma-257	118	6	λo(εo	λo(εo	VERB
ma-257	118	7	)	)	PUNCT
ma-257	118	8	and	and	CCONJ
ma-257	118	9	λo(εo	λo(εo	PROPN
ma-257	118	10	)	)	PUNCT
ma-257	118	11	by	by	ADP
ma-257	118	12	the	the	DET
ma-257	118	13	following	follow	VERB
ma-257	118	14	λo	λo	PROPN
ma-257	118	15	=	=	SYM
ma-257	118	16	min	min	PROPN
ma-257	118	17	{	{	PUNCT
ma-257	118	18	mεo	mεo	PROPN
ma-257	118	19	pa0f	pa0f	PROPN
ma-257	118	20	(	(	PUNCT
ma-257	118	21	ε	ε	PROPN
ma-257	118	22	1	1	NUM
ma-257	118	23	p−1	p−1	PROPN
ma-257	118	24	o	o	PROPN
ma-257	118	25	,	,	PUNCT
ma-257	118	26	ε	ε	PROPN
ma-257	118	27	1	1	NUM
ma-257	118	28	q−1	q−1	PROPN
ma-257	118	29	o	o	NOUN
ma-257	118	30	)	)	PUNCT
ma-257	118	31	,	,	PUNCT
ma-257	118	32	mεo	mεo	PROPN
ma-257	118	33	qb0g(ε	qb0g(ε	PROPN
ma-257	118	34	1	1	NUM
ma-257	118	35	p−1	p−1	PROPN
ma-257	118	36	o	o	NOUN
ma-257	118	37	,	,	PUNCT
ma-257	118	38	ε	ε	PROPN
ma-257	118	39	1	1	NUM
ma-257	119	1	q−1	q−1	PROPN
ma-257	119	2	o	o	NOUN
ma-257	119	3	)	)	PUNCT
ma-257	119	4	}	}	PUNCT
ma-257	119	5	,	,	PUNCT
ma-257	119	6	and	and	CCONJ
ma-257	119	7	λo	λo	PROPN
ma-257	119	8	=	=	PUNCT
ma-257	119	9	max	max	PROPN
ma-257	119	10	{	{	PUNCT
ma-257	120	1	ε	ε	PROPN
ma-257	120	2	α+p−1	α+p−1	VERB
ma-257	120	3	p−1	p−1	PROPN
ma-257	120	4	o	o	X
ma-257	120	5	λ1,p	λ1,p	PROPN
ma-257	120	6	pαoαpa0	pαoαpa0	PROPN
ma-257	120	7	,	,	PUNCT
ma-257	120	8	ε	ε	PROPN
ma-257	120	9	β+q−1	β+q−1	X
ma-257	121	1	q−1	q−1	PROPN
ma-257	121	2	o	o	NOUN
ma-257	122	1	λ1,q	λ1,q	NOUN
ma-257	122	2	qβoβqb0	qβoβqb0	NOUN
ma-257	122	3	,	,	PUNCT
ma-257	122	4	εo	εo	PROPN
ma-257	122	5	λ1,p	λ1,p	PROPN
ma-257	122	6	nαpa1	nαpa1	X
ma-257	122	7	,	,	PUNCT
ma-257	122	8	εo	εo	PROPN
ma-257	122	9	λ1,q	λ1,q	PROPN
ma-257	122	10	mβqb1	mβqb1	PROPN
ma-257	122	11	}	}	PUNCT
ma-257	122	12	.	.	PUNCT
ma-257	123	1	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	123	2	eur	eur	PROPN
ma-257	123	3	.	.	PUNCT
ma-257	124	1	j.	j.	PROPN
ma-257	124	2	math	math	PROPN
ma-257	124	3	.	.	PUNCT
ma-257	125	1	anal	anal	PROPN
ma-257	125	2	.	.	PUNCT
ma-257	126	1	10.28924	10.28924	NUM
ma-257	126	2	/	/	SYM
ma-257	126	3	ada	ada	PROPN
ma-257	126	4	/	/	SYM
ma-257	126	5	ma.5.1	ma.5.1	PROPN
ma-257	126	6	6	6	NUM
ma-257	126	7	example	example	NOUN
ma-257	126	8	2.1	2.1	NUM
ma-257	126	9	.	.	PUNCT
ma-257	127	1	assume	assume	VERB
ma-257	127	2	f	f	PROPN
ma-257	127	3	(	(	PUNCT
ma-257	127	4	υ1	υ1	ADJ
ma-257	127	5	,	,	PUNCT
ma-257	127	6	υ2	υ2	NOUN
ma-257	127	7	)	)	PUNCT
ma-257	127	8	=	=	PUNCT
ma-257	128	1	[	[	X
ma-257	128	2	υk1	υk1	NOUN
ma-257	128	3	2	2	NUM
ma-257	128	4	+	+	CCONJ
ma-257	128	5	(	(	PUNCT
ma-257	128	6	υ1υ2)l1	υ1υ2)l1	VERB
ma-257	128	7	−	−	NOUN
ma-257	128	8	1	1	NUM
ma-257	128	9	]	]	PUNCT
ma-257	128	10	,	,	PUNCT
ma-257	128	11	g(υ1	g(υ1	NOUN
ma-257	128	12	,	,	PUNCT
ma-257	128	13	υ2	υ2	NOUN
ma-257	128	14	)	)	PUNCT
ma-257	128	15	=	=	PUNCT
ma-257	129	1	[	[	X
ma-257	129	2	υk2	υk2	NOUN
ma-257	129	3	1	1	NUM
ma-257	129	4	+	+	CCONJ
ma-257	129	5	(	(	PUNCT
ma-257	129	6	υ1υ2	υ1υ2	NOUN
ma-257	129	7	)	)	PUNCT
ma-257	129	8	l2	l2	NOUN
ma-257	129	9	2	2	NUM
ma-257	129	10	−	−	NOUN
ma-257	129	11	1	1	NUM
ma-257	129	12	]	]	PUNCT
ma-257	129	13	where	where	SCONJ
ma-257	129	14	k1	k1	NOUN
ma-257	129	15	,	,	PUNCT
ma-257	129	16	k2	k2	PROPN
ma-257	129	17	,	,	PUNCT
ma-257	129	18	l1	l1	PROPN
ma-257	129	19	,	,	PUNCT
ma-257	129	20	l2	l2	NOUN
ma-257	129	21	are	be	AUX
ma-257	129	22	positive	positive	ADJ
ma-257	129	23	parameters	parameter	NOUN
ma-257	129	24	.	.	PUNCT
ma-257	130	1	thus	thus	ADV
ma-257	130	2	,	,	PUNCT
ma-257	130	3	f	f	PROPN
ma-257	130	4	,	,	PUNCT
ma-257	130	5	g	g	PROPN
ma-257	130	6	clearly	clearly	ADV
ma-257	130	7	satisfy	satisfy	VERB
ma-257	130	8	(	(	PUNCT
ma-257	130	9	s1	s1	NOUN
ma-257	130	10	)	)	PUNCT
ma-257	130	11	and	and	CCONJ
ma-257	130	12	(	(	PUNCT
ma-257	130	13	s2	s2	PROPN
ma-257	130	14	)	)	PUNCT
ma-257	130	15	if	if	SCONJ
ma-257	130	16	max{k2	max{k2	NOUN
ma-257	130	17	,	,	PUNCT
ma-257	130	18	l2	l2	NOUN
ma-257	130	19	}	}	PUNCT
ma-257	130	20	k1	k1	NOUN
ma-257	131	1	q−1	q−1	PROPN
ma-257	131	2	<	<	X
ma-257	131	3	p	p	X
ma-257	131	4	−	−	PROPN
ma-257	131	5	1	1	NUM
ma-257	131	6	,	,	PUNCT
ma-257	131	7	max{k2	max{k2	NOUN
ma-257	131	8	,	,	PUNCT
ma-257	131	9	l2	l2	NOUN
ma-257	131	10	}	}	PUNCT
ma-257	131	11	<	<	X
ma-257	131	12	q	q	X
ma-257	131	13	−	−	PROPN
ma-257	131	14	1	1	NUM
ma-257	131	15	and	and	CCONJ
ma-257	131	16	(	(	PUNCT
ma-257	131	17	max{k2	max{k2	NOUN
ma-257	131	18	,	,	PUNCT
ma-257	131	19	l2	l2	NOUN
ma-257	131	20	}	}	PUNCT
ma-257	131	21	1	1	NUM
ma-257	131	22	q−1	q−1	PROPN
ma-257	131	23	+	+	CCONJ
ma-257	131	24	1)l1	1)l1	NUM
ma-257	131	25	<	<	X
ma-257	131	26	p	p	X
ma-257	131	27	−	−	PROPN
ma-257	131	28	1	1	NUM
ma-257	131	29	such	such	ADJ
ma-257	131	30	that	that	SCONJ
ma-257	131	31	lim	lim	PROPN
ma-257	131	32	ξ→∞	ξ→∞	PROPN
ma-257	131	33	f	f	X
ma-257	131	34	(	(	PUNCT
ma-257	131	35	ξ	ξ	PROPN
ma-257	131	36	,	,	PUNCT
ma-257	131	37	m[g(ξ	m[g(ξ	NOUN
ma-257	131	38	,	,	PUNCT
ma-257	131	39	ξ	ξ	NOUN
ma-257	131	40	)	)	PUNCT
ma-257	131	41	]	]	PUNCT
ma-257	131	42	1	1	NUM
ma-257	131	43	q−1	q−1	PROPN
ma-257	131	44	)	)	PUNCT
ma-257	131	45	ξp−1	ξp−1	PROPN
ma-257	131	46	=	=	PUNCT
ma-257	131	47	0	0	NUM
ma-257	131	48	∀m	∀m	PROPN
ma-257	131	49	>	>	X
ma-257	131	50	0	0	PROPN
ma-257	131	51	,	,	PUNCT
ma-257	131	52	lim	lim	PROPN
ma-257	131	53	ξ→∞	ξ→∞	NUM
ma-257	131	54	g(ξ	g(ξ	PROPN
ma-257	131	55	,	,	PUNCT
ma-257	131	56	ξ	ξ	NOUN
ma-257	131	57	)	)	PUNCT
ma-257	131	58	ξq−1	ξq−1	NOUN
ma-257	131	59	=	=	PUNCT
ma-257	131	60	0	0	NUM
ma-257	131	61	,	,	PUNCT
ma-257	131	62	and	and	CCONJ
ma-257	131	63	lim	lim	PROPN
ma-257	131	64	ξ→∞	ξ→∞	PROPN
ma-257	131	65	g(ξ	g(ξ	PROPN
ma-257	131	66	,	,	PUNCT
ma-257	131	67	ξ	ξ	X
ma-257	131	68	)	)	PUNCT
ma-257	131	69	=	=	NOUN
ma-257	131	70	∞.	∞.	PROPN
ma-257	131	71	we	we	PRON
ma-257	131	72	can	can	AUX
ma-257	131	73	take	take	VERB
ma-257	131	74	εo	εo	ADV
ma-257	131	75	>	>	X
ma-257	131	76	0	0	PUNCT
ma-257	132	1	small	small	ADJ
ma-257	132	2	enough	enough	ADV
ma-257	132	3	that	that	SCONJ
ma-257	132	4	f	f	X
ma-257	132	5	,	,	PUNCT
ma-257	132	6	g	g	PROPN
ma-257	132	7	satisfy	satisfy	NOUN
ma-257	132	8	(	(	PUNCT
ma-257	132	9	s3	s3	PROPN
ma-257	132	10	)	)	PUNCT
ma-257	132	11	.	.	PUNCT
ma-257	133	1	remark	remark	VERB
ma-257	133	2	2.1	2.1	NUM
ma-257	133	3	.	.	PUNCT
ma-257	134	1	note	note	VERB
ma-257	134	2	that	that	SCONJ
ma-257	134	3	(	(	PUNCT
ma-257	134	4	s3	s3	PROPN
ma-257	134	5	)	)	PUNCT
ma-257	134	6	implies	imply	VERB
ma-257	134	7	λo	λo	VERB
ma-257	134	8	<	<	X
ma-257	134	9	λo	λo	X
ma-257	134	10	.	.	PUNCT
ma-257	135	1	here	here	ADV
ma-257	135	2	,	,	PUNCT
ma-257	135	3	we	we	PRON
ma-257	135	4	can	can	AUX
ma-257	135	5	establish	establish	VERB
ma-257	135	6	our	our	PRON
ma-257	135	7	existence	existence	NOUN
ma-257	135	8	results	result	NOUN
ma-257	135	9	.	.	PUNCT
ma-257	136	1	theorem	theorem	VERB
ma-257	136	2	2.1	2.1	NUM
ma-257	136	3	.	.	PUNCT
ma-257	137	1	suppose	suppose	VERB
ma-257	137	2	(	(	PUNCT
ma-257	137	3	s1)-(s3	s1)-(s3	NOUN
ma-257	137	4	)	)	PUNCT
ma-257	137	5	hold	hold	NOUN
ma-257	137	6	,	,	PUNCT
ma-257	137	7	hence	hence	ADV
ma-257	137	8	(	(	PUNCT
ma-257	137	9	1	1	X
ma-257	137	10	)	)	PUNCT
ma-257	137	11	has	have	VERB
ma-257	137	12	a	a	DET
ma-257	137	13	positive	positive	ADJ
ma-257	137	14	weak	weak	ADJ
ma-257	137	15	solution	solution	NOUN
ma-257	137	16	for	for	ADP
ma-257	137	17	every	every	DET
ma-257	137	18	λ	λ	PROPN
ma-257	137	19	∈	∈	PROPN
ma-257	137	20	[	[	X
ma-257	137	21	λo(εo	λo(εo	PROPN
ma-257	137	22	)	)	PUNCT
ma-257	137	23	,	,	PUNCT
ma-257	137	24	λo(εo	λo(εo	PROPN
ma-257	137	25	)	)	PUNCT
ma-257	137	26	]	]	PUNCT
ma-257	137	27	.	.	PUNCT
ma-257	138	1	proof	proof	NOUN
ma-257	138	2	.	.	PUNCT
ma-257	139	1	we	we	PRON
ma-257	139	2	shall	shall	AUX
ma-257	139	3	verify	verify	VERB
ma-257	139	4	that	that	SCONJ
ma-257	139	5	(	(	PUNCT
ma-257	139	6	ψ1	ψ1	NOUN
ma-257	139	7	,	,	PUNCT
ma-257	139	8	ψ2	ψ2	NOUN
ma-257	139	9	)	)	PUNCT
ma-257	139	10	=	=	SYM
ma-257	139	11	(	(	PUNCT
ma-257	139	12	ε	ε	PROPN
ma-257	139	13	1	1	NUM
ma-257	139	14	p−1	p−1	PROPN
ma-257	139	15	o	o	X
ma-257	139	16	ϕ	ϕ	X
ma-257	139	17	po	po	X
ma-257	139	18	α+1	α+1	NUM
ma-257	139	19	1,p	1,p	PROPN
ma-257	139	20	po	po	X
ma-257	139	21	,	,	PUNCT
ma-257	139	22	ε	ε	PROPN
ma-257	139	23	1	1	NUM
ma-257	140	1	q−1	q−1	NOUN
ma-257	140	2	o	o	NOUN
ma-257	140	3	ϕ	ϕ	X
ma-257	140	4	qo	qo	PROPN
ma-257	140	5	β+1	β+1	NUM
ma-257	140	6	1,q	1,q	PROPN
ma-257	140	7	qo	qo	NOUN
ma-257	140	8	)	)	PUNCT
ma-257	140	9	is	be	AUX
ma-257	140	10	a	a	DET
ma-257	140	11	positive	positive	ADJ
ma-257	140	12	weak	weak	ADJ
ma-257	140	13	subsolution	subsolution	NOUN
ma-257	140	14	of	of	ADP
ma-257	140	15	(	(	PUNCT
ma-257	140	16	1	1	NUM
ma-257	140	17	)	)	PUNCT
ma-257	140	18	.	.	PUNCT
ma-257	141	1	then	then	ADV
ma-257	141	2	∇ψ1	∇ψ1	VERB
ma-257	141	3	=	=	SYM
ma-257	141	4	ε	ε	PROPN
ma-257	141	5	1	1	NUM
ma-257	141	6	p−1	p−1	PROPN
ma-257	141	7	o	o	X
ma-257	141	8	∇ϕ	∇ϕ	PROPN
ma-257	141	9	po	po	PROPN
ma-257	141	10	α+1	α+1	NUM
ma-257	141	11	1,p	1,p	PROPN
ma-257	141	12	po	po	NOUN
ma-257	141	13	=	=	SYM
ma-257	141	14	ε	ε	PROPN
ma-257	141	15	1	1	NUM
ma-257	141	16	p−1	p−1	PROPN
ma-257	141	17	o	o	NOUN
ma-257	141	18	1	1	NUM
ma-257	141	19	+	+	NUM
ma-257	141	20	α	α	PRON
ma-257	141	21	ϕ	ϕ	X
ma-257	141	22	po−1−α	po−1−α	PROPN
ma-257	141	23	α+1	α+1	NUM
ma-257	141	24	1,p	1,p	PROPN
ma-257	141	25	∇ϕ1,p	∇ϕ1,p	NOUN
ma-257	141	26	,	,	PUNCT
ma-257	141	27	and	and	CCONJ
ma-257	141	28	∫	∫	PROPN
ma-257	141	29	ω	ω	NUM
ma-257	141	30	|∇ψ1|p−2∇ψ1	|∇ψ1|p−2∇ψ1	PROPN
ma-257	141	31	·	·	PUNCT
ma-257	141	32	∇ζ	∇ζ	NOUN
ma-257	141	33	dx	dx	NOUN
ma-257	141	34	=	=	PUNCT
ma-257	141	35	εo	εo	ADJ
ma-257	141	36	αp	αp	NOUN
ma-257	141	37	∫	∫	PROPN
ma-257	141	38	ω	ω	PROPN
ma-257	141	39	ϕ	ϕ	PROPN
ma-257	141	40	(	(	PUNCT
ma-257	141	41	1−	1−	NUM
ma-257	141	42	αp	αp	NOUN
ma-257	141	43	α+1	α+1	NUM
ma-257	141	44	)	)	PUNCT
ma-257	141	45	1,p	1,p	PROPN
ma-257	141	46	|∇ϕ1,p|p−2∇ϕ1,p	|∇ϕ1,p|p−2∇ϕ1,p	PROPN
ma-257	141	47	·	·	PUNCT
ma-257	142	1	∇ζdx	∇ζdx	NOUN
ma-257	142	2	=	=	PUNCT
ma-257	143	1	εo	εo	ADJ
ma-257	143	2	αp	αp	NOUN
ma-257	143	3	∫	∫	PROPN
ma-257	143	4	ω	ω	PROPN
ma-257	143	5	|∇ϕ1,p|p−2∇ϕ1,p	|∇ϕ1,p|p−2∇ϕ1,p	PROPN
ma-257	143	6	[	[	PUNCT
ma-257	143	7	∇	∇	X
ma-257	143	8	(	(	PUNCT
ma-257	143	9	ϕ	ϕ	X
ma-257	143	10	(	(	PUNCT
ma-257	143	11	1−	1−	NUM
ma-257	143	12	αp	αp	NOUN
ma-257	143	13	α+1	α+1	NUM
ma-257	143	14	)	)	PUNCT
ma-257	143	15	1,p	1,p	PROPN
ma-257	143	16	·	·	PUNCT
ma-257	143	17	ζ	ζ	NOUN
ma-257	143	18	)	)	PUNCT
ma-257	143	19	−	−	PROPN
ma-257	144	1	(	(	PUNCT
ma-257	144	2	1−	1−	NUM
ma-257	144	3	αp	αp	NOUN
ma-257	144	4	α+	α+	X
ma-257	144	5	1	1	X
ma-257	144	6	)	)	PUNCT
ma-257	144	7	ϕ	ϕ	NOUN
ma-257	144	8	−αp	−αp	X
ma-257	144	9	α+1	α+1	NUM
ma-257	144	10	1,p	1,p	PROPN
ma-257	144	11	∇ϕ1,p	∇ϕ1,p	NOUN
ma-257	144	12	·	·	PUNCT
ma-257	144	13	ζ	ζ	NOUN
ma-257	144	14	]	]	PUNCT
ma-257	144	15	dx	dx	PROPN
ma-257	145	1	=	=	PUNCT
ma-257	145	2	εo	εo	PROPN
ma-257	145	3	αp	αp	NOUN
ma-257	146	1	∫	∫	PROPN
ma-257	146	2	ω	ω	PROPN
ma-257	147	1	[	[	PUNCT
ma-257	147	2	|∇ϕ1,p|p−2∇ϕ1,p∇	|∇ϕ1,p|p−2∇ϕ1,p∇	PROPN
ma-257	147	3	(	(	PUNCT
ma-257	147	4	ϕ	ϕ	X
ma-257	147	5	(	(	PUNCT
ma-257	147	6	1−	1−	NUM
ma-257	147	7	αp	αp	NOUN
ma-257	147	8	α+1	α+1	NUM
ma-257	147	9	)	)	PUNCT
ma-257	147	10	1,p	1,p	PROPN
ma-257	147	11	·	·	PUNCT
ma-257	147	12	ζ	ζ	NOUN
ma-257	147	13	)	)	PUNCT
ma-257	147	14	−	−	PROPN
ma-257	147	15	(	(	PUNCT
ma-257	147	16	1−	1−	NUM
ma-257	147	17	αp	αp	NOUN
ma-257	147	18	α+	α+	X
ma-257	147	19	1	1	X
ma-257	147	20	)	)	PUNCT
ma-257	147	21	ϕ	ϕ	NOUN
ma-257	147	22	−αp	−αp	X
ma-257	147	23	α+1	α+1	NUM
ma-257	147	24	1,p	1,p	PROPN
ma-257	147	25	|∇ϕ1,p|p	|∇ϕ1,p|p	PART
ma-257	147	26	·	·	PUNCT
ma-257	147	27	ζ	ζ	X
ma-257	147	28	]	]	PUNCT
ma-257	147	29	dx	dx	PROPN
ma-257	147	30	=	=	PUNCT
ma-257	147	31	εo	εo	PROPN
ma-257	147	32	αp	αp	NOUN
ma-257	147	33	∫	∫	PROPN
ma-257	147	34	ω	ω	PROPN
ma-257	147	35	[	[	PUNCT
ma-257	147	36	λ1,p	λ1,p	PROPN
ma-257	147	37	ϕ	ϕ	X
ma-257	147	38	p	p	X
ma-257	147	39	α+1	α+1	NUM
ma-257	147	40	1,p	1,p	PROPN
ma-257	147	41	−	−	NOUN
ma-257	147	42	(	(	PUNCT
ma-257	147	43	1−	1−	NUM
ma-257	147	44	αp	αp	NOUN
ma-257	147	45	α+	α+	X
ma-257	147	46	1	1	X
ma-257	147	47	)	)	PUNCT
ma-257	147	48	ϕ	ϕ	NOUN
ma-257	147	49	−αp	−αp	X
ma-257	147	50	α+1	α+1	NUM
ma-257	147	51	1,p	1,p	PROPN
ma-257	147	52	|∇ϕ1,p|p	|∇ϕ1,p|p	X
ma-257	147	53	]	]	X
ma-257	147	54	ζ	ζ	X
ma-257	147	55	dx	dx	PROPN
ma-257	147	56	,	,	PUNCT
ma-257	147	57	then	then	ADV
ma-257	147	58	,	,	PUNCT
ma-257	147	59	∫	∫	PROPN
ma-257	147	60	ω	ω	NUM
ma-257	147	61	|∇ψ1|p−2∇ψ1	|∇ψ1|p−2∇ψ1	PROPN
ma-257	147	62	·	·	PUNCT
ma-257	147	63	∇ζ	∇ζ	NOUN
ma-257	147	64	dx	dx	NOUN
ma-257	147	65	=	=	PUNCT
ma-257	148	1	εo	εo	ADJ
ma-257	148	2	αp	αp	NOUN
ma-257	148	3	∫	∫	PROPN
ma-257	148	4	ω	ω	PROPN
ma-257	148	5	[	[	PUNCT
ma-257	148	6	λ1,p	λ1,p	PROPN
ma-257	148	7	ϕ	ϕ	X
ma-257	148	8	p	p	X
ma-257	148	9	α+1	α+1	NUM
ma-257	148	10	1,p	1,p	PROPN
ma-257	148	11	−	−	NOUN
ma-257	148	12	(	(	PUNCT
ma-257	148	13	1−	1−	NUM
ma-257	148	14	αp	αp	NOUN
ma-257	148	15	α+	α+	X
ma-257	148	16	1	1	X
ma-257	148	17	)	)	PUNCT
ma-257	148	18	ϕ	ϕ	NOUN
ma-257	148	19	−αp	−αp	X
ma-257	148	20	α+1	α+1	NUM
ma-257	148	21	1,p	1,p	PROPN
ma-257	148	22	|∇ϕ1,p|p	|∇ϕ1,p|p	X
ma-257	148	23	]	]	X
ma-257	148	24	ζ	ζ	X
ma-257	148	25	dx	dx	PROPN
ma-257	148	26	.	.	PUNCT
ma-257	149	1	(	(	PUNCT
ma-257	149	2	11	11	NUM
ma-257	149	3	)	)	PUNCT
ma-257	149	4	similarly,∫	similarly,∫	NOUN
ma-257	149	5	ω	ω	PROPN
ma-257	149	6	|∇ψ2|q−2∇ψ2	|∇ψ2|q−2∇ψ2	PROPN
ma-257	149	7	·	·	PUNCT
ma-257	150	1	∇ζ	∇ζ	NOUN
ma-257	150	2	dx	dx	NOUN
ma-257	150	3	=	=	PUNCT
ma-257	150	4	εo	εo	NOUN
ma-257	150	5	βq	βq	ADJ
ma-257	150	6	∫	∫	PROPN
ma-257	150	7	ω	ω	PROPN
ma-257	150	8	[	[	PUNCT
ma-257	150	9	λ1,q	λ1,q	NOUN
ma-257	150	10	ϕ	ϕ	X
ma-257	150	11	q	q	X
ma-257	150	12	β+1	β+1	NUM
ma-257	150	13	1,q	1,q	NUM
ma-257	150	14	−	−	NOUN
ma-257	150	15	(	(	PUNCT
ma-257	150	16	1−	1−	NUM
ma-257	150	17	βq	βq	ADJ
ma-257	150	18	β	β	X
ma-257	150	19	+	+	ADP
ma-257	150	20	1	1	X
ma-257	150	21	)	)	PUNCT
ma-257	150	22	ϕ	ϕ	NOUN
ma-257	150	23	−βq	−βq	PROPN
ma-257	150	24	β+1	β+1	NUM
ma-257	150	25	1,q	1,q	NUM
ma-257	150	26	|∇ϕ1,q|q	|∇ϕ1,q|q	NUM
ma-257	150	27	]	]	X
ma-257	150	28	ζ	ζ	X
ma-257	150	29	dx	dx	PROPN
ma-257	150	30	.	.	PUNCT
ma-257	151	1	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	151	2	eur	eur	PROPN
ma-257	151	3	.	.	PUNCT
ma-257	152	1	j.	j.	PROPN
ma-257	152	2	math	math	PROPN
ma-257	152	3	.	.	PUNCT
ma-257	153	1	anal	anal	PROPN
ma-257	153	2	.	.	PUNCT
ma-257	154	1	10.28924	10.28924	NUM
ma-257	154	2	/	/	SYM
ma-257	154	3	ada	ada	PROPN
ma-257	154	4	/	/	SYM
ma-257	154	5	ma.5.1	ma.5.1	PROPN
ma-257	154	6	7case(i	7case(i	NUM
ma-257	154	7	):	):	PUNCT
ma-257	154	8	when	when	SCONJ
ma-257	154	9	x	x	SYM
ma-257	154	10	∈	∈	PROPN
ma-257	154	11	ω̄δ	ω̄δ	NUM
ma-257	154	12	.	.	PUNCT
ma-257	155	1	put	put	VERB
ma-257	155	2	s	s	PART
ma-257	155	3	=	=	SYM
ma-257	155	4	α	α	PROPN
ma-257	155	5	,	,	PUNCT
ma-257	155	6	r	r	NOUN
ma-257	155	7	=	=	NOUN
ma-257	155	8	p	p	NOUN
ma-257	155	9	in	in	ADP
ma-257	155	10	(	(	PUNCT
ma-257	155	11	8)	8)	NUM
ma-257	155	12	,	,	PUNCT
ma-257	155	13	we	we	PRON
ma-257	155	14	have	have	VERB
ma-257	155	15	−p	−p	ADJ
ma-257	155	16	αp	αp	NOUN
ma-257	155	17	(	(	PUNCT
ma-257	155	18	1−	1−	NUM
ma-257	155	19	αp	αp	NOUN
ma-257	155	20	α+	α+	PUNCT
ma-257	155	21	1	1	NUM
ma-257	155	22	)	)	PUNCT
ma-257	155	23	|∇ϕ1,p|p	|∇ϕ1,p|p	PART
ma-257	155	24	≤	≤	NUM
ma-257	155	25	−m	−m	NOUN
ma-257	155	26	.	.	PUNCT
ma-257	156	1	hence	hence	ADV
ma-257	156	2	,	,	PUNCT
ma-257	156	3	−εo	−εo	PROPN
ma-257	156	4	ϕ	ϕ	NOUN
ma-257	156	5	−αp	−αp	X
ma-257	156	6	α+1	α+1	NUM
ma-257	156	7	1,p	1,p	PROPN
ma-257	156	8	αp	αp	NOUN
ma-257	156	9	(	(	PUNCT
ma-257	156	10	1−	1−	NUM
ma-257	156	11	αp	αp	NOUN
ma-257	156	12	α+	α+	PUNCT
ma-257	156	13	1	1	NUM
ma-257	156	14	)	)	PUNCT
ma-257	156	15	|∇ϕ1,p|p	|∇ϕ1,p|p	PART
ma-257	156	16	≤	≤	NOUN
ma-257	156	17	−mεo	−mεo	NOUN
ma-257	156	18	p	p	NOUN
ma-257	156	19	,	,	PUNCT
ma-257	156	20	and	and	CCONJ
ma-257	156	21	since	since	SCONJ
ma-257	156	22	λ	λ	PROPN
ma-257	156	23	≤	≤	X
ma-257	156	24	λo	λo	ADV
ma-257	156	25	,	,	PUNCT
ma-257	156	26	then	then	ADV
ma-257	156	27	λ	λ	X
ma-257	156	28	≤	≤	NUM
ma-257	156	29	mεo	mεo	VERB
ma-257	156	30	pa0f	pa0f	PROPN
ma-257	156	31	(	(	PUNCT
ma-257	156	32	ε	ε	PROPN
ma-257	156	33	1	1	NUM
ma-257	156	34	p−1	p−1	PROPN
ma-257	156	35	o	o	PROPN
ma-257	156	36	,	,	PUNCT
ma-257	156	37	ε	ε	PROPN
ma-257	156	38	1	1	NUM
ma-257	157	1	q−1	q−1	PROPN
ma-257	157	2	o	o	NOUN
ma-257	157	3	)	)	PUNCT
ma-257	157	4	.	.	PUNCT
ma-257	158	1	hence	hence	ADV
ma-257	158	2	,	,	PUNCT
ma-257	158	3	−mεo	−mεo	NOUN
ma-257	158	4	p	p	NOUN
ma-257	158	5	≤	≤	PROPN
ma-257	158	6	−λa0f	−λa0f	PUNCT
ma-257	158	7	(	(	PUNCT
ma-257	158	8	ε	ε	PROPN
ma-257	158	9	1	1	NUM
ma-257	158	10	p−1	p−1	PROPN
ma-257	158	11	o	o	PROPN
ma-257	158	12	,	,	PUNCT
ma-257	158	13	ε	ε	PROPN
ma-257	158	14	1	1	NUM
ma-257	158	15	q−1	q−1	PROPN
ma-257	158	16	o	o	NOUN
ma-257	158	17	)	)	PUNCT
ma-257	158	18	≤	≤	NOUN
ma-257	158	19	−λa0f	−λa0f	PUNCT
ma-257	158	20	(	(	PUNCT
ma-257	158	21	ε	ε	PROPN
ma-257	158	22	1	1	NUM
ma-257	158	23	p−1	p−1	PROPN
ma-257	158	24	o	o	X
ma-257	158	25	ϕ	ϕ	X
ma-257	158	26	po	po	X
ma-257	158	27	α+1	α+1	NUM
ma-257	158	28	1,p	1,p	PROPN
ma-257	158	29	po	po	X
ma-257	158	30	,	,	PUNCT
ma-257	158	31	ε	ε	PROPN
ma-257	158	32	1	1	NUM
ma-257	159	1	q−1	q−1	NOUN
ma-257	159	2	o	o	NOUN
ma-257	159	3	ϕ	ϕ	X
ma-257	159	4	qo	qo	PROPN
ma-257	159	5	β+1	β+1	NUM
ma-257	159	6	1,q	1,q	PROPN
ma-257	159	7	qo	qo	NOUN
ma-257	159	8	)	)	PUNCT
ma-257	159	9	,	,	PUNCT
ma-257	159	10	so	so	ADV
ma-257	159	11	,	,	PUNCT
ma-257	159	12	−εo	−εo	PROPN
ma-257	159	13	ϕ	ϕ	NOUN
ma-257	159	14	−αp	−αp	X
ma-257	159	15	α+1	α+1	NUM
ma-257	159	16	1,p	1,p	PROPN
ma-257	159	17	αp	αp	NOUN
ma-257	159	18	(	(	PUNCT
ma-257	159	19	1−	1−	NUM
ma-257	159	20	αp	αp	NOUN
ma-257	159	21	α+	α+	PUNCT
ma-257	159	22	1	1	NUM
ma-257	159	23	)	)	PUNCT
ma-257	159	24	|∇ϕ1,p|p	|∇ϕ1,p|p	PART
ma-257	159	25	≤	≤	NOUN
ma-257	159	26	−λa0f	−λa0f	PUNCT
ma-257	159	27	(	(	PUNCT
ma-257	159	28	ε	ε	PROPN
ma-257	159	29	1	1	NUM
ma-257	159	30	p−1	p−1	PROPN
ma-257	159	31	o	o	X
ma-257	159	32	ϕ	ϕ	X
ma-257	159	33	po	po	X
ma-257	159	34	α+1	α+1	NUM
ma-257	159	35	1,p	1,p	PROPN
ma-257	159	36	po	po	X
ma-257	159	37	,	,	PUNCT
ma-257	159	38	ε	ε	PROPN
ma-257	159	39	1	1	NUM
ma-257	160	1	q−1	q−1	NOUN
ma-257	160	2	o	o	NOUN
ma-257	160	3	ϕ	ϕ	X
ma-257	160	4	qo	qo	PROPN
ma-257	160	5	β+1	β+1	NUM
ma-257	160	6	1,q	1,q	PROPN
ma-257	160	7	qo	qo	NOUN
ma-257	160	8	)	)	PUNCT
ma-257	160	9	,	,	PUNCT
ma-257	160	10	(	(	PUNCT
ma-257	160	11	12	12	NUM
ma-257	160	12	)	)	PUNCT
ma-257	160	13	and	and	CCONJ
ma-257	160	14	since	since	SCONJ
ma-257	160	15	λ	λ	PROPN
ma-257	160	16	≥	≥	PRON
ma-257	160	17	λo	λo	INTJ
ma-257	160	18	,	,	PUNCT
ma-257	160	19	then	then	ADV
ma-257	160	20	λ	λ	X
ma-257	160	21	≥	≥	NOUN
ma-257	160	22	ε	ε	PROPN
ma-257	160	23	α+p−1	α+p−1	VERB
ma-257	160	24	p−1	p−1	PROPN
ma-257	160	25	o	o	X
ma-257	160	26	λ1,p	λ1,p	PROPN
ma-257	160	27	pαoαpa0	pαoαpa0	PROPN
ma-257	160	28	.	.	PUNCT
ma-257	161	1	hence	hence	ADV
ma-257	161	2	,	,	PUNCT
ma-257	161	3	εo	εo	PROPN
ma-257	161	4	λ1,p	λ1,p	PROPN
ma-257	161	5	ϕ	ϕ	X
ma-257	161	6	p	p	X
ma-257	161	7	α+1	α+1	NUM
ma-257	161	8	1,p	1,p	PROPN
ma-257	161	9	αp	αp	VERB
ma-257	161	10	≤	≤	NUM
ma-257	161	11	εo	εo	ADP
ma-257	161	12	λ1,p	λ1,p	PROPN
ma-257	161	13	αp	αp	NOUN
ma-257	161	14	≤	≤	NUM
ma-257	161	15	λa0	λa0	NOUN
ma-257	161	16	(	(	PUNCT
ma-257	161	17	ε	ε	PROPN
ma-257	161	18	1	1	NUM
ma-257	161	19	p−1	p−1	PROPN
ma-257	161	20	o	o	PROPN
ma-257	161	21	po	po	NOUN
ma-257	161	22	)	)	PUNCT
ma-257	161	23	α	α	PROPN
ma-257	161	24	≤	≤	NUM
ma-257	162	1	λa0	λa0	ADJ
ma-257	162	2	(	(	PUNCT
ma-257	162	3	ε	ε	PROPN
ma-257	162	4	1	1	NUM
ma-257	162	5	p−1	p−1	PROPN
ma-257	162	6	o	o	X
ma-257	162	7	ϕ	ϕ	X
ma-257	162	8	po	po	X
ma-257	162	9	α+1	α+1	NUM
ma-257	162	10	1,p	1,p	PROPN
ma-257	162	11	po	po	NOUN
ma-257	162	12	)	)	PUNCT
ma-257	162	13	α	α	PROPN
ma-257	162	14	.	.	PUNCT
ma-257	163	1	(	(	PUNCT
ma-257	163	2	13	13	NUM
ma-257	163	3	)	)	PUNCT
ma-257	163	4	using	use	VERB
ma-257	163	5	(	(	PUNCT
ma-257	163	6	12	12	NUM
ma-257	163	7	)	)	PUNCT
ma-257	163	8	and	and	CCONJ
ma-257	163	9	(	(	PUNCT
ma-257	163	10	13	13	NUM
ma-257	163	11	)	)	PUNCT
ma-257	163	12	in	in	ADP
ma-257	163	13	(	(	PUNCT
ma-257	163	14	11	11	NUM
ma-257	163	15	)	)	PUNCT
ma-257	163	16	,	,	PUNCT
ma-257	163	17	we	we	PRON
ma-257	163	18	see	see	VERB
ma-257	163	19	that	that	SCONJ
ma-257	163	20	∫	∫	PROPN
ma-257	163	21	ω	ω	NUM
ma-257	163	22	|∇ψ1|p−2∇ψ1	|∇ψ1|p−2∇ψ1	PROPN
ma-257	163	23	·	·	PUNCT
ma-257	164	1	∇ζ	∇ζ	PROPN
ma-257	164	2	dx	dx	PROPN
ma-257	164	3	≤	≤	NUM
ma-257	164	4	∫	∫	PROPN
ma-257	164	5	ω	ω	NOUN
ma-257	164	6	λa0	λa0	VERB
ma-257	164	7	ζ	ζ	X
ma-257	164	8	(	(	PUNCT
ma-257	164	9	ε	ε	PROPN
ma-257	164	10	1	1	NUM
ma-257	164	11	p−1	p−1	PROPN
ma-257	164	12	o	o	X
ma-257	164	13	ϕ	ϕ	X
ma-257	164	14	po	po	X
ma-257	164	15	α+1	α+1	NUM
ma-257	164	16	1,p	1,p	PROPN
ma-257	164	17	po	po	NOUN
ma-257	164	18	)	)	PUNCT
ma-257	164	19	α	α	PROPN
ma-257	164	20	dx	dx	PROPN
ma-257	165	1	−	−	PROPN
ma-257	165	2	∫	∫	PROPN
ma-257	165	3	ω	ω	NUM
ma-257	165	4	λa0f	λa0f	PROPN
ma-257	165	5	(	(	PUNCT
ma-257	165	6	ε	ε	PROPN
ma-257	165	7	1	1	NUM
ma-257	165	8	p−1	p−1	PROPN
ma-257	165	9	o	o	X
ma-257	165	10	ϕ	ϕ	X
ma-257	165	11	po	po	X
ma-257	165	12	α+1	α+1	NUM
ma-257	165	13	1,p	1,p	PROPN
ma-257	165	14	po	po	X
ma-257	165	15	,	,	PUNCT
ma-257	165	16	ε	ε	PROPN
ma-257	165	17	1	1	NUM
ma-257	166	1	q−1	q−1	NOUN
ma-257	166	2	o	o	NOUN
ma-257	166	3	ϕ	ϕ	X
ma-257	166	4	qo	qo	PROPN
ma-257	166	5	β+1	β+1	NUM
ma-257	166	6	1,q	1,q	PROPN
ma-257	166	7	qo	qo	NOUN
ma-257	166	8	)	)	PUNCT
ma-257	166	9	ζdx	ζdx	PROPN
ma-257	167	1	=	=	PROPN
ma-257	168	1	−λ	−λ	PROPN
ma-257	168	2	∫	∫	PROPN
ma-257	168	3	ω	ω	NUM
ma-257	168	4	a0[f	a0[f	X
ma-257	168	5	(	(	PUNCT
ma-257	168	6	ψ1	ψ1	NOUN
ma-257	168	7	,	,	PUNCT
ma-257	168	8	ψ2)−	ψ2)−	X
ma-257	168	9	1	1	NUM
ma-257	168	10	ψα1	ψα1	NOUN
ma-257	168	11	]	]	X
ma-257	168	12	ζ	ζ	X
ma-257	168	13	dx	dx	PROPN
ma-257	168	14	≤	≤	PROPN
ma-257	169	1	λ	λ	PROPN
ma-257	169	2	∫	∫	PROPN
ma-257	169	3	ω	ω	NUM
ma-257	169	4	a(x)[f	a(x)[f	PROPN
ma-257	169	5	(	(	PUNCT
ma-257	169	6	ψ1	ψ1	ADJ
ma-257	169	7	,	,	PUNCT
ma-257	169	8	ψ2)−	ψ2)−	X
ma-257	169	9	1	1	NUM
ma-257	169	10	ψα1	ψα1	NOUN
ma-257	169	11	]	]	X
ma-257	169	12	ζ	ζ	X
ma-257	169	13	dx	dx	PROPN
ma-257	169	14	.	.	PUNCT
ma-257	170	1	case(ii	case(ii	PROPN
ma-257	170	2	):	):	PUNCT
ma-257	170	3	when	when	SCONJ
ma-257	170	4	x	x	SYM
ma-257	170	5	∈	∈	NOUN
ma-257	170	6	ω−	ω−	ADP
ma-257	170	7	ω̄δ	ω̄δ	NUM
ma-257	170	8	;	;	PUNCT
ma-257	170	9	µ	µ	DET
ma-257	170	10	≤	≤	NUM
ma-257	170	11	ϕ1,p	ϕ1,p	PROPN
ma-257	170	12	≤	≤	NOUN
ma-257	170	13	1	1	NUM
ma-257	170	14	.	.	PUNCT
ma-257	171	1	since	since	SCONJ
ma-257	171	2	λ	λ	PROPN
ma-257	171	3	≥	≥	PRON
ma-257	171	4	λo	λo	INTJ
ma-257	171	5	,	,	PUNCT
ma-257	171	6	then	then	ADV
ma-257	171	7	εo	εo	ADP
ma-257	171	8	λ1,p	λ1,p	PROPN
ma-257	171	9	nαpa1	nαpa1	VERB
ma-257	171	10	≤	≤	PUNCT
ma-257	172	1	λ	λ	PROPN
ma-257	172	2	.	.	PUNCT
ma-257	172	3	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	172	4	eur	eur	PROPN
ma-257	172	5	.	.	PUNCT
ma-257	173	1	j.	j.	PROPN
ma-257	173	2	math	math	PROPN
ma-257	173	3	.	.	PUNCT
ma-257	174	1	anal	anal	PROPN
ma-257	174	2	.	.	PUNCT
ma-257	175	1	10.28924	10.28924	NUM
ma-257	175	2	/	/	SYM
ma-257	175	3	ada	ada	PROPN
ma-257	175	4	/	/	SYM
ma-257	175	5	ma.5.1	ma.5.1	PROPN
ma-257	175	6	8hence,∫	8hence,∫	PROPN
ma-257	175	7	ω	ω	NUM
ma-257	175	8	|∇ψ1|p−2∇ψ1	|∇ψ1|p−2∇ψ1	NOUN
ma-257	175	9	·	·	PUNCT
ma-257	175	10	∇ζ	∇ζ	NOUN
ma-257	175	11	dx	dx	NOUN
ma-257	176	1	=	=	PUNCT
ma-257	176	2	εo	εo	ADJ
ma-257	176	3	αp	αp	NOUN
ma-257	176	4	∫	∫	PROPN
ma-257	176	5	ω	ω	PROPN
ma-257	176	6	[	[	PUNCT
ma-257	176	7	λ1,p	λ1,p	PROPN
ma-257	176	8	φ	φ	PROPN
ma-257	176	9	p	p	NOUN
ma-257	176	10	α+1	α+1	NUM
ma-257	176	11	1,p	1,p	PROPN
ma-257	176	12	−	−	NOUN
ma-257	176	13	(	(	PUNCT
ma-257	176	14	1−	1−	NUM
ma-257	176	15	αp	αp	NOUN
ma-257	176	16	α+	α+	PUNCT
ma-257	176	17	1	1	X
ma-257	176	18	)	)	PUNCT
ma-257	176	19	φ	φ	NUM
ma-257	176	20	−αp	−αp	NOUN
ma-257	176	21	α+1	α+1	NUM
ma-257	176	22	1,p	1,p	PROPN
ma-257	176	23	|∇φ1,p|p	|∇φ1,p|p	NUM
ma-257	176	24	]	]	PUNCT
ma-257	176	25	ζdx	ζdx	PROPN
ma-257	176	26	≤	≤	PROPN
ma-257	177	1	εo	εo	ADP
ma-257	177	2	αp	αp	NOUN
ma-257	177	3	∫	∫	PROPN
ma-257	177	4	ω	ω	X
ma-257	178	1	λ1,p	λ1,p	PROPN
ma-257	178	2	φ	φ	PROPN
ma-257	178	3	p	p	NOUN
ma-257	178	4	α+1	α+1	NUM
ma-257	178	5	1,p	1,p	PROPN
ma-257	178	6	ζ	ζ	NOUN
ma-257	178	7	dx	dx	PROPN
ma-257	178	8	≤	≤	PROPN
ma-257	179	1	λ	λ	PROPN
ma-257	179	2	∫	∫	PROPN
ma-257	179	3	ω	ω	PROPN
ma-257	179	4	a1nζ	a1nζ	PUNCT
ma-257	179	5	dx	dx	PROPN
ma-257	179	6	=	=	PUNCT
ma-257	179	7	λ	λ	PROPN
ma-257	179	8	∫	∫	PROPN
ma-257	179	9	ω	ω	NUM
ma-257	179	10	a1	a1	PROPN
ma-257	179	11	[	[	PUNCT
ma-257	179	12	f	f	X
ma-257	179	13	(	(	PUNCT
ma-257	179	14	µε	µε	ADP
ma-257	179	15	1	1	NUM
ma-257	179	16	p−1	p−1	PROPN
ma-257	179	17	o	o	X
ma-257	179	18	po	po	NOUN
ma-257	179	19	,	,	PUNCT
ma-257	179	20	µε	µε	ADP
ma-257	179	21	1	1	NUM
ma-257	179	22	q−1	q−1	NOUN
ma-257	179	23	o	o	NOUN
ma-257	179	24	qo	qo	NOUN
ma-257	179	25	)	)	PUNCT
ma-257	179	26	−	−	PROPN
ma-257	180	1	(	(	PUNCT
ma-257	180	2	po	po	NOUN
ma-257	180	3	µε	µε	ADP
ma-257	180	4	1	1	NUM
ma-257	180	5	p−1	p−1	PROPN
ma-257	180	6	o	o	NOUN
ma-257	180	7	)	)	PUNCT
ma-257	181	1	α	α	X
ma-257	181	2	]	]	X
ma-257	181	3	ζ	ζ	X
ma-257	181	4	dx	dx	PROPN
ma-257	181	5	≤	≤	PROPN
ma-257	181	6	λ	λ	PROPN
ma-257	181	7	∫	∫	PROPN
ma-257	181	8	ω	ω	PROPN
ma-257	181	9	a1	a1	PROPN
ma-257	181	10	[	[	PUNCT
ma-257	181	11	f	f	X
ma-257	181	12	(	(	PUNCT
ma-257	181	13	ε	ε	PROPN
ma-257	181	14	1	1	NUM
ma-257	181	15	p−1	p−1	PROPN
ma-257	181	16	o	o	PROPN
ma-257	181	17	φ	φ	X
ma-257	181	18	po	po	PROPN
ma-257	181	19	α+1	α+1	NUM
ma-257	181	20	1,p	1,p	PROPN
ma-257	181	21	po	po	X
ma-257	181	22	,	,	PUNCT
ma-257	181	23	ε	ε	PROPN
ma-257	181	24	1	1	NUM
ma-257	182	1	q−1	q−1	NOUN
ma-257	182	2	o	o	NOUN
ma-257	182	3	φ	φ	PROPN
ma-257	182	4	qo	qo	PROPN
ma-257	182	5	β+1	β+1	PROPN
ma-257	182	6	1,q	1,q	PROPN
ma-257	182	7	qo	qo	NOUN
ma-257	182	8	)	)	PUNCT
ma-257	182	9	−	−	PROPN
ma-257	183	1	1	1	NUM
ma-257	183	2	(	(	PUNCT
ma-257	183	3	ε	ε	PROPN
ma-257	183	4	1	1	NUM
ma-257	184	1	p−1	p−1	PROPN
ma-257	184	2	o	o	PROPN
ma-257	184	3	φ	φ	X
ma-257	184	4	po	po	PROPN
ma-257	184	5	α+1	α+1	NUM
ma-257	184	6	1,p	1,p	PROPN
ma-257	184	7	po	po	X
ma-257	184	8	)	)	PUNCT
ma-257	184	9	α]ζ	α]ζ	PROPN
ma-257	184	10	dx	dx	PROPN
ma-257	185	1	=	=	SYM
ma-257	185	2	λ	λ	PROPN
ma-257	185	3	∫	∫	PROPN
ma-257	185	4	ω	ω	PROPN
ma-257	185	5	a1[f	a1[f	PROPN
ma-257	185	6	(	(	PUNCT
ma-257	185	7	ψ1	ψ1	NOUN
ma-257	185	8	,	,	PUNCT
ma-257	185	9	ψ2)−	ψ2)−	X
ma-257	185	10	1	1	NUM
ma-257	185	11	ψα1	ψα1	NOUN
ma-257	185	12	]	]	X
ma-257	185	13	ζ	ζ	X
ma-257	185	14	dx	dx	PROPN
ma-257	185	15	≤	≤	PROPN
ma-257	186	1	λ	λ	PROPN
ma-257	186	2	∫	∫	PROPN
ma-257	186	3	ω	ω	NUM
ma-257	186	4	a(x)[f	a(x)[f	PROPN
ma-257	186	5	(	(	PUNCT
ma-257	186	6	ψ1	ψ1	ADJ
ma-257	186	7	,	,	PUNCT
ma-257	186	8	ψ2)−	ψ2)−	X
ma-257	186	9	1	1	NUM
ma-257	186	10	ψα1	ψα1	NOUN
ma-257	186	11	]	]	X
ma-257	186	12	ζ	ζ	X
ma-257	186	13	dx	dx	PROPN
ma-257	186	14	.	.	PUNCT
ma-257	187	1	similarly	similarly	ADV
ma-257	187	2	,	,	PUNCT
ma-257	187	3	we	we	PRON
ma-257	187	4	can	can	AUX
ma-257	187	5	get	get	VERB
ma-257	187	6	also∫	also∫	NOUN
ma-257	188	1	ω	ω	PROPN
ma-257	188	2	|∇ψ2|q−2∇ψ2	|∇ψ2|q−2∇ψ2	PROPN
ma-257	188	3	·	·	PUNCT
ma-257	189	1	∇ζ	∇ζ	NOUN
ma-257	189	2	dx	dx	PROPN
ma-257	189	3	≤	≤	PROPN
ma-257	190	1	λ	λ	PROPN
ma-257	190	2	∫	∫	PROPN
ma-257	190	3	ω	ω	NUM
ma-257	190	4	b(x)[g(ψ1	b(x)[g(ψ1	NOUN
ma-257	190	5	,	,	PUNCT
ma-257	190	6	ψ2)−	ψ2)−	X
ma-257	190	7	1	1	NUM
ma-257	190	8	ψβ2	ψβ2	NOUN
ma-257	190	9	]	]	X
ma-257	190	10	ζ	ζ	X
ma-257	190	11	dx	dx	PROPN
ma-257	190	12	.	.	PUNCT
ma-257	191	1	thus	thus	ADV
ma-257	191	2	,	,	PUNCT
ma-257	191	3	(	(	PUNCT
ma-257	191	4	ψ1	ψ1	NOUN
ma-257	191	5	,	,	PUNCT
ma-257	191	6	ψ2	ψ2	NOUN
ma-257	191	7	)	)	PUNCT
ma-257	191	8	be	be	VERB
ma-257	191	9	a	a	DET
ma-257	191	10	positive	positive	ADJ
ma-257	191	11	weak	weak	ADJ
ma-257	191	12	subsolution	subsolution	NOUN
ma-257	191	13	of	of	ADP
ma-257	191	14	(	(	PUNCT
ma-257	191	15	1	1	NUM
ma-257	191	16	)	)	PUNCT
ma-257	191	17	.	.	PUNCT
ma-257	192	1	on	on	ADP
ma-257	192	2	the	the	DET
ma-257	192	3	other	other	ADJ
ma-257	192	4	side	side	NOUN
ma-257	192	5	,	,	PUNCT
ma-257	192	6	we	we	PRON
ma-257	192	7	will	will	AUX
ma-257	192	8	construct	construct	VERB
ma-257	192	9	a	a	DET
ma-257	192	10	positive	positive	ADJ
ma-257	192	11	weak	weak	ADJ
ma-257	192	12	supersolution	supersolution	NOUN
ma-257	192	13	of	of	ADP
ma-257	192	14	(	(	PUNCT
ma-257	192	15	1	1	NUM
ma-257	192	16	)	)	PUNCT
ma-257	192	17	.	.	PUNCT
ma-257	193	1	suppose	suppose	VERB
ma-257	193	2	(	(	PUNCT
ma-257	193	3	z1	z1	PROPN
ma-257	193	4	,	,	PUNCT
ma-257	193	5	z2	z2	NUM
ma-257	193	6	)	)	PUNCT
ma-257	193	7	=	=	PUNCT
ma-257	194	1	(	(	PUNCT
ma-257	194	2	c	c	NOUN
ma-257	194	3	ep(x	ep(x	NUM
ma-257	194	4	)	)	PUNCT
ma-257	194	5	,	,	PUNCT
ma-257	195	1	[	[	X
ma-257	195	2	λµbg(cµp	λµbg(cµp	NOUN
ma-257	195	3	,	,	PUNCT
ma-257	195	4	cµp	cµp	NOUN
ma-257	195	5	)	)	PUNCT
ma-257	195	6	]	]	PUNCT
ma-257	195	7	1	1	NUM
ma-257	195	8	q−1	q−1	PROPN
ma-257	195	9	eq(x	eq(x	NOUN
ma-257	195	10	)	)	PUNCT
ma-257	195	11	)	)	PUNCT
ma-257	195	12	where	where	SCONJ
ma-257	195	13	µa	µa	NOUN
ma-257	195	14	=	=	SYM
ma-257	195	15	‖a(x)‖∞	‖a(x)‖∞	ADJ
ma-257	195	16	,	,	PUNCT
ma-257	195	17	µb	µb	ADP
ma-257	195	18	=	=	PUNCT
ma-257	195	19	‖b(x)‖∞	‖b(x)‖∞	PROPN
ma-257	195	20	and	and	CCONJ
ma-257	195	21	µr	µr	ADP
ma-257	195	22	=	=	SYM
ma-257	195	23	‖er	‖er	X
ma-257	195	24	(	(	PUNCT
ma-257	195	25	x)‖∞	x)‖∞	NOUN
ma-257	195	26	for	for	ADP
ma-257	195	27	r	r	NOUN
ma-257	195	28	=	=	SYM
ma-257	195	29	p	p	NOUN
ma-257	195	30	,	,	PUNCT
ma-257	195	31	q.	q.	PROPN
ma-257	195	32	now	now	ADV
ma-257	195	33	by	by	ADP
ma-257	195	34	(	(	PUNCT
ma-257	195	35	s2	s2	PROPN
ma-257	195	36	)	)	PUNCT
ma-257	195	37	,	,	PUNCT
ma-257	195	38	we	we	PRON
ma-257	195	39	can	can	AUX
ma-257	195	40	takec	takec	VERB
ma-257	195	41	large	large	ADJ
ma-257	195	42	enough	enough	ADV
ma-257	195	43	such	such	ADJ
ma-257	195	44	that	that	DET
ma-257	195	45	cp−1	cp−1	PROPN
ma-257	195	46	≥	≥	NOUN
ma-257	195	47	λµaf	λµaf	PROPN
ma-257	195	48	(	(	PUNCT
ma-257	195	49	cµp	cµp	PROPN
ma-257	195	50	,	,	PUNCT
ma-257	195	51	[	[	X
ma-257	195	52	λµbg(cµp	λµbg(cµp	NOUN
ma-257	195	53	,	,	PUNCT
ma-257	195	54	cµp	cµp	NOUN
ma-257	195	55	)	)	PUNCT
ma-257	195	56	]	]	PUNCT
ma-257	195	57	1	1	NUM
ma-257	195	58	q−1µq	q−1µq	NOUN
ma-257	195	59	)	)	PUNCT
ma-257	195	60	,	,	PUNCT
ma-257	195	61	then	then	ADV
ma-257	195	62	,	,	PUNCT
ma-257	195	63	∫	∫	PROPN
ma-257	195	64	ω	ω	PROPN
ma-257	195	65	|∇z1|p−2∇z1	|∇z1|p−2∇z1	PROPN
ma-257	195	66	·	·	PUNCT
ma-257	195	67	∇ζ	∇ζ	NOUN
ma-257	195	68	dx	dx	PROPN
ma-257	195	69	=	=	SYM
ma-257	195	70	cp−1	cp−1	ADJ
ma-257	195	71	∫	∫	PROPN
ma-257	195	72	ω	ω	NUM
ma-257	195	73	|∇ep|p−2∇ep	|∇ep|p−2∇ep	ADJ
ma-257	195	74	·	·	PUNCT
ma-257	195	75	∇ζ	∇ζ	NOUN
ma-257	195	76	dx	dx	PROPN
ma-257	195	77	=	=	SYM
ma-257	195	78	cp−1	cp−1	ADJ
ma-257	195	79	∫	∫	PROPN
ma-257	195	80	ω	ω	PROPN
ma-257	195	81	ζ	ζ	PROPN
ma-257	195	82	dx	dx	PROPN
ma-257	195	83	≥	≥	PROPN
ma-257	196	1	∫	∫	PROPN
ma-257	196	2	ω	ω	PROPN
ma-257	196	3	λµaf	λµaf	PROPN
ma-257	196	4	(	(	PUNCT
ma-257	196	5	cµp	cµp	PROPN
ma-257	196	6	,	,	PUNCT
ma-257	196	7	[	[	X
ma-257	196	8	λµbg(cµp	λµbg(cµp	NOUN
ma-257	196	9	,	,	PUNCT
ma-257	196	10	cµp	cµp	NOUN
ma-257	196	11	)	)	PUNCT
ma-257	196	12	]	]	PUNCT
ma-257	196	13	1	1	NUM
ma-257	196	14	q−1µq	q−1µq	NOUN
ma-257	196	15	)	)	PUNCT
ma-257	196	16	·	·	PUNCT
ma-257	197	1	ζ	ζ	X
ma-257	197	2	dx	dx	PROPN
ma-257	197	3	≥	≥	PROPN
ma-257	197	4	λ	λ	X
ma-257	197	5	∫	∫	PROPN
ma-257	197	6	ω	ω	PROPN
ma-257	197	7	a(x)f	a(x)f	PROPN
ma-257	197	8	(	(	PUNCT
ma-257	197	9	c	c	PROPN
ma-257	197	10	ep(x	ep(x	NUM
ma-257	197	11	)	)	PUNCT
ma-257	197	12	,	,	PUNCT
ma-257	197	13	[	[	X
ma-257	197	14	λµbg(cµp	λµbg(cµp	NOUN
ma-257	197	15	,	,	PUNCT
ma-257	197	16	cµp	cµp	NOUN
ma-257	197	17	)	)	PUNCT
ma-257	197	18	]	]	PUNCT
ma-257	197	19	1	1	NUM
ma-257	197	20	q−1	q−1	PROPN
ma-257	197	21	eq(x	eq(x	NOUN
ma-257	197	22	)	)	PUNCT
ma-257	197	23	)	)	PUNCT
ma-257	197	24	·	·	PUNCT
ma-257	198	1	ζ	ζ	NOUN
ma-257	198	2	dx	dx	PROPN
ma-257	198	3	=	=	SYM
ma-257	198	4	λ	λ	PROPN
ma-257	198	5	∫	∫	PROPN
ma-257	198	6	ω	ω	PROPN
ma-257	198	7	a(x)f	a(x)f	PROPN
ma-257	198	8	(	(	PUNCT
ma-257	198	9	z1	z1	PROPN
ma-257	198	10	,	,	PUNCT
ma-257	198	11	z2	z2	PROPN
ma-257	198	12	)	)	PUNCT
ma-257	198	13	·	·	PUNCT
ma-257	199	1	ζ	ζ	NOUN
ma-257	199	2	dx	dx	PROPN
ma-257	199	3	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	199	4	eur	eur	PROPN
ma-257	199	5	.	.	PUNCT
ma-257	200	1	j.	j.	PROPN
ma-257	200	2	math	math	PROPN
ma-257	200	3	.	.	PUNCT
ma-257	201	1	anal	anal	PROPN
ma-257	201	2	.	.	PUNCT
ma-257	202	1	10.28924	10.28924	NUM
ma-257	202	2	/	/	SYM
ma-257	202	3	ada	ada	PROPN
ma-257	202	4	/	/	SYM
ma-257	202	5	ma.5.1	ma.5.1	PROPN
ma-257	202	6	9	9	NUM
ma-257	202	7	≥	≥	NUM
ma-257	202	8	λ	λ	X
ma-257	202	9	∫	∫	PROPN
ma-257	202	10	ω	ω	NUM
ma-257	202	11	a(x)[f	a(x)[f	PROPN
ma-257	202	12	(	(	PUNCT
ma-257	202	13	z1	z1	PROPN
ma-257	202	14	,	,	PUNCT
ma-257	202	15	z2)−	z2)−	PROPN
ma-257	202	16	1	1	NUM
ma-257	202	17	zα1	zα1	NOUN
ma-257	202	18	]	]	X
ma-257	202	19	ζ	ζ	X
ma-257	202	20	dx	dx	PROPN
ma-257	202	21	.	.	PUNCT
ma-257	203	1	also	also	ADV
ma-257	203	2	,	,	PUNCT
ma-257	203	3	by	by	ADP
ma-257	203	4	(	(	PUNCT
ma-257	203	5	s2	s2	PROPN
ma-257	203	6	)	)	PUNCT
ma-257	203	7	we	we	PRON
ma-257	203	8	can	can	AUX
ma-257	203	9	take	take	VERB
ma-257	203	10	cµp	cµp	PROPN
ma-257	203	11	≥	≥	PROPN
ma-257	203	12	µq[λµbg(cµp	µq[λµbg(cµp	PROPN
ma-257	203	13	,	,	PUNCT
ma-257	203	14	cµp	cµp	NOUN
ma-257	203	15	)	)	PUNCT
ma-257	203	16	]	]	PUNCT
ma-257	204	1	1	1	NUM
ma-257	204	2	q−1	q−1	PROPN
ma-257	204	3	.	.	PUNCT
ma-257	205	1	then∫	then∫	NOUN
ma-257	205	2	ω	ω	NUM
ma-257	205	3	|∇z2|q−2∇z2	|∇z2|q−2∇z2	PROPN
ma-257	205	4	·	·	PUNCT
ma-257	206	1	∇ζ	∇ζ	NOUN
ma-257	206	2	dx	dx	NOUN
ma-257	206	3	=	=	SYM
ma-257	206	4	λ	λ	PROPN
ma-257	206	5	∫	∫	PROPN
ma-257	206	6	ω	ω	PROPN
ma-257	206	7	µb	µb	ADP
ma-257	206	8	g(cµp	g(cµp	PROPN
ma-257	206	9	,	,	PUNCT
ma-257	206	10	cµp)|∇eq|q−2∇eq	cµp)|∇eq|q−2∇eq	NOUN
ma-257	206	11	·	·	PUNCT
ma-257	206	12	∇ζ	∇ζ	NOUN
ma-257	206	13	dx	dx	PROPN
ma-257	206	14	=	=	SYM
ma-257	206	15	λ	λ	PROPN
ma-257	206	16	∫	∫	PROPN
ma-257	206	17	ω	ω	PROPN
ma-257	206	18	µb	µb	ADP
ma-257	206	19	g(cµp	g(cµp	PROPN
ma-257	206	20	,	,	PUNCT
ma-257	206	21	cµp	cµp	NOUN
ma-257	206	22	)	)	PUNCT
ma-257	206	23	·	·	PUNCT
ma-257	207	1	ζ	ζ	X
ma-257	207	2	dx	dx	PROPN
ma-257	207	3	≥	≥	PROPN
ma-257	207	4	λ	λ	X
ma-257	207	5	∫	∫	PROPN
ma-257	207	6	ω	ω	X
ma-257	207	7	b(x)g	b(x)g	X
ma-257	207	8	(	(	PUNCT
ma-257	207	9	c	c	NOUN
ma-257	207	10	ep(x	ep(x	NUM
ma-257	207	11	)	)	PUNCT
ma-257	207	12	,	,	PUNCT
ma-257	208	1	[	[	X
ma-257	208	2	λµbg(cµp	λµbg(cµp	NOUN
ma-257	208	3	,	,	PUNCT
ma-257	208	4	cµp	cµp	NOUN
ma-257	208	5	)	)	PUNCT
ma-257	208	6	]	]	PUNCT
ma-257	208	7	1	1	NUM
ma-257	208	8	q−1	q−1	PROPN
ma-257	208	9	eq(x	eq(x	NOUN
ma-257	208	10	)	)	PUNCT
ma-257	208	11	)	)	PUNCT
ma-257	208	12	·	·	PUNCT
ma-257	209	1	ζ	ζ	X
ma-257	209	2	dx	dx	PROPN
ma-257	209	3	≥	≥	PROPN
ma-257	209	4	λ	λ	X
ma-257	209	5	∫	∫	PROPN
ma-257	209	6	ω	ω	PROPN
ma-257	209	7	b(x)g(z1	b(x)g(z1	PROPN
ma-257	209	8	,	,	PUNCT
ma-257	209	9	z2	z2	PROPN
ma-257	209	10	)	)	PUNCT
ma-257	209	11	·	·	PUNCT
ma-257	210	1	ζ	ζ	X
ma-257	210	2	dx	dx	PROPN
ma-257	210	3	≥	≥	PROPN
ma-257	210	4	λ	λ	X
ma-257	210	5	∫	∫	PROPN
ma-257	210	6	ω	ω	PROPN
ma-257	210	7	b(x)[g(z1	b(x)[g(z1	NOUN
ma-257	210	8	,	,	PUNCT
ma-257	210	9	z2)−	z2)−	PROPN
ma-257	210	10	1	1	NUM
ma-257	210	11	zβ2	zβ2	NOUN
ma-257	210	12	]	]	X
ma-257	210	13	ζ	ζ	X
ma-257	210	14	dx	dx	PROPN
ma-257	210	15	.	.	PUNCT
ma-257	211	1	thus	thus	ADV
ma-257	211	2	,	,	PUNCT
ma-257	211	3	(	(	PUNCT
ma-257	211	4	z1	z1	PROPN
ma-257	211	5	,	,	PUNCT
ma-257	211	6	z2	z2	PROPN
ma-257	211	7	)	)	PUNCT
ma-257	211	8	be	be	VERB
ma-257	211	9	a	a	DET
ma-257	211	10	positive	positive	ADJ
ma-257	211	11	weak	weak	ADJ
ma-257	211	12	supersolution	supersolution	NOUN
ma-257	211	13	of	of	ADP
ma-257	211	14	(	(	PUNCT
ma-257	211	15	1	1	NUM
ma-257	211	16	)	)	PUNCT
ma-257	211	17	for	for	ADP
ma-257	211	18	c	c	NOUN
ma-257	211	19	large	large	ADJ
ma-257	211	20	with	with	ADP
ma-257	211	21	ψ1	ψ1	ADJ
ma-257	211	22	≤	≤	ADJ
ma-257	211	23	z1	z1	NOUN
ma-257	211	24	and	and	CCONJ
ma-257	211	25	ψ2	ψ2	NOUN
ma-257	211	26	≤	≤	NOUN
ma-257	211	27	z2	z2	NOUN
ma-257	211	28	.	.	PUNCT
ma-257	212	1	thus	thus	ADV
ma-257	212	2	,	,	PUNCT
ma-257	212	3	there	there	PRON
ma-257	212	4	exists	exist	VERB
ma-257	212	5	a	a	DET
ma-257	212	6	positive	positive	ADJ
ma-257	212	7	weak	weak	ADJ
ma-257	212	8	solution	solution	NOUN
ma-257	212	9	(	(	PUNCT
ma-257	212	10	u	u	NOUN
ma-257	212	11	,	,	PUNCT
ma-257	212	12	v	v	NOUN
ma-257	212	13	)	)	PUNCT
ma-257	212	14	of	of	ADP
ma-257	212	15	(	(	PUNCT
ma-257	212	16	1	1	X
ma-257	212	17	)	)	PUNCT
ma-257	212	18	such	such	ADJ
ma-257	212	19	that	that	DET
ma-257	212	20	ψ1	ψ1	ADJ
ma-257	212	21	≤	≤	NUM
ma-257	212	22	u	u	NOUN
ma-257	212	23	≤	≤	NOUN
ma-257	212	24	z1	z1	NOUN
ma-257	212	25	and	and	CCONJ
ma-257	212	26	ψ2	ψ2	NOUN
ma-257	212	27	≤	≤	X
ma-257	212	28	v	v	PRON
ma-257	212	29	≤	≤	PROPN
ma-257	212	30	z2	z2	NUM
ma-257	212	31	.	.	PUNCT
ma-257	213	1	�	�	PROPN
ma-257	213	2	theorem	theorem	VERB
ma-257	213	3	2.2	2.2	NUM
ma-257	213	4	.	.	PUNCT
ma-257	214	1	let	let	AUX
ma-257	214	2	(	(	PUNCT
ma-257	214	3	s4	s4	PROPN
ma-257	214	4	)	)	PUNCT
ma-257	214	5	holds	hold	VERB
ma-257	214	6	with	with	ADP
ma-257	214	7	(	(	PUNCT
ma-257	214	8	p	p	NOUN
ma-257	214	9	−	−	PROPN
ma-257	214	10	1	1	NUM
ma-257	214	11	−	−	PROPN
ma-257	214	12	γ1)(q	γ1)(q	NOUN
ma-257	215	1	−	−	NUM
ma-257	215	2	1	1	NUM
ma-257	215	3	−	−	NOUN
ma-257	215	4	γ2	γ2	NOUN
ma-257	215	5	)	)	PUNCT
ma-257	215	6	=	=	SYM
ma-257	216	1	κ1κ2	κ1κ2	X
ma-257	216	2	and	and	CCONJ
ma-257	216	3	pκ1	pκ1	NOUN
ma-257	216	4	=	=	SYM
ma-257	216	5	q(p	q(p	PROPN
ma-257	216	6	−	−	NOUN
ma-257	216	7	1	1	NUM
ma-257	216	8	−	−	PROPN
ma-257	216	9	γ1	γ1	NOUN
ma-257	216	10	)	)	PUNCT
ma-257	216	11	.	.	PUNCT
ma-257	217	1	hence	hence	ADV
ma-257	217	2	(	(	PUNCT
ma-257	217	3	1	1	X
ma-257	217	4	)	)	PUNCT
ma-257	217	5	has	have	VERB
ma-257	217	6	no	no	DET
ma-257	217	7	positive	positive	ADJ
ma-257	217	8	weak	weak	ADJ
ma-257	217	9	solution	solution	NOUN
ma-257	217	10	if	if	SCONJ
ma-257	217	11	λ	λ	PROPN
ma-257	217	12	∈	∈	PROPN
ma-257	217	13	(	(	PUNCT
ma-257	217	14	λmax	λmax	NOUN
ma-257	217	15	,	,	PUNCT
ma-257	217	16	λmin	λmin	PROPN
ma-257	217	17	)	)	PUNCT
ma-257	217	18	,	,	PUNCT
ma-257	217	19	where	where	SCONJ
ma-257	217	20	λmax	λmax	NOUN
ma-257	217	21	=	=	SYM
ma-257	217	22	max	max	PROPN
ma-257	217	23	{	{	PUNCT
ma-257	217	24	λ1,p	λ1,p	PROPN
ma-257	217	25	−2	−2	PROPN
ma-257	217	26	t	t	PROPN
ma-257	217	27	,	,	PUNCT
ma-257	218	1	λ1,q	λ1,q	PROPN
ma-257	218	2	−2	−2	PROPN
ma-257	218	3	t	t	X
ma-257	218	4	}	}	PUNCT
ma-257	218	5	and	and	CCONJ
ma-257	218	6	λmin	λmin	NOUN
ma-257	218	7	=	=	SYM
ma-257	218	8	min	min	PROPN
ma-257	218	9	{	{	PUNCT
ma-257	218	10	λ1,p	λ1,p	PROPN
ma-257	218	11	2s	2s	PROPN
ma-257	218	12	,	,	PUNCT
ma-257	219	1	λ1,q	λ1,q	PROPN
ma-257	219	2	2s	2s	NOUN
ma-257	219	3	}	}	PUNCT
ma-257	219	4	with	with	ADP
ma-257	219	5	t	t	PROPN
ma-257	219	6	=	=	SYM
ma-257	219	7	min{f0a0	min{f0a0	PROPN
ma-257	219	8	,	,	PUNCT
ma-257	219	9	g0b0	g0b0	ADJ
ma-257	219	10	}	}	PUNCT
ma-257	219	11	and	and	CCONJ
ma-257	219	12	s	s	NOUN
ma-257	219	13	=	=	X
ma-257	219	14	max{f0a1	max{f0a1	PROPN
ma-257	219	15	,	,	PUNCT
ma-257	219	16	g0b1	g0b1	NOUN
ma-257	219	17	}	}	PUNCT
ma-257	219	18	.	.	PUNCT
ma-257	220	1	proof	proof	NOUN
ma-257	220	2	.	.	PUNCT
ma-257	221	1	let	let	VERB
ma-257	221	2	(	(	PUNCT
ma-257	221	3	u	u	NOUN
ma-257	221	4	,	,	PUNCT
ma-257	221	5	v	v	NOUN
ma-257	221	6	)	)	PUNCT
ma-257	221	7	be	be	AUX
ma-257	221	8	a	a	DET
ma-257	221	9	positive	positive	ADJ
ma-257	221	10	weak	weak	ADJ
ma-257	221	11	solution	solution	NOUN
ma-257	221	12	of	of	ADP
ma-257	221	13	(	(	PUNCT
ma-257	221	14	1	1	NUM
ma-257	221	15	)	)	PUNCT
ma-257	221	16	.	.	PUNCT
ma-257	222	1	proof	proof	NOUN
ma-257	222	2	’s	’s	PART
ma-257	222	3	idea	idea	NOUN
ma-257	222	4	is	be	AUX
ma-257	222	5	that	that	SCONJ
ma-257	222	6	a	a	DET
ma-257	222	7	contradiction	contradiction	NOUN
ma-257	222	8	will	will	AUX
ma-257	222	9	beobtained	beobtaine	VERB
ma-257	222	10	in	in	ADP
ma-257	222	11	the	the	DET
ma-257	222	12	end	end	NOUN
ma-257	222	13	.	.	PUNCT
ma-257	223	1	multiplying	multiply	VERB
ma-257	223	2	the	the	DET
ma-257	223	3	1st	1st	ADJ
ma-257	223	4	and	and	CCONJ
ma-257	223	5	2nd	2nd	ADJ
ma-257	223	6	equation	equation	NOUN
ma-257	223	7	of	of	ADP
ma-257	223	8	(	(	PUNCT
ma-257	223	9	1	1	NUM
ma-257	223	10	)	)	PUNCT
ma-257	223	11	by	by	ADP
ma-257	223	12	u	u	PROPN
ma-257	223	13	,	,	PUNCT
ma-257	223	14	v	v	NOUN
ma-257	223	15	,	,	PUNCT
ma-257	223	16	respectively	respectively	ADV
ma-257	223	17	.	.	PUNCT
ma-257	224	1	applyingyoung	applyingyoung	PROPN
ma-257	224	2	’s	’s	PART
ma-257	224	3	inequality	inequality	NOUN
ma-257	224	4	,	,	PUNCT
ma-257	224	5	so	so	ADV
ma-257	224	6	∫	∫	PROPN
ma-257	224	7	ω	ω	PROPN
ma-257	224	8	|∇u|pdx	|∇u|pdx	NUM
ma-257	224	9	≤	≤	PROPN
ma-257	224	10	λ	λ	PROPN
ma-257	224	11	∫	∫	PROPN
ma-257	224	12	ω	ω	PROPN
ma-257	224	13	f0a(x	f0a(x	NOUN
ma-257	224	14	)	)	PUNCT
ma-257	224	15	(	(	PUNCT
ma-257	224	16	up	up	ADP
ma-257	224	17	µ1	µ1	PROPN
ma-257	224	18	+	+	SYM
ma-257	224	19	vq	vq	PROPN
ma-257	224	20	µ2	µ2	PROPN
ma-257	224	21	)	)	PUNCT
ma-257	224	22	dx	dx	PROPN
ma-257	224	23	,	,	PUNCT
ma-257	224	24	(	(	PUNCT
ma-257	224	25	14	14	NUM
ma-257	224	26	)	)	PUNCT
ma-257	224	27	with	with	ADP
ma-257	224	28	µ1	µ1	PROPN
ma-257	224	29	=	=	SYM
ma-257	225	1	p	p	PROPN
ma-257	225	2	1+γ1	1+γ1	PROPN
ma-257	225	3	>	>	SYM
ma-257	225	4	1	1	NUM
ma-257	225	5	and	and	CCONJ
ma-257	225	6	µ2	µ2	PROPN
ma-257	225	7	=	=	SYM
ma-257	225	8	p	p	NOUN
ma-257	225	9	p−1−γ1	p−1−γ1	X
ma-257	225	10	>	>	X
ma-257	226	1	1	1	X
ma-257	226	2	.	.	PUNCT
ma-257	226	3	similarly	similarly	ADV
ma-257	226	4	,	,	PUNCT
ma-257	226	5	we	we	PRON
ma-257	226	6	have∫	have∫	VERB
ma-257	226	7	ω	ω	NUM
ma-257	226	8	|∇v	|∇v	NOUN
ma-257	226	9	|qdx	|qdx	PUNCT
ma-257	226	10	≤	≤	PROPN
ma-257	226	11	λ	λ	PROPN
ma-257	226	12	∫	∫	PROPN
ma-257	226	13	ω	ω	PROPN
ma-257	226	14	g0b(x	g0b(x	PROPN
ma-257	226	15	)	)	PUNCT
ma-257	226	16	(	(	PUNCT
ma-257	226	17	up	up	ADV
ma-257	226	18	ϑ1	ϑ1	PROPN
ma-257	226	19	+	+	CCONJ
ma-257	226	20	vq	vq	PROPN
ma-257	226	21	ϑ2	ϑ2	PROPN
ma-257	226	22	)	)	PUNCT
ma-257	226	23	dx	dx	PROPN
ma-257	226	24	,	,	PUNCT
ma-257	226	25	(	(	PUNCT
ma-257	226	26	15	15	NUM
ma-257	226	27	)	)	PUNCT
ma-257	226	28	with	with	ADP
ma-257	226	29	ϑ1	ϑ1	NOUN
ma-257	226	30	=	=	SYM
ma-257	226	31	q	q	X
ma-257	226	32	q−1−γ2	q−1−γ2	X
ma-257	226	33	>	>	SYM
ma-257	226	34	1	1	NUM
ma-257	226	35	and	and	CCONJ
ma-257	226	36	ϑ2	ϑ2	PROPN
ma-257	226	37	=	=	SYM
ma-257	226	38	q	q	PROPN
ma-257	227	1	1+γ2	1+γ2	NUM
ma-257	227	2	>	>	SYM
ma-257	227	3	1	1	X
ma-257	227	4	.	.	X
ma-257	227	5	note	note	VERB
ma-257	227	6	that	that	SCONJ
ma-257	227	7	λ1,p	λ1,p	PROPN
ma-257	227	8	∫	∫	PROPN
ma-257	227	9	ω	ω	PROPN
ma-257	227	10	updx	updx	VERB
ma-257	227	11	≤	≤	NUM
ma-257	227	12	∫	∫	PROPN
ma-257	227	13	ω	ω	PROPN
ma-257	227	14	|∇u|pdx	|∇u|pdx	PROPN
ma-257	227	15	,	,	PUNCT
ma-257	227	16	λ1,q	λ1,q	PROPN
ma-257	227	17	∫	∫	PROPN
ma-257	227	18	ω	ω	PROPN
ma-257	227	19	vqdx	vqdx	PROPN
ma-257	227	20	≤	≤	NUM
ma-257	227	21	∫	∫	PROPN
ma-257	228	1	ω	ω	NUM
ma-257	228	2	|∇v	|∇v	NOUN
ma-257	228	3	|qdx	|qdx	PROPN
ma-257	228	4	.	.	PUNCT
ma-257	229	1	(	(	PUNCT
ma-257	229	2	16	16	X
ma-257	229	3	)	)	PUNCT
ma-257	229	4	combining	combine	VERB
ma-257	229	5	(	(	PUNCT
ma-257	229	6	14)-(16	14)-(16	NUM
ma-257	229	7	)	)	PUNCT
ma-257	229	8	,	,	PUNCT
ma-257	229	9	we	we	PRON
ma-257	229	10	obtain	obtain	VERB
ma-257	229	11	λ1,p	λ1,p	PROPN
ma-257	229	12	∫	∫	PROPN
ma-257	229	13	ω	ω	PROPN
ma-257	229	14	updx	updx	ADJ
ma-257	229	15	+	+	X
ma-257	229	16	λ1,q	λ1,q	PROPN
ma-257	229	17	∫	∫	PROPN
ma-257	229	18	ω	ω	PROPN
ma-257	229	19	vqdx	vqdx	PROPN
ma-257	229	20	≤	≤	PUNCT
ma-257	229	21	λ	λ	PROPN
ma-257	229	22	[	[	PUNCT
ma-257	229	23	∫	∫	PROPN
ma-257	229	24	ω	ω	PROPN
ma-257	229	25	(	(	PUNCT
ma-257	229	26	f0a(x	f0a(x	NOUN
ma-257	229	27	)	)	PUNCT
ma-257	229	28	µ1	µ1	PROPN
ma-257	229	29	+	+	CCONJ
ma-257	229	30	g0b(x	g0b(x	ADJ
ma-257	229	31	)	)	PUNCT
ma-257	229	32	ϑ1	ϑ1	NOUN
ma-257	229	33	)	)	PUNCT
ma-257	229	34	updx	updx	VERB
ma-257	230	1	+	+	X
ma-257	230	2	∫	∫	PROPN
ma-257	230	3	ω	ω	PROPN
ma-257	230	4	(	(	PUNCT
ma-257	230	5	f0a(x	f0a(x	NOUN
ma-257	230	6	)	)	PUNCT
ma-257	230	7	µ2	µ2	PROPN
ma-257	230	8	+	+	CCONJ
ma-257	230	9	g0b(x	g0b(x	ADJ
ma-257	230	10	)	)	PUNCT
ma-257	230	11	ϑ2	ϑ2	PROPN
ma-257	230	12	)	)	PUNCT
ma-257	230	13	vqdx	vqdx	NOUN
ma-257	230	14	]	]	PUNCT
ma-257	230	15	.	.	PUNCT
ma-257	231	1	(	(	PUNCT
ma-257	231	2	17	17	NUM
ma-257	231	3	)	)	PUNCT
ma-257	231	4	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	231	5	eur	eur	PROPN
ma-257	231	6	.	.	PUNCT
ma-257	232	1	j.	j.	PROPN
ma-257	232	2	math	math	PROPN
ma-257	232	3	.	.	PUNCT
ma-257	233	1	anal	anal	PROPN
ma-257	233	2	.	.	PUNCT
ma-257	234	1	10.28924	10.28924	NUM
ma-257	234	2	/	/	SYM
ma-257	234	3	ada	ada	PROPN
ma-257	234	4	/	/	SYM
ma-257	234	5	ma.5.1	ma.5.1	PROPN
ma-257	234	6	10case(i	10case(i	NUM
ma-257	234	7	):	):	PUNCT
ma-257	234	8	when	when	SCONJ
ma-257	234	9	x	x	SYM
ma-257	234	10	∈	∈	PROPN
ma-257	234	11	ω̄δ	ω̄δ	NUM
ma-257	234	12	;	;	PUNCT
ma-257	234	13	a(x	a(x	PROPN
ma-257	234	14	)	)	PUNCT
ma-257	234	15	≤	≤	PUNCT
ma-257	234	16	−a0	−a0	PROPN
ma-257	234	17	,	,	PUNCT
ma-257	234	18	b(x	b(x	NOUN
ma-257	234	19	)	)	PUNCT
ma-257	234	20	≤	≤	PUNCT
ma-257	235	1	−b0	−b0	PROPN
ma-257	235	2	,	,	PUNCT
ma-257	235	3	hence	hence	ADV
ma-257	235	4	(	(	PUNCT
ma-257	235	5	λ1,p	λ1,p	PROPN
ma-257	235	6	+	+	CCONJ
ma-257	235	7	2λt	2λt	NOUN
ma-257	235	8	)	)	PUNCT
ma-257	235	9	∫	∫	PROPN
ma-257	236	1	ω	ω	PROPN
ma-257	236	2	updx	updx	ADJ
ma-257	236	3	+	+	X
ma-257	236	4	(	(	PUNCT
ma-257	236	5	λ1,q	λ1,q	NOUN
ma-257	236	6	+	+	CCONJ
ma-257	236	7	2λt	2λt	ADJ
ma-257	236	8	)	)	PUNCT
ma-257	236	9	∫	∫	PROPN
ma-257	237	1	ω	ω	PROPN
ma-257	237	2	vqdx	vqdx	PROPN
ma-257	237	3	≤	≤	PROPN
ma-257	237	4	0	0	NUM
ma-257	237	5	,	,	PUNCT
ma-257	237	6	(	(	PUNCT
ma-257	237	7	18	18	NUM
ma-257	237	8	)	)	PUNCT
ma-257	237	9	where	where	SCONJ
ma-257	237	10	t	t	NOUN
ma-257	237	11	=	=	SYM
ma-257	237	12	min{f0a0	min{f0a0	PROPN
ma-257	237	13	,	,	PUNCT
ma-257	237	14	g0b0	g0b0	ADJ
ma-257	237	15	}	}	PUNCT
ma-257	237	16	,	,	PUNCT
ma-257	237	17	that	that	PRON
ma-257	237	18	is	be	AUX
ma-257	237	19	a	a	DET
ma-257	237	20	contradiction	contradiction	NOUN
ma-257	237	21	when	when	SCONJ
ma-257	237	22	λ	λ	X
ma-257	237	23	>	>	X
ma-257	237	24	λmax	λmax	PROPN
ma-257	237	25	.case(ii	.case(ii	PUNCT
ma-257	237	26	):	):	PUNCT
ma-257	237	27	when	when	SCONJ
ma-257	237	28	x	x	SYM
ma-257	237	29	∈	∈	NOUN
ma-257	237	30	ω−	ω−	ADP
ma-257	237	31	ω̄δ	ω̄δ	NOUN
ma-257	237	32	;	;	PUNCT
ma-257	237	33	a(x	a(x	PROPN
ma-257	237	34	)	)	PUNCT
ma-257	237	35	≤	≤	NUM
ma-257	237	36	a1	a1	NOUN
ma-257	237	37	,	,	PUNCT
ma-257	237	38	b(x	b(x	NOUN
ma-257	237	39	)	)	PUNCT
ma-257	237	40	≤	≤	NUM
ma-257	237	41	b1	b1	NOUN
ma-257	237	42	,	,	PUNCT
ma-257	237	43	hence	hence	ADV
ma-257	237	44	(	(	PUNCT
ma-257	237	45	λ1,p	λ1,p	PROPN
ma-257	237	46	−	−	PROPN
ma-257	237	47	2λs	2λs	NOUN
ma-257	237	48	)	)	PUNCT
ma-257	237	49	∫	∫	PROPN
ma-257	238	1	ω	ω	PROPN
ma-257	238	2	updx	updx	ADJ
ma-257	238	3	+	+	X
ma-257	238	4	(	(	PUNCT
ma-257	238	5	λ1,q	λ1,q	NOUN
ma-257	238	6	−	−	NOUN
ma-257	238	7	2λs	2λs	ADJ
ma-257	238	8	)	)	PUNCT
ma-257	238	9	∫	∫	PROPN
ma-257	239	1	ω	ω	NUM
ma-257	239	2	vqdx	vqdx	PROPN
ma-257	239	3	≤	≤	PROPN
ma-257	239	4	0	0	NUM
ma-257	239	5	,	,	PUNCT
ma-257	239	6	(	(	PUNCT
ma-257	239	7	19	19	NUM
ma-257	239	8	)	)	PUNCT
ma-257	239	9	where	where	SCONJ
ma-257	239	10	s	s	NOUN
ma-257	239	11	=	=	SYM
ma-257	239	12	max{f0a1	max{f0a1	PROPN
ma-257	239	13	,	,	PUNCT
ma-257	239	14	g0b1	g0b1	NOUN
ma-257	239	15	}	}	PUNCT
ma-257	239	16	,	,	PUNCT
ma-257	239	17	that	that	PRON
ma-257	239	18	is	be	AUX
ma-257	239	19	a	a	DET
ma-257	239	20	contradiction	contradiction	NOUN
ma-257	239	21	when	when	SCONJ
ma-257	239	22	λ	λ	X
ma-257	239	23	<	<	X
ma-257	239	24	λmin	λmin	PROPN
ma-257	239	25	.	.	PUNCT
ma-257	240	1	�	�	PROPN
ma-257	240	2	3	3	NUM
ma-257	240	3	.	.	PUNCT
ma-257	240	4	stability	stability	NOUN
ma-257	240	5	and	and	CCONJ
ma-257	240	6	instability	instability	NOUN
ma-257	240	7	results	result	NOUN
ma-257	240	8	now	now	ADV
ma-257	240	9	,	,	PUNCT
ma-257	240	10	we	we	PRON
ma-257	240	11	study	study	VERB
ma-257	240	12	the	the	DET
ma-257	240	13	stability	stability	NOUN
ma-257	240	14	and	and	CCONJ
ma-257	240	15	instability	instability	NOUN
ma-257	240	16	results	result	NOUN
ma-257	240	17	of	of	ADP
ma-257	240	18	positive	positive	ADJ
ma-257	240	19	weak	weak	ADJ
ma-257	240	20	solution	solution	NOUN
ma-257	240	21	for	for	ADP
ma-257	240	22	(	(	PUNCT
ma-257	240	23	1	1	NUM
ma-257	240	24	)	)	PUNCT
ma-257	240	25	with	with	ADP
ma-257	240	26	differentchoices	differentchoice	NOUN
ma-257	240	27	of	of	ADP
ma-257	240	28	f	f	PROPN
ma-257	240	29	and	and	CCONJ
ma-257	240	30	g	g	PROPN
ma-257	240	31	(	(	PUNCT
ma-257	240	32	see	see	VERB
ma-257	240	33	[	[	X
ma-257	240	34	28,29	28,29	NUM
ma-257	240	35	]	]	PUNCT
ma-257	240	36	)	)	PUNCT
ma-257	240	37	.	.	PUNCT
ma-257	241	1	suppose	suppose	VERB
ma-257	241	2	(	(	PUNCT
ma-257	241	3	u	u	NOUN
ma-257	241	4	,	,	PUNCT
ma-257	241	5	v	v	NOUN
ma-257	241	6	)	)	PUNCT
ma-257	241	7	be	be	VERB
ma-257	241	8	any	any	DET
ma-257	241	9	positive	positive	ADJ
ma-257	241	10	weak	weak	ADJ
ma-257	241	11	solution	solution	NOUN
ma-257	241	12	of	of	ADP
ma-257	241	13	(	(	PUNCT
ma-257	241	14	1	1	NUM
ma-257	241	15	)	)	PUNCT
ma-257	241	16	,	,	PUNCT
ma-257	241	17	hence	hence	ADV
ma-257	241	18	the	the	DET
ma-257	241	19	linearized	linearize	VERB
ma-257	241	20	system	system	NOUN
ma-257	241	21	associated	associate	VERB
ma-257	241	22	with(1	with(1	PROPN
ma-257	241	23	)	)	PUNCT
ma-257	241	24	is	be	AUX
ma-257	241	25	defined	define	VERB
ma-257	241	26	as	as	ADP
ma-257	241	27	follows:	follows:	PROPN
ma-257	241	28	−(p	−(p	NOUN
ma-257	241	29	−	−	PROPN
ma-257	241	30	1)div(|∇u|p−2∇ϕ)−	1)div(|∇u|p−2∇ϕ)−	PROPN
ma-257	241	31	λa(x	λa(x	NOUN
ma-257	241	32	)	)	PUNCT
ma-257	242	1	[	[	X
ma-257	242	2	(	(	PUNCT
ma-257	242	3	fu	fu	NOUN
ma-257	242	4	+	+	CCONJ
ma-257	242	5	α	α	NOUN
ma-257	242	6	uα+1	uα+1	NOUN
ma-257	242	7	)	)	PUNCT
ma-257	242	8	ϕ+	ϕ+	CCONJ
ma-257	242	9	fvψ	fvψ	NOUN
ma-257	242	10	]	]	PUNCT
ma-257	243	1	=	=	PUNCT
ma-257	243	2	µϕ	µϕ	ADJ
ma-257	243	3	,	,	PUNCT
ma-257	243	4	x	x	SYM
ma-257	243	5	∈	∈	PROPN
ma-257	243	6	ω	ω	PROPN
ma-257	243	7	,	,	PUNCT
ma-257	243	8	−(q	−(q	VERB
ma-257	243	9	−	−	ADP
ma-257	243	10	1)div(|∇v	1)div(|∇v	NOUN
ma-257	243	11	|q−2∇ψ)−	|q−2∇ψ)−	NOUN
ma-257	243	12	λb(x	λb(x	NUM
ma-257	243	13	)	)	PUNCT
ma-257	243	14	[	[	PUNCT
ma-257	243	15	guϕ+	guϕ+	ADJ
ma-257	243	16	(	(	PUNCT
ma-257	243	17	gv	gv	ADP
ma-257	243	18	+	+	CCONJ
ma-257	243	19	β	β	X
ma-257	243	20	vβ+1	vβ+1	X
ma-257	243	21	)	)	PUNCT
ma-257	243	22	ψ	ψ	X
ma-257	243	23	]	]	X
ma-257	243	24	=	=	SYM
ma-257	243	25	µψ	µψ	PROPN
ma-257	243	26	,	,	PUNCT
ma-257	243	27	x	x	SYM
ma-257	243	28	∈	∈	PROPN
ma-257	243	29	ω	ω	PROPN
ma-257	243	30	,	,	PUNCT
ma-257	243	31	ϕ	ϕ	X
ma-257	243	32	=	=	SYM
ma-257	243	33	0	0	PUNCT
ma-257	243	34	=	=	SYM
ma-257	243	35	ψ	ψ	NOUN
ma-257	243	36	,	,	PUNCT
ma-257	243	37	x	x	SYM
ma-257	243	38	∈	∈	PROPN
ma-257	243	39	∂ω	∂ω	PROPN
ma-257	243	40	,	,	PUNCT
ma-257	243	41	(	(	PUNCT
ma-257	243	42	20	20	NUM
ma-257	243	43	)	)	PUNCT
ma-257	243	44	where	where	SCONJ
ma-257	243	45	subscripts	subscript	NOUN
ma-257	243	46	refer	refer	VERB
ma-257	243	47	to	to	ADP
ma-257	243	48	the	the	DET
ma-257	243	49	partial	partial	ADJ
ma-257	243	50	derivative	derivative	NOUN
ma-257	243	51	of	of	ADP
ma-257	243	52	f	f	PROPN
ma-257	243	53	or	or	CCONJ
ma-257	243	54	g	g	PROPN
ma-257	243	55	(	(	PUNCT
ma-257	243	56	see	see	VERB
ma-257	243	57	[	[	X
ma-257	243	58	30	30	NUM
ma-257	243	59	]	]	PUNCT
ma-257	243	60	)	)	PUNCT
ma-257	243	61	.	.	PUNCT
ma-257	244	1	let	let	AUX
ma-257	244	2	µ1	µ1	PROPN
ma-257	244	3	be	be	AUX
ma-257	244	4	the	the	DET
ma-257	244	5	first	first	ADJ
ma-257	244	6	eigenvalueand	eigenvalueand	NOUN
ma-257	244	7	(	(	PUNCT
ma-257	244	8	ϕ1	ϕ1	NOUN
ma-257	244	9	,	,	PUNCT
ma-257	244	10	ψ1	ψ1	NOUN
ma-257	244	11	)	)	PUNCT
ma-257	244	12	be	be	VERB
ma-257	244	13	the	the	DET
ma-257	244	14	corresponding	corresponding	ADJ
ma-257	244	15	eigenfunction	eigenfunction	NOUN
ma-257	244	16	of	of	ADP
ma-257	244	17	(	(	PUNCT
ma-257	244	18	20	20	NUM
ma-257	244	19	)	)	PUNCT
ma-257	244	20	such	such	ADJ
ma-257	244	21	that	that	DET
ma-257	244	22	ϕ1	ϕ1	NOUN
ma-257	244	23	,	,	PUNCT
ma-257	244	24	ψ1	ψ1	NOUN
ma-257	244	25	>	>	X
ma-257	244	26	0	0	PUNCT
ma-257	244	27	in	in	ADP
ma-257	244	28	ω	ω	PROPN
ma-257	244	29	.	.	PUNCT
ma-257	245	1	definition	definition	NOUN
ma-257	245	2	3.1	3.1	NUM
ma-257	245	3	.	.	PUNCT
ma-257	246	1	we	we	PRON
ma-257	246	2	say	say	VERB
ma-257	246	3	(	(	PUNCT
ma-257	246	4	u	u	NOUN
ma-257	246	5	,	,	PUNCT
ma-257	246	6	v	v	NOUN
ma-257	246	7	)	)	PUNCT
ma-257	246	8	is	be	AUX
ma-257	246	9	a	a	DET
ma-257	246	10	stable	stable	ADJ
ma-257	246	11	solution	solution	NOUN
ma-257	246	12	of	of	ADP
ma-257	246	13	(	(	PUNCT
ma-257	246	14	1	1	X
ma-257	246	15	)	)	PUNCT
ma-257	246	16	if	if	SCONJ
ma-257	246	17	all	all	PRON
ma-257	246	18	eigenvalues	eigenvalue	VERB
ma-257	246	19	of	of	ADP
ma-257	246	20	(	(	PUNCT
ma-257	246	21	20	20	NUM
ma-257	246	22	)	)	PUNCT
ma-257	246	23	are	be	AUX
ma-257	246	24	strictly	strictly	ADV
ma-257	246	25	positive	positive	ADJ
ma-257	246	26	,	,	PUNCT
ma-257	246	27	which	which	PRON
ma-257	246	28	can	can	AUX
ma-257	246	29	be	be	AUX
ma-257	246	30	implied	imply	VERB
ma-257	246	31	if	if	SCONJ
ma-257	246	32	the	the	DET
ma-257	246	33	first	first	ADJ
ma-257	246	34	eigenvalue	eigenvalue	PROPN
ma-257	246	35	µ1	µ1	PROPN
ma-257	246	36	>	>	X
ma-257	246	37	0	0	PROPN
ma-257	246	38	.	.	PUNCT
ma-257	247	1	otherwise	otherwise	ADV
ma-257	247	2	(	(	PUNCT
ma-257	247	3	u	u	NOUN
ma-257	247	4	,	,	PUNCT
ma-257	247	5	v	v	NOUN
ma-257	247	6	)	)	PUNCT
ma-257	247	7	is	be	AUX
ma-257	247	8	unstable	unstable	ADJ
ma-257	247	9	.	.	PUNCT
ma-257	248	1	our	our	PRON
ma-257	248	2	assumptions	assumption	NOUN
ma-257	248	3	are	be	AUX
ma-257	248	4	as	as	SCONJ
ma-257	248	5	follows	follow	VERB
ma-257	248	6	:	:	PUNCT
ma-257	248	7	(	(	PUNCT
ma-257	248	8	t1	t1	NOUN
ma-257	248	9	):	):	PUNCT
ma-257	248	10	for	for	ADP
ma-257	248	11	u	u	NOUN
ma-257	248	12	,	,	PUNCT
ma-257	248	13	v	v	X
ma-257	248	14	>	>	X
ma-257	248	15	0	0	NUM
ma-257	248	16	,	,	PUNCT
ma-257	248	17	the	the	DET
ma-257	248	18	functions	function	NOUN
ma-257	248	19	fv	fv	X
ma-257	248	20	,	,	PUNCT
ma-257	248	21	gu	gu	PROPN
ma-257	248	22	are	be	AUX
ma-257	248	23	positive	positive	ADJ
ma-257	248	24	.	.	PUNCT
ma-257	249	1	(	(	PUNCT
ma-257	249	2	t2	t2	NOUN
ma-257	249	3	):	):	PUNCT
ma-257	249	4	for	for	ADP
ma-257	249	5	every	every	DET
ma-257	249	6	v	v	NOUN
ma-257	249	7	>	>	X
ma-257	249	8	0	0	NUM
ma-257	249	9	,	,	PUNCT
ma-257	249	10	the	the	DET
ma-257	249	11	function	function	NOUN
ma-257	249	12	(	(	PUNCT
ma-257	249	13	f	f	PROPN
ma-257	249	14	(	(	PUNCT
ma-257	249	15	u	u	PROPN
ma-257	249	16	,	,	PUNCT
ma-257	249	17	v)−	v)−	PROPN
ma-257	249	18	u−α	u−α	X
ma-257	249	19	)	)	PUNCT
ma-257	250	1	/up−1	/up−1	X
ma-257	250	2	is	be	AUX
ma-257	250	3	strictly	strictly	ADV
ma-257	250	4	increasing	increase	VERB
ma-257	250	5	at	at	ADP
ma-257	250	6	u.	u.	PROPN
ma-257	250	7	(	(	PUNCT
ma-257	250	8	t3	t3	PROPN
ma-257	250	9	):	):	PUNCT
ma-257	250	10	for	for	ADP
ma-257	250	11	every	every	DET
ma-257	250	12	u	u	NOUN
ma-257	250	13	>	>	X
ma-257	250	14	0	0	PROPN
ma-257	250	15	,	,	PUNCT
ma-257	250	16	the	the	DET
ma-257	250	17	function	function	NOUN
ma-257	250	18	(	(	PUNCT
ma-257	250	19	g(u	g(u	PROPN
ma-257	250	20	,	,	PUNCT
ma-257	250	21	v)−	v)−	ADJ
ma-257	250	22	v−β	v−β	NOUN
ma-257	250	23	)	)	PUNCT
ma-257	251	1	/vq−1	/vq−1	PRON
ma-257	251	2	is	be	AUX
ma-257	251	3	strictly	strictly	ADV
ma-257	251	4	increasing	increase	VERB
ma-257	251	5	at	at	ADP
ma-257	251	6	v	v	NUM
ma-257	251	7	.	.	PUNCT
ma-257	252	1	theorem	theorem	NOUN
ma-257	252	2	3.1	3.1	NUM
ma-257	252	3	.	.	PUNCT
ma-257	253	1	suppose	suppose	VERB
ma-257	253	2	that	that	SCONJ
ma-257	253	3	(	(	PUNCT
ma-257	253	4	t1)-(t3	t1)-(t3	NOUN
ma-257	253	5	)	)	PUNCT
ma-257	253	6	are	be	AUX
ma-257	253	7	satisfied	satisfied	ADJ
ma-257	253	8	,	,	PUNCT
ma-257	253	9	hence	hence	ADV
ma-257	253	10	every	every	DET
ma-257	253	11	positive	positive	ADJ
ma-257	253	12	weak	weak	ADJ
ma-257	253	13	solution	solution	NOUN
ma-257	253	14	of	of	ADP
ma-257	253	15	(	(	PUNCT
ma-257	253	16	1	1	NUM
ma-257	253	17	)	)	PUNCT
ma-257	253	18	is	be	AUX
ma-257	253	19	stable	stable	ADJ
ma-257	253	20	in	in	ADP
ma-257	253	21	ω̄δ	ω̄δ	NOUN
ma-257	253	22	and	and	CCONJ
ma-257	253	23	unstable	unstable	ADJ
ma-257	253	24	in	in	ADP
ma-257	253	25	ω−	ω−	PROPN
ma-257	253	26	ω̄δ	ω̄δ	PRON
ma-257	253	27	.	.	PUNCT
ma-257	254	1	proof	proof	NOUN
ma-257	254	2	.	.	PUNCT
ma-257	255	1	let	let	VERB
ma-257	255	2	(	(	PUNCT
ma-257	255	3	uo	uo	INTJ
ma-257	255	4	,	,	PUNCT
ma-257	255	5	vo	vo	NOUN
ma-257	255	6	)	)	PUNCT
ma-257	255	7	be	be	AUX
ma-257	255	8	any	any	DET
ma-257	255	9	positive	positive	ADJ
ma-257	255	10	weak	weak	ADJ
ma-257	255	11	solution	solution	NOUN
ma-257	255	12	of	of	ADP
ma-257	255	13	(	(	PUNCT
ma-257	255	14	1	1	NUM
ma-257	255	15	)	)	PUNCT
ma-257	255	16	.	.	PUNCT
ma-257	256	1	multiplying	multiply	VERB
ma-257	256	2	the	the	DET
ma-257	256	3	1st	1st	ADJ
ma-257	256	4	and	and	CCONJ
ma-257	256	5	2nd	2nd	ADJ
ma-257	256	6	equation	equation	NOUN
ma-257	256	7	of(1	of(1	NOUN
ma-257	256	8	)	)	PUNCT
ma-257	256	9	by	by	ADP
ma-257	256	10	(	(	PUNCT
ma-257	256	11	p	p	X
ma-257	256	12	−	−	PROPN
ma-257	256	13	1)ϕ1	1)ϕ1	NUM
ma-257	256	14	,	,	PUNCT
ma-257	256	15	(	(	PUNCT
ma-257	256	16	q	q	PROPN
ma-257	256	17	−	−	PROPN
ma-257	256	18	1)ψ1	1)ψ1	PROPN
ma-257	256	19	,	,	PUNCT
ma-257	256	20	respectively	respectively	ADV
ma-257	256	21	and	and	CCONJ
ma-257	256	22	integrating	integrate	VERB
ma-257	256	23	over	over	ADP
ma-257	256	24	ω	ω	NOUN
ma-257	256	25	,	,	PUNCT
ma-257	256	26	so	so	ADV
ma-257	256	27	−	−	PROPN
ma-257	257	1	(	(	PUNCT
ma-257	257	2	p	p	NOUN
ma-257	257	3	−	−	PROPN
ma-257	257	4	1	1	NUM
ma-257	257	5	)	)	PUNCT
ma-257	257	6	∫	∫	PROPN
ma-257	258	1	ω	ω	NUM
ma-257	258	2	ϕ1(x)div(|∇uo	ϕ1(x)div(|∇uo	PROPN
ma-257	259	1	|p−2∇uo)dx	|p−2∇uo)dx	NOUN
ma-257	259	2	=	=	PUNCT
ma-257	259	3	(	(	PUNCT
ma-257	259	4	p	p	NOUN
ma-257	259	5	−	−	PROPN
ma-257	259	6	1)λ	1)λ	NUM
ma-257	259	7	∫	∫	PROPN
ma-257	259	8	ω	ω	NUM
ma-257	259	9	ϕ1(x)a(x	ϕ1(x)a(x	PROPN
ma-257	259	10	)	)	PUNCT
ma-257	260	1	[	[	PUNCT
ma-257	260	2	f	f	X
ma-257	260	3	(	(	PUNCT
ma-257	260	4	uo	uo	INTJ
ma-257	260	5	,	,	PUNCT
ma-257	260	6	vo)−	vo)−	NOUN
ma-257	260	7	1	1	NUM
ma-257	260	8	uoα	uoα	NOUN
ma-257	260	9	]	]	PUNCT
ma-257	260	10	dx	dx	PROPN
ma-257	260	11	,	,	PUNCT
ma-257	260	12	(	(	PUNCT
ma-257	260	13	21	21	NUM
ma-257	260	14	)	)	PUNCT
ma-257	260	15	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	260	16	eur	eur	PROPN
ma-257	260	17	.	.	PUNCT
ma-257	261	1	j.	j.	PROPN
ma-257	261	2	math	math	PROPN
ma-257	261	3	.	.	PUNCT
ma-257	262	1	anal	anal	PROPN
ma-257	262	2	.	.	PUNCT
ma-257	263	1	10.28924	10.28924	NUM
ma-257	263	2	/	/	SYM
ma-257	263	3	ada	ada	PROPN
ma-257	263	4	/	/	SYM
ma-257	263	5	ma.5.1	ma.5.1	PROPN
ma-257	263	6	11and	11and	NOUN
ma-257	263	7	−	−	PROPN
ma-257	264	1	(	(	PUNCT
ma-257	264	2	q	q	NOUN
ma-257	264	3	−	−	PROPN
ma-257	264	4	1	1	NUM
ma-257	264	5	)	)	PUNCT
ma-257	264	6	∫	∫	PROPN
ma-257	265	1	ω	ω	NUM
ma-257	265	2	ψ1(x)div(|∇vo	ψ1(x)div(|∇vo	PROPN
ma-257	266	1	|q−2∇vo)dx	|q−2∇vo)dx	VERB
ma-257	266	2	=	=	SYM
ma-257	266	3	(	(	PUNCT
ma-257	266	4	q	q	NOUN
ma-257	266	5	−	−	PROPN
ma-257	266	6	1)λ	1)λ	NUM
ma-257	266	7	∫	∫	PROPN
ma-257	266	8	ω	ω	NUM
ma-257	266	9	ψ1(x)b(x	ψ1(x)b(x	PROPN
ma-257	266	10	)	)	PUNCT
ma-257	267	1	[	[	PUNCT
ma-257	267	2	g(uo	g(uo	NOUN
ma-257	267	3	,	,	PUNCT
ma-257	267	4	vo)−	vo)−	NOUN
ma-257	267	5	1	1	NUM
ma-257	267	6	voβ	voβ	NOUN
ma-257	267	7	]	]	PUNCT
ma-257	267	8	dx	dx	PROPN
ma-257	267	9	.	.	PUNCT
ma-257	268	1	(	(	PUNCT
ma-257	268	2	22	22	NUM
ma-257	268	3	)	)	PUNCT
ma-257	268	4	similarly	similarly	ADV
ma-257	268	5	,	,	PUNCT
ma-257	268	6	multiplying	multiply	VERB
ma-257	268	7	the	the	DET
ma-257	268	8	1st	1st	ADJ
ma-257	268	9	and	and	CCONJ
ma-257	268	10	2nd	2nd	ADJ
ma-257	268	11	equation	equation	NOUN
ma-257	268	12	of	of	ADP
ma-257	268	13	(	(	PUNCT
ma-257	268	14	20	20	NUM
ma-257	268	15	)	)	PUNCT
ma-257	268	16	by	by	ADP
ma-257	268	17	−uo	−uo	NOUN
ma-257	268	18	,	,	PUNCT
ma-257	268	19	−vo	−vo	ADJ
ma-257	268	20	,	,	PUNCT
ma-257	268	21	respectively	respectively	ADV
ma-257	268	22	and	and	CCONJ
ma-257	268	23	integratingover	integratingover	PROPN
ma-257	268	24	ω	ω	PROPN
ma-257	268	25	,	,	PUNCT
ma-257	268	26	so	so	ADV
ma-257	268	27	−µ1	−µ1	PROPN
ma-257	268	28	∫	∫	PROPN
ma-257	268	29	ω	ω	PROPN
ma-257	268	30	uoϕ1(x)dx	uoϕ1(x)dx	ADP
ma-257	268	31	=(	=(	PROPN
ma-257	269	1	p	p	NOUN
ma-257	269	2	−	−	PROPN
ma-257	269	3	1	1	NUM
ma-257	269	4	)	)	PUNCT
ma-257	269	5	∫	∫	PROPN
ma-257	270	1	ω	ω	NUM
ma-257	270	2	uo	uo	NOUN
ma-257	270	3	div(|∇uo	div(|∇uo	X
ma-257	271	1	|p−2∇ϕ1)dx	|p−2∇ϕ1)dx	ADJ
ma-257	272	1	+	+	CCONJ
ma-257	272	2	λ	λ	PROPN
ma-257	272	3	∫	∫	PROPN
ma-257	272	4	ω	ω	PROPN
ma-257	272	5	ϕ1(x)a(x	ϕ1(x)a(x	PROPN
ma-257	272	6	)	)	PUNCT
ma-257	273	1	[	[	PUNCT
ma-257	273	2	uo	uo	NOUN
ma-257	273	3	fu	fu	NOUN
ma-257	273	4	+	+	CCONJ
ma-257	273	5	α	α	NOUN
ma-257	273	6	uoα	uoα	NOUN
ma-257	273	7	]	]	PUNCT
ma-257	273	8	dx	dx	PROPN
ma-257	274	1	+	+	CCONJ
ma-257	274	2	λ	λ	PROPN
ma-257	274	3	∫	∫	PROPN
ma-257	274	4	ω	ω	PROPN
ma-257	274	5	ψ1(x)a(x)fvuo	ψ1(x)a(x)fvuo	PROPN
ma-257	274	6	dx	dx	PROPN
ma-257	274	7	,	,	PUNCT
ma-257	274	8	(	(	PUNCT
ma-257	274	9	23	23	NUM
ma-257	274	10	)	)	PUNCT
ma-257	274	11	and	and	CCONJ
ma-257	274	12	−µ1	−µ1	PROPN
ma-257	274	13	∫	∫	PROPN
ma-257	274	14	ω	ω	PROPN
ma-257	274	15	voψ1(x)dx	voψ1(x)dx	NOUN
ma-257	274	16	=(	=(	PROPN
ma-257	274	17	q	q	NOUN
ma-257	274	18	−	−	PROPN
ma-257	274	19	1	1	NUM
ma-257	274	20	)	)	PUNCT
ma-257	274	21	∫	∫	PROPN
ma-257	275	1	ω	ω	NUM
ma-257	275	2	vo	vo	PROPN
ma-257	276	1	div(|∇vo	div(|∇vo	PROPN
ma-257	276	2	|q−2∇ψ1)dx	|q−2∇ψ1)dx	VERB
ma-257	276	3	+	+	CCONJ
ma-257	276	4	λ	λ	PROPN
ma-257	276	5	∫	∫	PROPN
ma-257	276	6	ω	ω	X
ma-257	276	7	ϕ1(x)b(x	ϕ1(x)b(x	PROPN
ma-257	276	8	)	)	PUNCT
ma-257	276	9	[	[	PUNCT
ma-257	276	10	vogv	vogv	VERB
ma-257	276	11	+	+	CCONJ
ma-257	276	12	β	β	NOUN
ma-257	276	13	voβ	voβ	NOUN
ma-257	276	14	]	]	PUNCT
ma-257	276	15	dx	dx	PROPN
ma-257	277	1	+	+	CCONJ
ma-257	278	1	λ	λ	PROPN
ma-257	278	2	∫	∫	PROPN
ma-257	278	3	ω	ω	PROPN
ma-257	278	4	ψ1(x)b(x)guvo	ψ1(x)b(x)guvo	PROPN
ma-257	278	5	dx	dx	PROPN
ma-257	278	6	.	.	PUNCT
ma-257	279	1	(	(	PUNCT
ma-257	279	2	24	24	NUM
ma-257	279	3	)	)	PUNCT
ma-257	279	4	combining	combine	VERB
ma-257	279	5	(	(	PUNCT
ma-257	279	6	21	21	NUM
ma-257	279	7	)	)	PUNCT
ma-257	279	8	to	to	ADP
ma-257	279	9	(	(	PUNCT
ma-257	279	10	24	24	NUM
ma-257	279	11	)	)	PUNCT
ma-257	279	12	,	,	PUNCT
ma-257	279	13	we	we	PRON
ma-257	279	14	get	get	VERB
ma-257	279	15	−	−	NOUN
ma-257	280	1	(	(	PUNCT
ma-257	280	2	p	p	NOUN
ma-257	280	3	−	−	PROPN
ma-257	280	4	1	1	NUM
ma-257	280	5	)	)	PUNCT
ma-257	280	6	∫	∫	PROPN
ma-257	280	7	ω	ω	PROPN
ma-257	280	8	[	[	PUNCT
ma-257	280	9	ϕ1(x)div(|∇uo	ϕ1(x)div(|∇uo	X
ma-257	280	10	|p−2∇uo)−	|p−2∇uo)−	NUM
ma-257	280	11	uo	uo	X
ma-257	280	12	div(|∇uo	div(|∇uo	ADV
ma-257	280	13	|p−2∇ϕ1	|p−2∇ϕ1	ADJ
ma-257	280	14	)	)	PUNCT
ma-257	280	15	]	]	PUNCT
ma-257	281	1	dx	dx	PROPN
ma-257	282	1	−	−	PROPN
ma-257	283	1	(	(	PUNCT
ma-257	283	2	q	q	NOUN
ma-257	283	3	−	−	PROPN
ma-257	283	4	1	1	NUM
ma-257	283	5	)	)	PUNCT
ma-257	283	6	∫	∫	PROPN
ma-257	283	7	ω	ω	PROPN
ma-257	283	8	[	[	PUNCT
ma-257	283	9	ψ1(x)div(|∇vo	ψ1(x)div(|∇vo	NOUN
ma-257	283	10	|q−2∇vo)−	|q−2∇vo)−	X
ma-257	283	11	vo	vo	NOUN
ma-257	283	12	div(|∇vo	div(|∇vo	PROPN
ma-257	283	13	|q−2∇ψ1	|q−2∇ψ1	NOUN
ma-257	283	14	)	)	PUNCT
ma-257	283	15	]	]	PUNCT
ma-257	284	1	dx	dx	PROPN
ma-257	285	1	+	+	CCONJ
ma-257	285	2	λ	λ	PROPN
ma-257	285	3	∫	∫	PROPN
ma-257	285	4	ω	ω	PROPN
ma-257	285	5	ϕ1(x)a(x	ϕ1(x)a(x	PROPN
ma-257	285	6	)	)	PUNCT
ma-257	286	1	[	[	PUNCT
ma-257	286	2	uo	uo	NOUN
ma-257	286	3	fu	fu	NOUN
ma-257	286	4	+	+	CCONJ
ma-257	286	5	α	α	NOUN
ma-257	286	6	uoα	uoα	NOUN
ma-257	286	7	]	]	PUNCT
ma-257	286	8	dx	dx	PROPN
ma-257	287	1	+	+	CCONJ
ma-257	287	2	λ	λ	PROPN
ma-257	287	3	∫	∫	PROPN
ma-257	287	4	ω	ω	NUM
ma-257	287	5	ψ1(x)b(x	ψ1(x)b(x	PROPN
ma-257	287	6	)	)	PUNCT
ma-257	288	1	[	[	PUNCT
ma-257	288	2	vogv	vogv	VERB
ma-257	288	3	+	+	CCONJ
ma-257	288	4	β	β	NOUN
ma-257	288	5	voβ	voβ	NOUN
ma-257	288	6	]	]	X
ma-257	288	7	dx	dx	PROPN
ma-257	289	1	−	−	PROPN
ma-257	289	2	(	(	PUNCT
ma-257	289	3	p	p	NOUN
ma-257	289	4	−	−	PROPN
ma-257	289	5	1)λ	1)λ	NUM
ma-257	289	6	∫	∫	PROPN
ma-257	289	7	ω	ω	NUM
ma-257	289	8	ϕ1(x)a(x	ϕ1(x)a(x	PROPN
ma-257	289	9	)	)	PUNCT
ma-257	290	1	[	[	PUNCT
ma-257	290	2	f	f	X
ma-257	290	3	(	(	PUNCT
ma-257	290	4	uo	uo	INTJ
ma-257	290	5	,	,	PUNCT
ma-257	290	6	vo)−	vo)−	NOUN
ma-257	290	7	1	1	NUM
ma-257	290	8	uoα	uoα	NOUN
ma-257	290	9	]	]	PUNCT
ma-257	290	10	dx	dx	PROPN
ma-257	290	11	−	−	PROPN
ma-257	290	12	(	(	PUNCT
ma-257	290	13	q	q	PROPN
ma-257	290	14	−	−	PROPN
ma-257	290	15	1)λ	1)λ	NUM
ma-257	290	16	∫	∫	PROPN
ma-257	290	17	ω	ω	PROPN
ma-257	290	18	ψ1(x)b(x)[g(uo	ψ1(x)b(x)[g(uo	NOUN
ma-257	290	19	,	,	PUNCT
ma-257	290	20	vo)−	vo)−	NOUN
ma-257	290	21	1	1	NUM
ma-257	290	22	voβ	voβ	NOUN
ma-257	290	23	]	]	X
ma-257	290	24	dx	dx	PROPN
ma-257	291	1	+	+	CCONJ
ma-257	291	2	λ	λ	PROPN
ma-257	291	3	∫	∫	PROPN
ma-257	291	4	ω	ω	PROPN
ma-257	291	5	a(x)ψ1(x)fvuo	a(x)ψ1(x)fvuo	PROPN
ma-257	291	6	dx	dx	PROPN
ma-257	292	1	+	+	CCONJ
ma-257	292	2	λ	λ	PROPN
ma-257	292	3	∫	∫	PROPN
ma-257	292	4	ω	ω	PROPN
ma-257	292	5	b(x)ϕ1(x)guvo	b(x)ϕ1(x)guvo	PROPN
ma-257	293	1	dx	dx	PROPN
ma-257	293	2	=	=	SYM
ma-257	293	3	−µ1	−µ1	PROPN
ma-257	293	4	∫	∫	PROPN
ma-257	293	5	ω	ω	PROPN
ma-257	294	1	[	[	X
ma-257	294	2	uoϕ1(x	uoϕ1(x	NOUN
ma-257	294	3	)	)	PUNCT
ma-257	294	4	+	+	CCONJ
ma-257	294	5	voψ1(x)]dx	voψ1(x)]dx	PROPN
ma-257	294	6	.	.	X
ma-257	294	7	(	(	PUNCT
ma-257	294	8	25	25	NUM
ma-257	294	9	)	)	PUNCT
ma-257	294	10	using	use	VERB
ma-257	294	11	green	green	PROPN
ma-257	294	12	’s	’s	PART
ma-257	294	13	first	first	ADJ
ma-257	294	14	identity	identity	NOUN
ma-257	294	15	,	,	PUNCT
ma-257	294	16	then∫	then∫	NOUN
ma-257	294	17	ω	ω	NUM
ma-257	295	1	uo	uo	NOUN
ma-257	295	2	div(|∇uo	div(|∇uo	X
ma-257	295	3	|p−2∇ϕ1)dx	|p−2∇ϕ1)dx	PROPN
ma-257	295	4	=	=	PUNCT
ma-257	295	5	∫	∫	PROPN
ma-257	295	6	ω	ω	NUM
ma-257	295	7	ϕ1(x)div(|∇uo	ϕ1(x)div(|∇uo	X
ma-257	295	8	|p−2∇uo)dx	|p−2∇uo)dx	NOUN
ma-257	295	9	,	,	PUNCT
ma-257	295	10	(	(	PUNCT
ma-257	295	11	26	26	NUM
ma-257	295	12	)	)	PUNCT
ma-257	295	13	and	and	CCONJ
ma-257	295	14	∫	∫	PROPN
ma-257	295	15	ω	ω	PROPN
ma-257	295	16	vo	vo	PROPN
ma-257	295	17	div(|∇vo	div(|∇vo	PROPN
ma-257	295	18	|q−2∇ψ1)dx	|q−2∇ψ1)dx	VERB
ma-257	295	19	=	=	SYM
ma-257	295	20	∫	∫	PROPN
ma-257	295	21	ω	ω	NUM
ma-257	295	22	ψ1(x)div(|∇vo	ψ1(x)div(|∇vo	PROPN
ma-257	296	1	|q−2∇vo)dx	|q−2∇vo)dx	PROPN
ma-257	296	2	.	.	PUNCT
ma-257	297	1	(	(	PUNCT
ma-257	297	2	27	27	NUM
ma-257	297	3	)	)	PUNCT
ma-257	297	4	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	297	5	eur	eur	PROPN
ma-257	297	6	.	.	PUNCT
ma-257	298	1	j.	j.	PROPN
ma-257	298	2	math	math	PROPN
ma-257	298	3	.	.	PUNCT
ma-257	299	1	anal	anal	PROPN
ma-257	299	2	.	.	PUNCT
ma-257	300	1	10.28924	10.28924	NUM
ma-257	300	2	/	/	SYM
ma-257	300	3	ada	ada	PROPN
ma-257	300	4	/	/	SYM
ma-257	300	5	ma.5.1	ma.5.1	PROPN
ma-257	300	6	12by	12by	NOUN
ma-257	300	7	using	use	VERB
ma-257	300	8	(	(	PUNCT
ma-257	300	9	26	26	NUM
ma-257	300	10	)	)	PUNCT
ma-257	300	11	and	and	CCONJ
ma-257	300	12	(	(	PUNCT
ma-257	300	13	27	27	NUM
ma-257	300	14	)	)	PUNCT
ma-257	300	15	in	in	ADP
ma-257	300	16	(	(	PUNCT
ma-257	300	17	25	25	NUM
ma-257	300	18	)	)	PUNCT
ma-257	300	19	,	,	PUNCT
ma-257	300	20	then	then	ADV
ma-257	300	21	λ	λ	PROPN
ma-257	300	22	∫	∫	PROPN
ma-257	300	23	ω	ω	PROPN
ma-257	300	24	ϕ1(x)a(x	ϕ1(x)a(x	PROPN
ma-257	300	25	)	)	PUNCT
ma-257	301	1	[	[	PUNCT
ma-257	301	2	uo	uo	NOUN
ma-257	301	3	fu	fu	NOUN
ma-257	301	4	−	−	PROPN
ma-257	302	1	(	(	PUNCT
ma-257	302	2	p	p	PROPN
ma-257	302	3	−	−	PROPN
ma-257	302	4	1)f	1)f	NUM
ma-257	302	5	(	(	PUNCT
ma-257	302	6	uo	uo	INTJ
ma-257	302	7	,	,	PUNCT
ma-257	302	8	vo	vo	NOUN
ma-257	302	9	)	)	PUNCT
ma-257	303	1	+	+	CCONJ
ma-257	303	2	α+	α+	PUNCT
ma-257	303	3	p	p	NOUN
ma-257	303	4	−	−	PROPN
ma-257	303	5	1	1	NUM
ma-257	303	6	uoα	uoα	NOUN
ma-257	303	7	]	]	PUNCT
ma-257	303	8	dx	dx	PROPN
ma-257	304	1	+	+	CCONJ
ma-257	304	2	λ	λ	PROPN
ma-257	304	3	∫	∫	PROPN
ma-257	304	4	ω	ω	NUM
ma-257	304	5	ψ1(x)b(x	ψ1(x)b(x	PROPN
ma-257	304	6	)	)	PUNCT
ma-257	305	1	[	[	PUNCT
ma-257	305	2	vogv	vogv	VERB
ma-257	305	3	−	−	PROPN
ma-257	305	4	(	(	PUNCT
ma-257	305	5	q	q	NOUN
ma-257	305	6	−	−	PROPN
ma-257	305	7	1)g(uo	1)g(uo	NUM
ma-257	305	8	,	,	PUNCT
ma-257	305	9	vo	vo	NOUN
ma-257	305	10	)	)	PUNCT
ma-257	306	1	+	+	NUM
ma-257	306	2	β	β	X
ma-257	306	3	+	+	NOUN
ma-257	306	4	q	q	NOUN
ma-257	306	5	−	−	NUM
ma-257	306	6	1	1	NUM
ma-257	306	7	voβ	voβ	NOUN
ma-257	306	8	]	]	PUNCT
ma-257	306	9	dx	dx	PROPN
ma-257	307	1	+	+	CCONJ
ma-257	308	1	λ	λ	PROPN
ma-257	308	2	∫	∫	PROPN
ma-257	308	3	ω	ω	PROPN
ma-257	308	4	ψ1(x)a(x)fvuodx	ψ1(x)a(x)fvuodx	PROPN
ma-257	308	5	+	+	CCONJ
ma-257	308	6	λ	λ	PROPN
ma-257	308	7	∫	∫	PROPN
ma-257	308	8	ω	ω	PROPN
ma-257	308	9	ϕ1(x)b(x)guvodx	ϕ1(x)b(x)guvodx	PROPN
ma-257	308	10	=	=	SYM
ma-257	308	11	−µ1	−µ1	PROPN
ma-257	308	12	∫	∫	PROPN
ma-257	308	13	ω	ω	PROPN
ma-257	309	1	[	[	X
ma-257	309	2	uoϕ1(x	uoϕ1(x	NOUN
ma-257	309	3	)	)	PUNCT
ma-257	309	4	+	+	CCONJ
ma-257	310	1	voψ1(x)]dx	voψ1(x)]dx	PROPN
ma-257	310	2	.	.	X
ma-257	310	3	(	(	PUNCT
ma-257	310	4	28	28	NUM
ma-257	310	5	)	)	PUNCT
ma-257	310	6	also	also	ADV
ma-257	310	7	,	,	PUNCT
ma-257	310	8	since	since	SCONJ
ma-257	310	9	(	(	PUNCT
ma-257	310	10	f	f	X
ma-257	310	11	(	(	PUNCT
ma-257	310	12	uo	uo	INTJ
ma-257	310	13	,	,	PUNCT
ma-257	310	14	vo)−	vo)−	NOUN
ma-257	310	15	u−αo	u−αo	NOUN
ma-257	310	16	)	)	PUNCT
ma-257	310	17	/uo	/uo	PUNCT
ma-257	311	1	p−1	p−1	PROPN
ma-257	311	2	is	be	AUX
ma-257	311	3	strictly	strictly	ADV
ma-257	311	4	increasing	increase	VERB
ma-257	311	5	at	at	ADP
ma-257	311	6	uo	uo	NOUN
ma-257	311	7	∀vo	∀vo	NOUN
ma-257	311	8	>	>	PUNCT
ma-257	311	9	0	0	NUM
ma-257	311	10	,	,	PUNCT
ma-257	311	11	then	then	ADV
ma-257	311	12	for	for	ADP
ma-257	311	13	uo	uo	PROPN
ma-257	311	14	,	,	PUNCT
ma-257	311	15	vo	vo	X
ma-257	311	16	>	>	X
ma-257	311	17	0	0	NUM
ma-257	311	18	uo	uo	PROPN
ma-257	311	19	fu	fu	NOUN
ma-257	311	20	−	−	PROPN
ma-257	312	1	(	(	PUNCT
ma-257	312	2	p	p	PROPN
ma-257	312	3	−	−	PROPN
ma-257	312	4	1)f	1)f	NUM
ma-257	312	5	(	(	PUNCT
ma-257	312	6	uo	uo	INTJ
ma-257	312	7	,	,	PUNCT
ma-257	312	8	vo	vo	NOUN
ma-257	312	9	)	)	PUNCT
ma-257	313	1	+	+	CCONJ
ma-257	313	2	(	(	PUNCT
ma-257	313	3	α+	α+	PRON
ma-257	313	4	p	p	NOUN
ma-257	313	5	−	−	PROPN
ma-257	313	6	1)uo	1)uo	PROPN
ma-257	313	7	−α	−α	PROPN
ma-257	313	8	uop	uop	NOUN
ma-257	313	9	>	>	X
ma-257	313	10	0	0	NUM
ma-257	313	11	,	,	PUNCT
ma-257	313	12	(	(	PUNCT
ma-257	313	13	29	29	NUM
ma-257	313	14	)	)	PUNCT
ma-257	313	15	and	and	CCONJ
ma-257	313	16	since	since	SCONJ
ma-257	313	17	(	(	PUNCT
ma-257	313	18	g(uo	g(uo	NOUN
ma-257	313	19	,	,	PUNCT
ma-257	313	20	vo)−	vo)−	NOUN
ma-257	313	21	vo−β	vo−β	NOUN
ma-257	313	22	)	)	PUNCT
ma-257	313	23	/vo	/vo	PUNCT
ma-257	314	1	q−1	q−1	PRON
ma-257	314	2	is	be	AUX
ma-257	314	3	strictly	strictly	ADV
ma-257	314	4	increasing	increase	VERB
ma-257	314	5	at	at	ADP
ma-257	314	6	vo	vo	NOUN
ma-257	314	7	∀uo	∀uo	X
ma-257	314	8	>	>	X
ma-257	314	9	0	0	NUM
ma-257	314	10	,	,	PUNCT
ma-257	314	11	then	then	ADV
ma-257	314	12	for	for	ADP
ma-257	314	13	uo	uo	PROPN
ma-257	314	14	,	,	PUNCT
ma-257	314	15	vo	vo	X
ma-257	314	16	>	>	X
ma-257	314	17	0	0	NUM
ma-257	314	18	vogv	vogv	VERB
ma-257	314	19	−	−	PROPN
ma-257	315	1	(	(	PUNCT
ma-257	315	2	q	q	NOUN
ma-257	315	3	−	−	PROPN
ma-257	315	4	1)g(uo	1)g(uo	NUM
ma-257	315	5	,	,	PUNCT
ma-257	315	6	vo	vo	NOUN
ma-257	315	7	)	)	PUNCT
ma-257	316	1	+	+	CCONJ
ma-257	316	2	(	(	PUNCT
ma-257	316	3	β	β	X
ma-257	316	4	+	+	NOUN
ma-257	316	5	q	q	ADJ
ma-257	316	6	−	−	PROPN
ma-257	316	7	1)vo	1)vo	NUM
ma-257	316	8	−β	−β	PROPN
ma-257	316	9	voq	voq	VERB
ma-257	316	10	>	>	X
ma-257	316	11	0	0	X
ma-257	316	12	.	.	PUNCT
ma-257	317	1	(	(	PUNCT
ma-257	317	2	30	30	X
ma-257	317	3	)	)	PUNCT
ma-257	317	4	case(i	case(i	PROPN
ma-257	317	5	):	):	PUNCT
ma-257	317	6	when	when	SCONJ
ma-257	317	7	x	x	SYM
ma-257	317	8	∈	∈	PROPN
ma-257	317	9	ω̄δ	ω̄δ	NUM
ma-257	317	10	;	;	PUNCT
ma-257	317	11	a(x	a(x	NOUN
ma-257	317	12	)	)	PUNCT
ma-257	317	13	,	,	PUNCT
ma-257	317	14	b(x	b(x	NOUN
ma-257	317	15	)	)	PUNCT
ma-257	317	16	<	<	X
ma-257	317	17	0	0	X
ma-257	317	18	.	.	PUNCT
ma-257	318	1	thus	thus	ADV
ma-257	318	2	substituting	substitute	VERB
ma-257	318	3	(	(	PUNCT
ma-257	318	4	29)-(30	29)-(30	NUM
ma-257	318	5	)	)	PUNCT
ma-257	318	6	in	in	ADP
ma-257	318	7	(	(	PUNCT
ma-257	318	8	28	28	NUM
ma-257	318	9	)	)	PUNCT
ma-257	318	10	,	,	PUNCT
ma-257	318	11	so	so	ADV
ma-257	318	12	−	−	PUNCT
ma-257	318	13	µ1	µ1	PROPN
ma-257	318	14	∫	∫	PROPN
ma-257	318	15	ω	ω	PROPN
ma-257	319	1	[	[	NOUN
ma-257	319	2	uoϕ1(x	uoϕ1(x	NOUN
ma-257	319	3	)	)	PUNCT
ma-257	319	4	+	+	CCONJ
ma-257	319	5	voψ1(x)]dx	voψ1(x)]dx	X
ma-257	319	6	<	<	X
ma-257	319	7	0	0	NUM
ma-257	319	8	,	,	PUNCT
ma-257	319	9	(	(	PUNCT
ma-257	319	10	31	31	NUM
ma-257	319	11	)	)	PUNCT
ma-257	319	12	then	then	ADV
ma-257	319	13	,	,	PUNCT
ma-257	319	14	µ1	µ1	PROPN
ma-257	319	15	>	>	X
ma-257	319	16	0	0	PUNCT
ma-257	320	1	and	and	CCONJ
ma-257	320	2	the	the	DET
ma-257	320	3	solution	solution	NOUN
ma-257	320	4	is	be	AUX
ma-257	320	5	stable.case(ii	stable.case(ii	ADJ
ma-257	320	6	):	):	PUNCT
ma-257	320	7	when	when	SCONJ
ma-257	320	8	x	x	SYM
ma-257	320	9	∈	∈	NOUN
ma-257	320	10	ω−	ω−	ADP
ma-257	320	11	ω̄δ	ω̄δ	NOUN
ma-257	320	12	;	;	PUNCT
ma-257	320	13	a(x	a(x	NOUN
ma-257	320	14	)	)	PUNCT
ma-257	320	15	,	,	PUNCT
ma-257	320	16	b(x	b(x	NOUN
ma-257	320	17	)	)	PUNCT
ma-257	320	18	>	>	X
ma-257	320	19	0	0	X
ma-257	320	20	.	.	PUNCT
ma-257	321	1	thus	thus	ADV
ma-257	321	2	substituting	substitute	VERB
ma-257	321	3	(	(	PUNCT
ma-257	321	4	29)-(30	29)-(30	NUM
ma-257	321	5	)	)	PUNCT
ma-257	321	6	in	in	ADP
ma-257	321	7	(	(	PUNCT
ma-257	321	8	28	28	NUM
ma-257	321	9	)	)	PUNCT
ma-257	321	10	,	,	PUNCT
ma-257	321	11	so	so	ADV
ma-257	321	12	−	−	PUNCT
ma-257	321	13	µ1	µ1	PROPN
ma-257	321	14	∫	∫	PROPN
ma-257	321	15	ω	ω	PROPN
ma-257	322	1	[	[	NOUN
ma-257	322	2	uoϕ1(x	uoϕ1(x	NOUN
ma-257	322	3	)	)	PUNCT
ma-257	322	4	+	+	CCONJ
ma-257	322	5	voψ1(x)]dx	voψ1(x)]dx	X
ma-257	322	6	>	>	X
ma-257	322	7	0	0	NUM
ma-257	322	8	,	,	PUNCT
ma-257	322	9	(	(	PUNCT
ma-257	322	10	32	32	NUM
ma-257	322	11	)	)	PUNCT
ma-257	322	12	then	then	ADV
ma-257	322	13	µ1	µ1	VERB
ma-257	322	14	<	<	X
ma-257	322	15	0	0	NUM
ma-257	322	16	and	and	CCONJ
ma-257	322	17	the	the	DET
ma-257	322	18	solution	solution	NOUN
ma-257	322	19	is	be	AUX
ma-257	322	20	unstable	unstable	ADJ
ma-257	322	21	.	.	PUNCT
ma-257	323	1	�	�	PROPN
ma-257	323	2	remark	remark	VERB
ma-257	323	3	3.1	3.1	NUM
ma-257	323	4	.	.	PUNCT
ma-257	324	1	by	by	ADP
ma-257	324	2	replacing	replace	VERB
ma-257	324	3	assumptions	assumption	NOUN
ma-257	324	4	(	(	PUNCT
ma-257	324	5	t1)-(t3	t1)-(t3	NUM
ma-257	324	6	)	)	PUNCT
ma-257	324	7	with	with	ADP
ma-257	324	8	next	next	ADJ
ma-257	324	9	:	:	PUNCT
ma-257	324	10	(	(	PUNCT
ma-257	324	11	l1	l1	PROPN
ma-257	324	12	):	):	PUNCT
ma-257	324	13	for	for	ADP
ma-257	324	14	u	u	NOUN
ma-257	324	15	,	,	PUNCT
ma-257	324	16	v	v	X
ma-257	324	17	>	>	X
ma-257	324	18	0	0	NUM
ma-257	324	19	,	,	PUNCT
ma-257	324	20	the	the	DET
ma-257	324	21	functions	function	NOUN
ma-257	324	22	fv	fv	X
ma-257	324	23	,	,	PUNCT
ma-257	324	24	gu	gu	PROPN
ma-257	324	25	are	be	AUX
ma-257	324	26	negative	negative	ADJ
ma-257	324	27	.	.	PUNCT
ma-257	325	1	(	(	PUNCT
ma-257	325	2	l2	l2	NOUN
ma-257	325	3	):	):	PUNCT
ma-257	325	4	for	for	ADP
ma-257	325	5	every	every	DET
ma-257	325	6	v	v	NOUN
ma-257	325	7	>	>	X
ma-257	325	8	0	0	NUM
ma-257	325	9	,	,	PUNCT
ma-257	325	10	the	the	DET
ma-257	325	11	function	function	NOUN
ma-257	325	12	(	(	PUNCT
ma-257	325	13	f	f	X
ma-257	325	14	(	(	PUNCT
ma-257	325	15	u	u	PROPN
ma-257	325	16	,	,	PUNCT
ma-257	325	17	v)−	v)−	PROPN
ma-257	325	18	u−α	u−α	X
ma-257	325	19	)	)	PUNCT
ma-257	325	20	/up−1	/up−1	X
ma-257	325	21	is	be	AUX
ma-257	325	22	strictly	strictly	ADV
ma-257	325	23	decreasing	decrease	VERB
ma-257	325	24	at	at	ADP
ma-257	325	25	u.	u.	PROPN
ma-257	325	26	(	(	PUNCT
ma-257	325	27	l3	l3	PROPN
ma-257	325	28	):	):	PUNCT
ma-257	325	29	for	for	ADP
ma-257	325	30	every	every	DET
ma-257	325	31	u	u	NOUN
ma-257	325	32	>	>	X
ma-257	325	33	0	0	PROPN
ma-257	325	34	,	,	PUNCT
ma-257	325	35	the	the	DET
ma-257	325	36	function	function	NOUN
ma-257	325	37	(	(	PUNCT
ma-257	325	38	g(u	g(u	PROPN
ma-257	325	39	,	,	PUNCT
ma-257	325	40	v)−	v)−	ADJ
ma-257	325	41	v−β	v−β	NOUN
ma-257	325	42	)	)	PUNCT
ma-257	326	1	/vq−1	/vq−1	PRON
ma-257	326	2	is	be	AUX
ma-257	326	3	strictly	strictly	ADV
ma-257	326	4	decreasing	decrease	VERB
ma-257	326	5	at	at	ADP
ma-257	326	6	v	v	NUM
ma-257	326	7	.	.	PUNCT
ma-257	327	1	we	we	PRON
ma-257	327	2	deduce	deduce	VERB
ma-257	327	3	the	the	DET
ma-257	327	4	following	follow	VERB
ma-257	327	5	:	:	PUNCT
ma-257	327	6	corollary	corollary	ADJ
ma-257	327	7	3.1	3.1	NUM
ma-257	327	8	.	.	PUNCT
ma-257	327	9	suppose	suppose	VERB
ma-257	327	10	that	that	SCONJ
ma-257	327	11	(	(	PUNCT
ma-257	327	12	l1)-(l3	l1)-(l3	NOUN
ma-257	327	13	)	)	PUNCT
ma-257	327	14	are	be	AUX
ma-257	327	15	satisfied	satisfied	ADJ
ma-257	327	16	,	,	PUNCT
ma-257	327	17	hence	hence	ADV
ma-257	327	18	every	every	DET
ma-257	327	19	positive	positive	ADJ
ma-257	327	20	weak	weak	ADJ
ma-257	327	21	solution	solution	NOUN
ma-257	327	22	of	of	ADP
ma-257	327	23	(	(	PUNCT
ma-257	327	24	1	1	NUM
ma-257	327	25	)	)	PUNCT
ma-257	327	26	is	be	AUX
ma-257	327	27	unstable	unstable	ADJ
ma-257	327	28	in	in	ADP
ma-257	327	29	ω̄δ	ω̄δ	PRON
ma-257	327	30	and	and	CCONJ
ma-257	327	31	stable	stable	ADJ
ma-257	327	32	in	in	ADP
ma-257	327	33	ω−	ω−	PROPN
ma-257	327	34	ω̄δ	ω̄δ	PRON
ma-257	327	35	.	.	PUNCT
ma-257	328	1	proof	proof	NOUN
ma-257	328	2	.	.	PUNCT
ma-257	329	1	in	in	ADP
ma-257	329	2	the	the	DET
ma-257	329	3	same	same	ADJ
ma-257	329	4	way	way	NOUN
ma-257	329	5	that	that	PRON
ma-257	329	6	theorem	theorem	VERB
ma-257	329	7	3.1	3.1	NUM
ma-257	329	8	is	be	AUX
ma-257	329	9	proved	prove	VERB
ma-257	329	10	,	,	PUNCT
ma-257	329	11	the	the	DET
ma-257	329	12	proof	proof	ADJ
ma-257	329	13	procedure	procedure	NOUN
ma-257	329	14	is	be	AUX
ma-257	329	15	similar	similar	ADJ
ma-257	329	16	.	.	PUNCT
ma-257	330	1	�	�	PROPN
ma-257	330	2	remark	remark	VERB
ma-257	330	3	3.2	3.2	NUM
ma-257	330	4	.	.	PUNCT
ma-257	331	1	as	as	SCONJ
ma-257	331	2	shown	show	VERB
ma-257	331	3	in	in	ADP
ma-257	331	4	the	the	DET
ma-257	331	5	preceding	precede	VERB
ma-257	331	6	theorem	theorem	NOUN
ma-257	331	7	and	and	CCONJ
ma-257	331	8	corollary	corollary	ADJ
ma-257	331	9	,	,	PUNCT
ma-257	331	10	the	the	DET
ma-257	331	11	stability	stability	NOUN
ma-257	331	12	results	result	NOUN
ma-257	331	13	of	of	ADP
ma-257	331	14	positive	positive	ADJ
ma-257	331	15	weak	weak	ADJ
ma-257	331	16	solution	solution	NOUN
ma-257	331	17	are	be	AUX
ma-257	331	18	dependent	dependent	ADJ
ma-257	331	19	on	on	ADP
ma-257	331	20	the	the	DET
ma-257	331	21	domain	domain	NOUN
ma-257	331	22	,	,	PUNCT
ma-257	331	23	in	in	ADP
ma-257	331	24	addition	addition	NOUN
ma-257	331	25	to	to	ADP
ma-257	331	26	the	the	DET
ma-257	331	27	provided	provide	VERB
ma-257	331	28	assumptions	assumption	NOUN
ma-257	331	29	.	.	PUNCT
ma-257	332	1	remark	remark	NOUN
ma-257	332	2	3.3	3.3	NUM
ma-257	332	3	.	.	PUNCT
ma-257	333	1	if	if	SCONJ
ma-257	333	2	p	p	NOUN
ma-257	333	3	=	=	X
ma-257	333	4	q	q	NOUN
ma-257	333	5	=	=	SYM
ma-257	333	6	2	2	NUM
ma-257	333	7	in	in	ADP
ma-257	333	8	system	system	NOUN
ma-257	333	9	(	(	PUNCT
ma-257	333	10	1	1	NUM
ma-257	333	11	)	)	PUNCT
ma-257	333	12	,	,	PUNCT
ma-257	333	13	we	we	PRON
ma-257	333	14	get	get	VERB
ma-257	333	15	the	the	DET
ma-257	333	16	results	result	NOUN
ma-257	333	17	of	of	ADP
ma-257	333	18	the	the	DET
ma-257	333	19	system	system	NOUN
ma-257	333	20	which	which	PRON
ma-257	333	21	have	have	AUX
ma-257	333	22	been	be	AUX
ma-257	333	23	studied	study	VERB
ma-257	333	24	in	in	ADP
ma-257	333	25	[	[	PUNCT
ma-257	333	26	9	9	NUM
ma-257	333	27	]	]	PUNCT
ma-257	333	28	.	.	PUNCT
ma-257	334	1	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	334	2	eur	eur	PROPN
ma-257	334	3	.	.	PUNCT
ma-257	335	1	j.	j.	PROPN
ma-257	335	2	math	math	PROPN
ma-257	335	3	.	.	PUNCT
ma-257	336	1	anal	anal	PROPN
ma-257	336	2	.	.	PUNCT
ma-257	337	1	10.28924	10.28924	NUM
ma-257	337	2	/	/	SYM
ma-257	337	3	ada	ada	PROPN
ma-257	337	4	/	/	SYM
ma-257	337	5	ma.5.1	ma.5.1	PROPN
ma-257	337	6	13references	13reference	NOUN
ma-257	338	1	[	[	X
ma-257	338	2	1	1	NUM
ma-257	338	3	]	]	X
ma-257	338	4	r.	r.	PROPN
ma-257	338	5	aris	aris	PROPN
ma-257	338	6	,	,	PUNCT
ma-257	338	7	introduction	introduction	NOUN
ma-257	338	8	to	to	ADP
ma-257	338	9	the	the	DET
ma-257	338	10	analysis	analysis	NOUN
ma-257	338	11	of	of	ADP
ma-257	338	12	chemical	chemical	ADJ
ma-257	338	13	reactors	reactor	NOUN
ma-257	338	14	,	,	PUNCT
ma-257	338	15	prentice	prentice	NOUN
ma-257	338	16	-	-	PUNCT
ma-257	338	17	hall	hall	NOUN
ma-257	338	18	(	(	PUNCT
ma-257	338	19	1965).[2	1965).[2	PROPN
ma-257	338	20	]	]	X
ma-257	338	21	s.	s.	PROPN
ma-257	338	22	cui	cui	PROPN
ma-257	338	23	,	,	PUNCT
ma-257	338	24	existence	existence	NOUN
ma-257	338	25	and	and	CCONJ
ma-257	338	26	nonexistence	nonexistence	NOUN
ma-257	338	27	of	of	ADP
ma-257	338	28	positive	positive	ADJ
ma-257	338	29	solutions	solution	NOUN
ma-257	338	30	for	for	ADP
ma-257	338	31	singular	singular	PROPN
ma-257	338	32	semilinear	semilinear	PROPN
ma-257	338	33	elliptic	elliptic	ADJ
ma-257	338	34	boundary	boundary	ADJ
ma-257	338	35	value	value	NOUN
ma-257	338	36	problems	problem	NOUN
ma-257	338	37	,	,	PUNCT
ma-257	338	38	nonlinear	nonlinear	ADJ
ma-257	338	39	anal	anal	NOUN
ma-257	338	40	.	.	PUNCT
ma-257	339	1	41	41	NUM
ma-257	339	2	(	(	PUNCT
ma-257	339	3	1	1	NUM
ma-257	339	4	-	-	SYM
ma-257	339	5	2	2	NUM
ma-257	339	6	)	)	PUNCT
ma-257	339	7	(	(	PUNCT
ma-257	339	8	2000	2000	NUM
ma-257	339	9	)	)	PUNCT
ma-257	339	10	149–176	149–176	NUM
ma-257	339	11	.	.	PUNCT
ma-257	340	1	https://doi.org/10.1016/s0362-546x(98)00271-5.[3	https://doi.org/10.1016/s0362-546x(98)00271-5.[3	NOUN
ma-257	340	2	]	]	X
ma-257	340	3	s.	s.	PROPN
ma-257	340	4	khafagy	khafagy	PROPN
ma-257	340	5	,	,	PUNCT
ma-257	340	6	e.	e.	PROPN
ma-257	340	7	a.	a.	PROPN
ma-257	340	8	el	el	PROPN
ma-257	340	9	-	-	PUNCT
ma-257	340	10	zahrani	zahrani	PROPN
ma-257	340	11	,	,	PUNCT
ma-257	340	12	h.	h.	PROPN
ma-257	340	13	m.	m.	PROPN
ma-257	340	14	serag	serag	PROPN
ma-257	340	15	,	,	PUNCT
ma-257	340	16	existence	existence	NOUN
ma-257	340	17	and	and	CCONJ
ma-257	340	18	uniqueness	uniqueness	NOUN
ma-257	340	19	of	of	ADP
ma-257	340	20	weak	weak	ADJ
ma-257	340	21	solution	solution	NOUN
ma-257	340	22	for	for	ADP
ma-257	340	23	nonlinear	nonlinear	ADJ
ma-257	340	24	weighted	weight	VERB
ma-257	340	25	(	(	PUNCT
ma-257	340	26	p	p	X
ma-257	340	27	,	,	PUNCT
ma-257	340	28	q)-laplacian	q)-laplacian	PUNCT
ma-257	340	29	system	system	NOUN
ma-257	340	30	with	with	ADP
ma-257	340	31	application	application	NOUN
ma-257	340	32	on	on	ADP
ma-257	340	33	an	an	DET
ma-257	340	34	optimal	optimal	ADJ
ma-257	340	35	control	control	NOUN
ma-257	340	36	problem	problem	NOUN
ma-257	340	37	,	,	PUNCT
ma-257	340	38	jordan	jordan	PROPN
ma-257	340	39	j.	j.	PROPN
ma-257	340	40	math	math	PROPN
ma-257	340	41	.	.	PUNCT
ma-257	341	1	stat	stat	PROPN
ma-257	341	2	.	.	PUNCT
ma-257	342	1	15	15	NUM
ma-257	342	2	(	(	PUNCT
ma-257	342	3	4a	4a	NUM
ma-257	342	4	)	)	PUNCT
ma-257	342	5	(	(	PUNCT
ma-257	342	6	2022	2022	NUM
ma-257	342	7	)	)	PUNCT
ma-257	342	8	983–998.[4	983–998.[4	NUM
ma-257	342	9	]	]	X
ma-257	342	10	e.	e.	PROPN
ma-257	342	11	k.	k.	PROPN
ma-257	342	12	lee	lee	PROPN
ma-257	342	13	,	,	PUNCT
ma-257	342	14	r.	r.	PROPN
ma-257	342	15	shivaji	shivaji	PROPN
ma-257	342	16	,	,	PUNCT
ma-257	342	17	j.	j.	PROPN
ma-257	342	18	ye	ye	PROPN
ma-257	342	19	,	,	PUNCT
ma-257	342	20	classes	class	NOUN
ma-257	342	21	of	of	ADP
ma-257	342	22	infinite	infinite	ADJ
ma-257	342	23	semipositone	semipositone	NOUN
ma-257	342	24	systems	system	NOUN
ma-257	342	25	,	,	PUNCT
ma-257	342	26	proc	proc	NOUN
ma-257	342	27	.	.	PUNCT
ma-257	342	28	r.	r.	PROPN
ma-257	342	29	soc	soc	PROPN
ma-257	342	30	.	.	PUNCT
ma-257	343	1	edinburgh	edinburgh	PROPN
ma-257	343	2	sect	sect	PROPN
ma-257	343	3	.	.	PUNCT
ma-257	344	1	a	a	DET
ma-257	344	2	math	math	NOUN
ma-257	344	3	.	.	PUNCT
ma-257	345	1	139	139	NUM
ma-257	345	2	(	(	PUNCT
ma-257	345	3	4)(2009	4)(2009	NUM
ma-257	345	4	)	)	PUNCT
ma-257	345	5	853–865	853–865	NUM
ma-257	345	6	.	.	PUNCT
ma-257	346	1	https://doi.org/10.1017/s0308210508000255.[5	https://doi.org/10.1017/s0308210508000255.[5	PROPN
ma-257	346	2	]	]	PUNCT
ma-257	346	3	s.	s.	PROPN
ma-257	346	4	khafagy	khafagy	PROPN
ma-257	346	5	,	,	PUNCT
ma-257	346	6	z.	z.	PROPN
ma-257	346	7	sadeghi	sadeghi	PROPN
ma-257	346	8	,	,	PUNCT
ma-257	346	9	existence	existence	NOUN
ma-257	346	10	of	of	ADP
ma-257	346	11	positive	positive	ADJ
ma-257	346	12	weak	weak	ADJ
ma-257	346	13	solution	solution	NOUN
ma-257	346	14	for	for	ADP
ma-257	346	15	a	a	DET
ma-257	346	16	weighted	weight	VERB
ma-257	346	17	system	system	NOUN
ma-257	346	18	of	of	ADP
ma-257	346	19	autocatalytic	autocatalytic	ADJ
ma-257	346	20	reaction	reaction	NOUN
ma-257	346	21	steadystate	steadystate	NOUN
ma-257	346	22	type	type	NOUN
ma-257	346	23	,	,	PUNCT
ma-257	346	24	eur	eur	PROPN
ma-257	346	25	.	.	PUNCT
ma-257	347	1	j.	j.	PROPN
ma-257	347	2	math	math	PROPN
ma-257	347	3	.	.	PUNCT
ma-257	348	1	appl	appl	PROPN
ma-257	348	2	.	.	PROPN
ma-257	349	1	4	4	NUM
ma-257	349	2	(	(	PUNCT
ma-257	349	3	11	11	NUM
ma-257	349	4	)	)	PUNCT
ma-257	349	5	(	(	PUNCT
ma-257	349	6	2024	2024	NUM
ma-257	349	7	)	)	PUNCT
ma-257	349	8	1	1	NUM
ma-257	349	9	-	-	SYM
ma-257	349	10	7	7	NUM
ma-257	349	11	.	.	PUNCT
ma-257	350	1	https://doi.org/10.28919/ejma.2024.4.11[6	https://doi.org/10.28919/ejma.2024.4.11[6	PROPN
ma-257	350	2	]	]	PUNCT
ma-257	350	3	m.	m.	PROPN
ma-257	350	4	ramaswamy	ramaswamy	PROPN
ma-257	350	5	,	,	PUNCT
ma-257	350	6	r.	r.	PROPN
ma-257	350	7	shivaji	shivaji	PROPN
ma-257	350	8	,	,	PUNCT
ma-257	350	9	j.	j.	PROPN
ma-257	350	10	ye	ye	PROPN
ma-257	350	11	,	,	PUNCT
ma-257	350	12	positive	positive	ADJ
ma-257	350	13	solutions	solution	NOUN
ma-257	350	14	for	for	ADP
ma-257	350	15	a	a	DET
ma-257	350	16	class	class	NOUN
ma-257	350	17	of	of	ADP
ma-257	350	18	infinite	infinite	ADJ
ma-257	350	19	semipositone	semipositone	NOUN
ma-257	350	20	problems	problem	NOUN
ma-257	350	21	,	,	PUNCT
ma-257	350	22	differential	differential	ADJ
ma-257	350	23	integralequations	integralequation	NOUN
ma-257	350	24	20	20	NUM
ma-257	350	25	(	(	PUNCT
ma-257	350	26	12	12	NUM
ma-257	350	27	)	)	PUNCT
ma-257	350	28	(	(	PUNCT
ma-257	350	29	2007	2007	NUM
ma-257	350	30	)	)	PUNCT
ma-257	350	31	1423–1433	1423–1433	NUM
ma-257	350	32	.	.	PUNCT
ma-257	351	1	https://doi.org/10.57262/die/1356039073.[7	https://doi.org/10.57262/die/1356039073.[7	X
ma-257	351	2	]	]	X
ma-257	351	3	s.	s.	PROPN
ma-257	351	4	rasouli	rasouli	PROPN
ma-257	351	5	,	,	PUNCT
ma-257	351	6	b.	b.	PROPN
ma-257	351	7	salehi	salehi	PROPN
ma-257	351	8	,	,	PUNCT
ma-257	351	9	on	on	ADP
ma-257	351	10	the	the	DET
ma-257	351	11	existence	existence	NOUN
ma-257	351	12	of	of	ADP
ma-257	351	13	positive	positive	ADJ
ma-257	351	14	weak	weak	ADJ
ma-257	351	15	solutions	solution	NOUN
ma-257	351	16	for	for	ADP
ma-257	351	17	a	a	DET
ma-257	351	18	class	class	NOUN
ma-257	351	19	of	of	ADP
ma-257	351	20	chemically	chemically	ADV
ma-257	351	21	reacting	react	VERB
ma-257	351	22	systems	system	NOUN
ma-257	351	23	withsign	withsign	NOUN
ma-257	351	24	-	-	PUNCT
ma-257	351	25	changing	change	VERB
ma-257	351	26	weights	weight	NOUN
ma-257	351	27	,	,	PUNCT
ma-257	351	28	bull	bull	NOUN
ma-257	351	29	.	.	PUNCT
ma-257	352	1	transilv	transilv	PROPN
ma-257	352	2	.	.	PUNCT
ma-257	353	1	univ	univ	PROPN
ma-257	353	2	.	.	PUNCT
ma-257	353	3	brasov	brasov	PROPN
ma-257	353	4	.	.	PUNCT
ma-257	354	1	ser	ser	PROPN
ma-257	354	2	.	.	PUNCT
ma-257	355	1	iii	iii	PROPN
ma-257	355	2	math	math	NOUN
ma-257	355	3	.	.	PUNCT
ma-257	356	1	comput	comput	NOUN
ma-257	356	2	.	.	PUNCT
ma-257	357	1	sci	sci	PROPN
ma-257	357	2	.	.	PROPN
ma-257	358	1	9	9	NUM
ma-257	359	1	(	(	PUNCT
ma-257	359	2	2	2	NUM
ma-257	359	3	)	)	PUNCT
ma-257	359	4	(	(	PUNCT
ma-257	360	1	2016	2016	NUM
ma-257	360	2	)	)	PUNCT
ma-257	360	3	71–78.[8	71–78.[8	PROPN
ma-257	360	4	]	]	PUNCT
ma-257	361	1	s.	s.	PROPN
ma-257	361	2	rasouli	rasouli	PROPN
ma-257	361	3	,	,	PUNCT
ma-257	361	4	b.	b.	PROPN
ma-257	361	5	salehi	salehi	PROPN
ma-257	361	6	,	,	PUNCT
ma-257	361	7	positive	positive	ADJ
ma-257	361	8	solutions	solution	NOUN
ma-257	361	9	for	for	ADP
ma-257	361	10	a	a	DET
ma-257	361	11	class	class	NOUN
ma-257	361	12	of	of	ADP
ma-257	361	13	chemically	chemically	ADV
ma-257	361	14	reacting	react	VERB
ma-257	361	15	systems	system	NOUN
ma-257	361	16	with	with	ADP
ma-257	361	17	sign	sign	NOUN
ma-257	361	18	-	-	PUNCT
ma-257	361	19	changing	change	VERB
ma-257	361	20	weights	weight	NOUN
ma-257	361	21	,	,	PUNCT
ma-257	361	22	world	world	NOUN
ma-257	361	23	j.	j.	PROPN
ma-257	361	24	model	model	PROPN
ma-257	361	25	.	.	PUNCT
ma-257	362	1	simul	simul	PROPN
ma-257	362	2	.	.	PROPN
ma-257	363	1	11	11	NUM
ma-257	363	2	(	(	PUNCT
ma-257	363	3	1	1	NUM
ma-257	363	4	)	)	PUNCT
ma-257	363	5	(	(	PUNCT
ma-257	363	6	2015	2015	NUM
ma-257	363	7	)	)	PUNCT
ma-257	363	8	15–19.[9	15–19.[9	NUM
ma-257	363	9	]	]	PUNCT
ma-257	363	10	s.	s.	PROPN
ma-257	363	11	khafagy	khafagy	PROPN
ma-257	363	12	,	,	PUNCT
ma-257	363	13	a.	a.	PROPN
ma-257	363	14	ezzat	ezzat	PROPN
ma-257	363	15	mohamed	mohamed	PROPN
ma-257	363	16	,	,	PUNCT
ma-257	363	17	existence	existence	NOUN
ma-257	363	18	and	and	CCONJ
ma-257	363	19	stability	stability	NOUN
ma-257	363	20	of	of	ADP
ma-257	363	21	positive	positive	ADJ
ma-257	363	22	weak	weak	ADJ
ma-257	363	23	solutions	solution	NOUN
ma-257	363	24	for	for	ADP
ma-257	363	25	a	a	DET
ma-257	363	26	class	class	NOUN
ma-257	363	27	of	of	ADP
ma-257	363	28	chemically	chemically	ADV
ma-257	363	29	reactingsystems	reactingsystem	NOUN
ma-257	363	30	,	,	PUNCT
ma-257	363	31	eur	eur	PROPN
ma-257	363	32	.	.	PUNCT
ma-257	364	1	j.	j.	PROPN
ma-257	364	2	math	math	PROPN
ma-257	364	3	.	.	PUNCT
ma-257	365	1	appl	appl	PROPN
ma-257	365	2	4	4	NUM
ma-257	365	3	(	(	PUNCT
ma-257	365	4	2	2	NUM
ma-257	365	5	)	)	PUNCT
ma-257	365	6	(	(	PUNCT
ma-257	365	7	2024	2024	NUM
ma-257	365	8	)	)	PUNCT
ma-257	365	9	1–12	1–12	NOUN
ma-257	365	10	.	.	PUNCT
ma-257	366	1	https://doi.org/10.28919/ejma.2024.4.2.[10	https://doi.org/10.28919/ejma.2024.4.2.[10	PROPN
ma-257	366	2	]	]	PUNCT
ma-257	366	3	i.	i.	PROPN
ma-257	366	4	ali	ali	PROPN
ma-257	366	5	,	,	PUNCT
ma-257	366	6	a.	a.	NOUN
ma-257	366	7	castro	castro	PROPN
ma-257	366	8	,	,	PUNCT
ma-257	366	9	r.	r.	PROPN
ma-257	366	10	shivaji	shivaji	PROPN
ma-257	366	11	,	,	PUNCT
ma-257	366	12	uniqueness	uniqueness	NOUN
ma-257	366	13	and	and	CCONJ
ma-257	366	14	stability	stability	NOUN
ma-257	366	15	of	of	ADP
ma-257	366	16	nonnegative	nonnegative	ADJ
ma-257	366	17	solutions	solution	NOUN
ma-257	366	18	for	for	ADP
ma-257	366	19	semipositone	semipositone	NOUN
ma-257	366	20	problems	problem	NOUN
ma-257	366	21	in	in	ADP
ma-257	366	22	a	a	DET
ma-257	366	23	ball	ball	NOUN
ma-257	366	24	,	,	PUNCT
ma-257	366	25	proc	proc	NOUN
ma-257	366	26	.	.	PUNCT
ma-257	367	1	amer	amer	PROPN
ma-257	367	2	.	.	PUNCT
ma-257	367	3	math	math	PROPN
ma-257	367	4	.	.	PUNCT
ma-257	368	1	soc	soc	PROPN
ma-257	368	2	.	.	PUNCT
ma-257	369	1	117	117	NUM
ma-257	369	2	(	(	PUNCT
ma-257	369	3	3	3	NUM
ma-257	369	4	)	)	PUNCT
ma-257	369	5	(	(	PUNCT
ma-257	369	6	1993	1993	NUM
ma-257	369	7	)	)	PUNCT
ma-257	370	1	775–782	775–782	NUM
ma-257	370	2	.	.	PUNCT
ma-257	371	1	https://doi.org/10.1090/s0002-9939-1993-1116249-5.[11	https://doi.org/10.1090/s0002-9939-1993-1116249-5.[11	PROPN
ma-257	371	2	]	]	PUNCT
ma-257	371	3	k.	k.	PROPN
ma-257	371	4	brown	brown	PROPN
ma-257	371	5	,	,	PUNCT
ma-257	371	6	r.	r.	PROPN
ma-257	371	7	shivaji	shivaji	PROPN
ma-257	371	8	,	,	PUNCT
ma-257	371	9	instability	instability	NOUN
ma-257	371	10	of	of	ADP
ma-257	371	11	nonnegative	nonnegative	ADJ
ma-257	371	12	solutions	solution	NOUN
ma-257	371	13	for	for	ADP
ma-257	371	14	a	a	DET
ma-257	371	15	class	class	NOUN
ma-257	371	16	of	of	ADP
ma-257	371	17	semipositone	semipositone	NOUN
ma-257	371	18	problems	problem	NOUN
ma-257	371	19	,	,	PUNCT
ma-257	371	20	proc	proc	PROPN
ma-257	371	21	.	.	PUNCT
ma-257	372	1	amer	amer	PROPN
ma-257	372	2	.	.	PUNCT
ma-257	373	1	math.soc	math.soc	X
ma-257	373	2	.	.	PROPN
ma-257	373	3	112	112	NUM
ma-257	373	4	(	(	PUNCT
ma-257	373	5	1	1	NUM
ma-257	373	6	)	)	PUNCT
ma-257	373	7	(	(	PUNCT
ma-257	373	8	1991	1991	NUM
ma-257	373	9	)	)	PUNCT
ma-257	374	1	121–124	121–124	NUM
ma-257	374	2	.	.	PUNCT
ma-257	375	1	https://doi.org/10.1090/s0002-9939-1991-1043405-5.[12	https://doi.org/10.1090/s0002-9939-1991-1043405-5.[12	PROPN
ma-257	375	2	]	]	PUNCT
ma-257	375	3	a.	a.	NOUN
ma-257	375	4	tertikas	tertikas	PROPN
ma-257	375	5	,	,	PUNCT
ma-257	375	6	stability	stability	NOUN
ma-257	375	7	and	and	CCONJ
ma-257	375	8	instability	instability	NOUN
ma-257	375	9	of	of	ADP
ma-257	375	10	positive	positive	ADJ
ma-257	375	11	solutions	solution	NOUN
ma-257	375	12	of	of	ADP
ma-257	375	13	semipositone	semipositone	NOUN
ma-257	375	14	problems	problem	NOUN
ma-257	375	15	,	,	PUNCT
ma-257	375	16	proc	proc	PROPN
ma-257	375	17	.	.	PUNCT
ma-257	376	1	amer	amer	PROPN
ma-257	376	2	.	.	PUNCT
ma-257	376	3	math	math	PROPN
ma-257	376	4	.	.	PUNCT
ma-257	377	1	soc	soc	PROPN
ma-257	377	2	.	.	PUNCT
ma-257	378	1	114	114	NUM
ma-257	378	2	(	(	PUNCT
ma-257	378	3	4)(1992	4)(1992	NUM
ma-257	378	4	)	)	PUNCT
ma-257	378	5	1035–1040	1035–1040	NUM
ma-257	378	6	.	.	PUNCT
ma-257	379	1	https://doi.org/10.1090/s0002-9939-1992-1092928-2.[13	https://doi.org/10.1090/s0002-9939-1992-1092928-2.[13	PROPN
ma-257	379	2	]	]	PUNCT
ma-257	379	3	g.	g.	PROPN
ma-257	379	4	afrouzi	afrouzi	PROPN
ma-257	379	5	,	,	PUNCT
ma-257	379	6	z.	z.	PROPN
ma-257	379	7	sadeeghi	sadeeghi	PROPN
ma-257	379	8	,	,	PUNCT
ma-257	379	9	stability	stability	NOUN
ma-257	379	10	results	result	VERB
ma-257	379	11	for	for	ADP
ma-257	379	12	a	a	DET
ma-257	379	13	class	class	NOUN
ma-257	379	14	of	of	ADP
ma-257	379	15	elliptic	elliptic	ADJ
ma-257	379	16	problems	problem	NOUN
ma-257	379	17	,	,	PUNCT
ma-257	379	18	int	int	NOUN
ma-257	379	19	.	.	PUNCT
ma-257	380	1	j.	j.	PROPN
ma-257	380	2	nonlinear	nonlinear	PROPN
ma-257	380	3	sci	sci	PROPN
ma-257	380	4	.	.	PROPN
ma-257	380	5	6	6	NUM
ma-257	380	6	(	(	PUNCT
ma-257	380	7	2	2	NUM
ma-257	380	8	)	)	PUNCT
ma-257	380	9	(	(	PUNCT
ma-257	380	10	2008	2008	NUM
ma-257	380	11	)	)	PUNCT
ma-257	380	12	114–117.[14	114–117.[14	PROPN
ma-257	380	13	]	]	PUNCT
ma-257	380	14	j.	j.	PROPN
ma-257	380	15	karátson	karátson	PROPN
ma-257	380	16	,	,	PUNCT
ma-257	380	17	p.	p.	PROPN
ma-257	380	18	simon	simon	PROPN
ma-257	380	19	,	,	PUNCT
ma-257	380	20	on	on	ADP
ma-257	380	21	the	the	DET
ma-257	380	22	stability	stability	NOUN
ma-257	380	23	properties	property	NOUN
ma-257	380	24	of	of	ADP
ma-257	380	25	nonnegative	nonnegative	ADJ
ma-257	380	26	solutions	solution	NOUN
ma-257	380	27	of	of	ADP
ma-257	380	28	semilinear	semilinear	ADJ
ma-257	380	29	problems	problem	NOUN
ma-257	380	30	with	with	ADP
ma-257	380	31	convex	convex	PROPN
ma-257	380	32	orconcave	orconcave	ADJ
ma-257	380	33	nonlinearity	nonlinearity	NOUN
ma-257	380	34	,	,	PUNCT
ma-257	380	35	j.	j.	PROPN
ma-257	380	36	comput	comput	PROPN
ma-257	380	37	.	.	PUNCT
ma-257	381	1	appl	appl	PROPN
ma-257	381	2	.	.	PROPN
ma-257	381	3	math	math	PROPN
ma-257	381	4	.	.	PUNCT
ma-257	382	1	131	131	NUM
ma-257	382	2	(	(	PUNCT
ma-257	382	3	2001	2001	NUM
ma-257	382	4	)	)	PUNCT
ma-257	383	1	497–501	497–501	NUM
ma-257	383	2	.	.	PUNCT
ma-257	384	1	https://doi.org/10.1016/s0377-0427(00	https://doi.org/10.1016/s0377-0427(00	PROPN
ma-257	384	2	)	)	PUNCT
ma-257	384	3	00714	00714	NUM
ma-257	384	4	-	-	SYM
ma-257	384	5	7.[15	7.[15	NUM
ma-257	384	6	]	]	PUNCT
ma-257	385	1	p.	p.	PROPN
ma-257	385	2	korman	korman	PROPN
ma-257	385	3	,	,	PUNCT
ma-257	385	4	j.	j.	PROPN
ma-257	385	5	shi	shi	PROPN
ma-257	385	6	,	,	PUNCT
ma-257	385	7	instability	instability	NOUN
ma-257	385	8	and	and	CCONJ
ma-257	385	9	exact	exact	ADJ
ma-257	385	10	multiplicity	multiplicity	NOUN
ma-257	385	11	of	of	ADP
ma-257	385	12	solutions	solution	NOUN
ma-257	385	13	of	of	ADP
ma-257	385	14	semilinear	semilinear	PROPN
ma-257	385	15	equations	equation	NOUN
ma-257	385	16	,	,	PUNCT
ma-257	385	17	electron	electron	PROPN
ma-257	385	18	.	.	PUNCT
ma-257	386	1	j.	j.	PROPN
ma-257	386	2	differ	differ	VERB
ma-257	386	3	.	.	PUNCT
ma-257	387	1	equ	equ	PROPN
ma-257	387	2	.	.	PUNCT
ma-257	388	1	conf5	conf5	NOUN
ma-257	388	2	(	(	PUNCT
ma-257	388	3	2000	2000	NUM
ma-257	388	4	)	)	PUNCT
ma-257	388	5	311–322.[16	311–322.[16	PROPN
ma-257	388	6	]	]	X
ma-257	388	7	c.	c.	PROPN
ma-257	388	8	maya	maya	PROPN
ma-257	388	9	,	,	PUNCT
ma-257	388	10	r.	r.	PROPN
ma-257	388	11	shivaji	shivaji	PROPN
ma-257	388	12	,	,	PUNCT
ma-257	388	13	instability	instability	NOUN
ma-257	388	14	of	of	ADP
ma-257	388	15	nonnegative	nonnegative	ADJ
ma-257	388	16	solutions	solution	NOUN
ma-257	388	17	for	for	ADP
ma-257	388	18	a	a	DET
ma-257	388	19	class	class	NOUN
ma-257	388	20	of	of	ADP
ma-257	388	21	semilinear	semilinear	PROPN
ma-257	388	22	elliptic	elliptic	ADJ
ma-257	388	23	boundary	boundary	ADJ
ma-257	388	24	value	value	NOUN
ma-257	388	25	problems	problem	NOUN
ma-257	388	26	,	,	PUNCT
ma-257	388	27	j.	j.	PROPN
ma-257	388	28	comput	comput	PROPN
ma-257	388	29	.	.	PUNCT
ma-257	389	1	appl	appl	PROPN
ma-257	389	2	.	.	PROPN
ma-257	389	3	math	math	NOUN
ma-257	389	4	.	.	PUNCT
ma-257	390	1	88	88	NUM
ma-257	390	2	(	(	PUNCT
ma-257	390	3	1	1	NUM
ma-257	390	4	)	)	PUNCT
ma-257	390	5	(	(	PUNCT
ma-257	390	6	1998	1998	NUM
ma-257	390	7	)	)	PUNCT
ma-257	390	8	125–128	125–128	NUM
ma-257	390	9	.	.	PUNCT
ma-257	391	1	https://doi.org/10.1016/s0377-0427(97)00209-4.[17	https://doi.org/10.1016/s0377-0427(97)00209-4.[17	PROPN
ma-257	391	2	]	]	PUNCT
ma-257	391	3	i.	i.	NOUN
ma-257	391	4	voros	voros	PROPN
ma-257	391	5	,	,	PUNCT
ma-257	391	6	stability	stability	NOUN
ma-257	391	7	properties	property	NOUN
ma-257	391	8	of	of	ADP
ma-257	391	9	non	non	ADJ
ma-257	391	10	-	-	ADJ
ma-257	391	11	negative	negative	ADJ
ma-257	391	12	solutions	solution	NOUN
ma-257	391	13	of	of	ADP
ma-257	391	14	semilinear	semilinear	PROPN
ma-257	391	15	symmetric	symmetric	ADJ
ma-257	391	16	cooperative	cooperative	ADJ
ma-257	391	17	systems	system	NOUN
ma-257	391	18	.	.	PUNCT
ma-257	391	19	,	,	PUNCT
ma-257	391	20	electron	electron	PROPN
ma-257	391	21	.	.	PUNCT
ma-257	391	22	j.differ	j.differ	NOUN
ma-257	391	23	.	.	PUNCT
ma-257	392	1	equ	equ	PROPN
ma-257	392	2	.	.	PROPN
ma-257	392	3	2004	2004	NUM
ma-257	392	4	(	(	PUNCT
ma-257	392	5	2004	2004	NUM
ma-257	392	6	)	)	PUNCT
ma-257	392	7	1–6.[18	1–6.[18	NUM
ma-257	392	8	]	]	X
ma-257	392	9	s.	s.	PROPN
ma-257	392	10	khafagy	khafagy	PROPN
ma-257	392	11	,	,	PUNCT
ma-257	392	12	a.	a.	PROPN
ma-257	392	13	ezzat	ezzat	PROPN
ma-257	392	14	mohamed	mohamed	PROPN
ma-257	392	15	,	,	PUNCT
ma-257	392	16	uniqueness	uniqueness	NOUN
ma-257	392	17	of	of	ADP
ma-257	392	18	weak	weak	ADJ
ma-257	392	19	solution	solution	NOUN
ma-257	392	20	for	for	ADP
ma-257	392	21	nonlocal	nonlocal	ADJ
ma-257	392	22	(	(	PUNCT
ma-257	392	23	p	p	NOUN
ma-257	392	24	,	,	PUNCT
ma-257	392	25	q)-kirchhoff	q)-kirchhoff	NOUN
ma-257	392	26	system	system	NOUN
ma-257	392	27	,	,	PUNCT
ma-257	392	28	open	open	ADJ
ma-257	392	29	j.	j.	PROPN
ma-257	392	30	math.anal	math.anal	PROPN
ma-257	392	31	.	.	PROPN
ma-257	392	32	8	8	NUM
ma-257	392	33	(	(	PUNCT
ma-257	392	34	2	2	NUM
ma-257	392	35	)	)	PUNCT
ma-257	392	36	(	(	PUNCT
ma-257	392	37	2024	2024	NUM
ma-257	392	38	)	)	PUNCT
ma-257	392	39	1	1	NUM
ma-257	392	40	-	-	SYM
ma-257	392	41	9	9	NUM
ma-257	392	42	.	.	PUNCT
ma-257	392	43	https://doi:10.30538	https://doi:10.30538	NOUN
ma-257	392	44	/	/	SYM
ma-257	392	45	psrp	psrp	NOUN
ma-257	392	46	-	-	PUNCT
ma-257	392	47	oma2024.0138[19	oma2024.0138[19	NOUN
ma-257	392	48	]	]	PUNCT
ma-257	392	49	s.	s.	PROPN
ma-257	392	50	khafagy	khafagy	PROPN
ma-257	392	51	,	,	PUNCT
ma-257	392	52	a	a	DET
ma-257	392	53	direct	direct	ADJ
ma-257	392	54	proof	proof	NOUN
ma-257	392	55	of	of	ADP
ma-257	392	56	stability	stability	NOUN
ma-257	392	57	of	of	ADP
ma-257	392	58	nonnegative	nonnegative	ADJ
ma-257	392	59	weak	weak	ADJ
ma-257	392	60	solution	solution	NOUN
ma-257	392	61	for	for	ADP
ma-257	392	62	fractional	fractional	ADJ
ma-257	392	63	p	p	NOUN
ma-257	392	64	-	-	PUNCT
ma-257	392	65	laplacian	laplacian	ADJ
ma-257	392	66	problem	problem	NOUN
ma-257	392	67	with	with	ADP
ma-257	392	68	concavenonlinearity	concavenonlinearity	NOUN
ma-257	392	69	,	,	PUNCT
ma-257	392	70	open	open	ADJ
ma-257	392	71	j.	j.	PROPN
ma-257	392	72	math	math	PROPN
ma-257	392	73	.	.	PUNCT
ma-257	393	1	anal	anal	PROPN
ma-257	393	2	.	.	PUNCT
ma-257	394	1	8	8	NUM
ma-257	394	2	(	(	PUNCT
ma-257	394	3	1	1	NUM
ma-257	394	4	)	)	PUNCT
ma-257	394	5	(	(	PUNCT
ma-257	394	6	2024	2024	NUM
ma-257	394	7	)	)	PUNCT
ma-257	394	8	76	76	NUM
ma-257	394	9	-	-	SYM
ma-257	394	10	79.[20	79.[20	PROPN
ma-257	394	11	]	]	X
ma-257	394	12	s.	s.	PROPN
ma-257	394	13	khafagy	khafagy	PROPN
ma-257	394	14	,	,	PUNCT
ma-257	394	15	s.	s.	PROPN
ma-257	394	16	rasouli	rasouli	PROPN
ma-257	394	17	,	,	PUNCT
ma-257	394	18	h.	h.	PROPN
ma-257	394	19	serag	serag	PROPN
ma-257	394	20	,	,	PUNCT
ma-257	394	21	existence	existence	NOUN
ma-257	394	22	results	result	VERB
ma-257	394	23	for	for	ADP
ma-257	394	24	fractional	fractional	ADJ
ma-257	394	25	fisher	fisher	PROPN
ma-257	394	26	-	-	PUNCT
ma-257	394	27	kolmogoroff	kolmogoroff	NOUN
ma-257	394	28	steady	steady	ADJ
ma-257	394	29	state	state	NOUN
ma-257	394	30	problem	problem	NOUN
ma-257	394	31	,	,	PUNCT
ma-257	394	32	eur	eur	PROPN
ma-257	394	33	.	.	PUNCT
ma-257	394	34	j.math	j.math	PROPN
ma-257	394	35	.	.	PUNCT
ma-257	395	1	appl	appl	PROPN
ma-257	395	2	.	.	PROPN
ma-257	396	1	3	3	NUM
ma-257	396	2	(	(	PUNCT
ma-257	396	3	20	20	NUM
ma-257	396	4	)	)	PUNCT
ma-257	396	5	(	(	PUNCT
ma-257	396	6	2023	2023	NUM
ma-257	396	7	)	)	PUNCT
ma-257	396	8	1–7	1–7	X
ma-257	396	9	.	.	PUNCT
ma-257	396	10	https://doi.org/10.28919/ejma.2023.3.20.[21	https://doi.org/10.28919/ejma.2023.3.20.[21	PROPN
ma-257	396	11	]	]	PUNCT
ma-257	397	1	c.	c.	PROPN
ma-257	397	2	atkinson	atkinson	PROPN
ma-257	397	3	,	,	PUNCT
ma-257	397	4	k.	k.	PROPN
ma-257	397	5	el	el	PROPN
ma-257	397	6	-	-	PUNCT
ma-257	397	7	ali	ali	PROPN
ma-257	397	8	,	,	PUNCT
ma-257	397	9	some	some	DET
ma-257	397	10	boundary	boundary	ADJ
ma-257	397	11	value	value	NOUN
ma-257	397	12	problems	problem	NOUN
ma-257	397	13	for	for	ADP
ma-257	397	14	the	the	DET
ma-257	397	15	bingham	bingham	PROPN
ma-257	397	16	model	model	NOUN
ma-257	397	17	,	,	PUNCT
ma-257	397	18	j.	j.	PROPN
ma-257	397	19	non	non	ADJ
ma-257	397	20	-	-	ADJ
ma-257	397	21	newtonian	newtonian	ADJ
ma-257	397	22	fluid	fluid	ADJ
ma-257	397	23	mech	mech	NOUN
ma-257	397	24	.	.	PUNCT
ma-257	398	1	41	41	NUM
ma-257	398	2	(	(	PUNCT
ma-257	398	3	3)(1992	3)(1992	NUM
ma-257	398	4	)	)	PUNCT
ma-257	398	5	339–363	339–363	NUM
ma-257	398	6	.	.	PUNCT
ma-257	399	1	https://doi.org/10.1016/0377-0257(92)87006-w	https://doi.org/10.1016/0377-0257(92)87006-w	NOUN
ma-257	399	2	.	.	PUNCT
ma-257	400	1	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	400	2	https://doi.org/10.1016/s0362-546x(98)00271-5	https://doi.org/10.1016/s0362-546x(98)00271-5	PROPN
ma-257	400	3	https://doi.org/10.1017/s0308210508000255	https://doi.org/10.1017/s0308210508000255	VERB
ma-257	400	4	https://doi.org/10.28919/ejma.2024.4.11	https://doi.org/10.28919/ejma.2024.4.11	ADV
ma-257	400	5	https://doi.org/10.57262/die/1356039073	https://doi.org/10.57262/die/1356039073	NOUN
ma-257	400	6	https://doi.org/10.28919/ejma.2024.4.2	https://doi.org/10.28919/ejma.2024.4.2	PROPN
ma-257	400	7	https://doi.org/10.1090/s0002-9939-1993-1116249-5	https://doi.org/10.1090/s0002-9939-1993-1116249-5	PROPN
ma-257	401	1	https://doi.org/10.1090/s0002-9939-1991-1043405-5	https://doi.org/10.1090/s0002-9939-1991-1043405-5	AUX
ma-257	401	2	https://doi.org/10.1090/s0002-9939-1992-1092928-2	https://doi.org/10.1090/s0002-9939-1992-1092928-2	PROPN
ma-257	401	3	https://doi.org/10.1016/s0377-0427(00)00714-7	https://doi.org/10.1016/s0377-0427(00)00714-7	PROPN
ma-257	401	4	https://doi.org/10.1016/s0377-0427(00)00714-7	https://doi.org/10.1016/s0377-0427(00)00714-7	PROPN
ma-257	401	5	https://doi.org/10.1016/s0377-0427(97)00209-4	https://doi.org/10.1016/s0377-0427(97)00209-4	NOUN
ma-257	401	6	https://doi:10.30538	https://doi:10.30538	NOUN
ma-257	401	7	/	/	SYM
ma-257	401	8	psrp	psrp	PROPN
ma-257	401	9	-	-	PUNCT
ma-257	401	10	oma2024.0138	oma2024.0138	PROPN
ma-257	401	11	https://doi.org/10.28919/ejma.2023.3.20	https://doi.org/10.28919/ejma.2023.3.20	X
ma-257	401	12	https://doi.org/10.1016/0377-0257(92)87006-w	https://doi.org/10.1016/0377-0257(92)87006-w	PROPN
ma-257	401	13	eur	eur	PROPN
ma-257	401	14	.	.	PUNCT
ma-257	402	1	j.	j.	PROPN
ma-257	402	2	math	math	PROPN
ma-257	402	3	.	.	PUNCT
ma-257	403	1	anal	anal	PROPN
ma-257	403	2	.	.	PUNCT
ma-257	404	1	10.28924	10.28924	NUM
ma-257	404	2	/	/	SYM
ma-257	404	3	ada	ada	PROPN
ma-257	404	4	/	/	SYM
ma-257	404	5	ma.5.1	ma.5.1	PROPN
ma-257	404	6	14	14	NUM
ma-257	405	1	[	[	X
ma-257	405	2	22	22	NUM
ma-257	405	3	]	]	X
ma-257	405	4	s.	s.	PROPN
ma-257	405	5	khafagy	khafagy	PROPN
ma-257	405	6	,	,	PUNCT
ma-257	405	7	h.	h.	PROPN
ma-257	405	8	serag	serag	PROPN
ma-257	405	9	,	,	PUNCT
ma-257	405	10	stability	stability	NOUN
ma-257	405	11	results	result	VERB
ma-257	405	12	for	for	ADP
ma-257	405	13	a	a	DET
ma-257	405	14	singular	singular	ADJ
ma-257	405	15	system	system	NOUN
ma-257	405	16	of	of	ADP
ma-257	405	17	generalized	generalize	VERB
ma-257	405	18	p	p	PROPN
ma-257	405	19	-	-	PUNCT
ma-257	405	20	fisher	fisher	NOUN
ma-257	405	21	kolmogoroff	kolmogoroff	PROPN
ma-257	405	22	steady	steady	PROPN
ma-257	405	23	statetype	statetype	PROPN
ma-257	405	24	,	,	PUNCT
ma-257	405	25	j.	j.	PROPN
ma-257	405	26	math	math	PROPN
ma-257	405	27	.	.	PUNCT
ma-257	406	1	sci	sci	PROPN
ma-257	406	2	.	.	PROPN
ma-257	407	1	optim	optim	PROPN
ma-257	407	2	.	.	NOUN
ma-257	408	1	2	2	NUM
ma-257	408	2	(	(	PUNCT
ma-257	408	3	1	1	NUM
ma-257	408	4	)	)	PUNCT
ma-257	408	5	(	(	PUNCT
ma-257	408	6	2024	2024	NUM
ma-257	408	7	)	)	PUNCT
ma-257	408	8	97	97	NUM
ma-257	408	9	-	-	SYM
ma-257	408	10	103.[23	103.[23	NUM
ma-257	408	11	]	]	X
ma-257	409	1	g.	g.	PROPN
ma-257	409	2	afrouzi	afrouzi	PROPN
ma-257	409	3	,	,	PUNCT
ma-257	409	4	s.	s.	PROPN
ma-257	409	5	rasouli	rasouli	PROPN
ma-257	409	6	,	,	PUNCT
ma-257	409	7	stability	stability	NOUN
ma-257	409	8	properties	property	NOUN
ma-257	409	9	of	of	ADP
ma-257	409	10	non	non	ADJ
ma-257	409	11	-	-	ADJ
ma-257	409	12	negative	negative	ADJ
ma-257	409	13	solutions	solution	NOUN
ma-257	409	14	to	to	ADP
ma-257	409	15	a	a	DET
ma-257	409	16	non	non	ADJ
ma-257	409	17	-	-	ADJ
ma-257	409	18	autonomous	autonomous	ADJ
ma-257	409	19	p	p	ADJ
ma-257	409	20	-	-	PUNCT
ma-257	409	21	laplacian	laplacian	ADJ
ma-257	409	22	equation	equation	NOUN
ma-257	409	23	,	,	PUNCT
ma-257	409	24	chaos	chaos	NOUN
ma-257	409	25	solitons	soliton	NOUN
ma-257	409	26	fractals	fractal	VERB
ma-257	409	27	29	29	NUM
ma-257	409	28	(	(	PUNCT
ma-257	409	29	5	5	NUM
ma-257	409	30	)	)	PUNCT
ma-257	409	31	(	(	PUNCT
ma-257	409	32	2006	2006	NUM
ma-257	409	33	)	)	PUNCT
ma-257	409	34	1095–1099	1095–1099	NUM
ma-257	409	35	.	.	PUNCT
ma-257	410	1	https://doi.org/10.1016/j.chaos.2005.08.165.[24	https://doi.org/10.1016/j.chaos.2005.08.165.[24	PROPN
ma-257	410	2	]	]	PUNCT
ma-257	410	3	j.	j.	PROPN
ma-257	410	4	karátson	karátson	PROPN
ma-257	410	5	,	,	PUNCT
ma-257	410	6	p.	p.	PROPN
ma-257	410	7	simon	simon	PROPN
ma-257	410	8	,	,	PUNCT
ma-257	410	9	on	on	ADP
ma-257	410	10	the	the	DET
ma-257	410	11	linearized	linearize	VERB
ma-257	410	12	stability	stability	NOUN
ma-257	410	13	of	of	ADP
ma-257	410	14	positive	positive	ADJ
ma-257	410	15	solutions	solution	NOUN
ma-257	410	16	of	of	ADP
ma-257	410	17	quasilinear	quasilinear	NOUN
ma-257	410	18	problems	problem	NOUN
ma-257	410	19	with	with	ADP
ma-257	410	20	p	p	NOUN
ma-257	410	21	-	-	PUNCT
ma-257	410	22	convex	convex	NOUN
ma-257	410	23	or	or	CCONJ
ma-257	410	24	p	p	NOUN
ma-257	410	25	-	-	PUNCT
ma-257	410	26	concave	concave	NOUN
ma-257	410	27	nonlinearity	nonlinearity	NOUN
ma-257	410	28	,	,	PUNCT
ma-257	410	29	nonlinear	nonlinear	ADJ
ma-257	410	30	anal	anal	NOUN
ma-257	410	31	.	.	PUNCT
ma-257	411	1	theory	theory	NOUN
ma-257	411	2	methods	method	NOUN
ma-257	411	3	appl	appl	PROPN
ma-257	411	4	.	.	PUNCT
ma-257	412	1	47	47	NUM
ma-257	412	2	(	(	PUNCT
ma-257	412	3	7	7	NUM
ma-257	412	4	)	)	PUNCT
ma-257	412	5	(	(	PUNCT
ma-257	412	6	2001	2001	NUM
ma-257	412	7	)	)	PUNCT
ma-257	412	8	4513–4520	4513–4520	NUM
ma-257	412	9	.	.	PUNCT
ma-257	413	1	https://doi.org/10	https://doi.org/10	PROPN
ma-257	413	2	.	.	PUNCT
ma-257	414	1	1016	1016	NUM
ma-257	414	2	/	/	SYM
ma-257	414	3	s0362	s0362	PROPN
ma-257	414	4	-	-	PUNCT
ma-257	414	5	546x(01)00564	546x(01)00564	PROPN
ma-257	414	6	-	-	PUNCT
ma-257	414	7	8.[25	8.[25	NUM
ma-257	414	8	]	]	PUNCT
ma-257	414	9	s.	s.	PROPN
ma-257	414	10	khafagy	khafagy	PROPN
ma-257	414	11	,	,	PUNCT
ma-257	414	12	h.	h.	PROPN
ma-257	414	13	serag	serag	PROPN
ma-257	414	14	,	,	PUNCT
ma-257	414	15	stability	stability	NOUN
ma-257	414	16	results	result	NOUN
ma-257	414	17	of	of	ADP
ma-257	414	18	positive	positive	ADJ
ma-257	414	19	weak	weak	ADJ
ma-257	414	20	solution	solution	NOUN
ma-257	414	21	for	for	ADP
ma-257	414	22	singular	singular	ADJ
ma-257	414	23	p	p	PROPN
ma-257	414	24	-	-	PUNCT
ma-257	414	25	laplacian	laplacian	ADJ
ma-257	414	26	nonlinear	nonlinear	ADJ
ma-257	414	27	system	system	NOUN
ma-257	414	28	,	,	PUNCT
ma-257	415	1	j.	j.	PROPN
ma-257	415	2	appl.math	appl.math	PROPN
ma-257	415	3	.	.	PUNCT
ma-257	415	4	inform	inform	NOUN
ma-257	415	5	.	.	PUNCT
ma-257	416	1	36	36	NUM
ma-257	416	2	(	(	PUNCT
ma-257	416	3	3	3	NUM
ma-257	416	4	)	)	PUNCT
ma-257	416	5	(	(	PUNCT
ma-257	416	6	2018	2018	NUM
ma-257	416	7	)	)	PUNCT
ma-257	417	1	173–179	173–179	NUM
ma-257	417	2	.	.	PUNCT
ma-257	418	1	https://doi.org/10.14317/jami.2018.173.[26	https://doi.org/10.14317/jami.2018.173.[26	PROPN
ma-257	418	2	]	]	PUNCT
ma-257	418	3	s.	s.	PROPN
ma-257	418	4	khafagy	khafagy	PROPN
ma-257	418	5	,	,	PUNCT
ma-257	418	6	h.	h.	PROPN
ma-257	418	7	serag	serag	PROPN
ma-257	418	8	,	,	PUNCT
ma-257	418	9	on	on	ADP
ma-257	418	10	the	the	DET
ma-257	418	11	stability	stability	NOUN
ma-257	418	12	of	of	ADP
ma-257	418	13	positive	positive	ADJ
ma-257	418	14	weak	weak	ADJ
ma-257	418	15	solution	solution	NOUN
ma-257	418	16	for	for	ADP
ma-257	418	17	(	(	PUNCT
ma-257	418	18	p	p	X
ma-257	418	19	,	,	PUNCT
ma-257	418	20	q)-laplacian	q)-laplacian	PUNCT
ma-257	418	21	nonlinear	nonlinear	ADJ
ma-257	418	22	system	system	NOUN
ma-257	418	23	,	,	PUNCT
ma-257	418	24	appl	appl	NOUN
ma-257	418	25	.	.	PUNCT
ma-257	419	1	math.e	math.e	NOUN
ma-257	419	2	-	-	PUNCT
ma-257	419	3	notes	note	NOUN
ma-257	419	4	20	20	NUM
ma-257	419	5	(	(	PUNCT
ma-257	419	6	2020	2020	NUM
ma-257	419	7	)	)	PUNCT
ma-257	419	8	108–114.[27	108–114.[27	PROPN
ma-257	419	9	]	]	PUNCT
ma-257	419	10	l.	l.	PROPN
ma-257	419	11	c.	c.	PROPN
ma-257	419	12	evans	evans	PROPN
ma-257	419	13	,	,	PUNCT
ma-257	419	14	partial	partial	ADJ
ma-257	419	15	differential	differential	NOUN
ma-257	419	16	equations	equation	NOUN
ma-257	419	17	,	,	PUNCT
ma-257	419	18	vol	vol	NOUN
ma-257	419	19	.	.	PROPN
ma-257	419	20	19	19	NUM
ma-257	419	21	,	,	PUNCT
ma-257	419	22	american	american	PROPN
ma-257	419	23	mathematical	mathematical	ADJ
ma-257	419	24	society	society	NOUN
ma-257	419	25	,	,	PUNCT
ma-257	419	26	2022	2022	NUM
ma-257	419	27	.	.	PUNCT
ma-257	420	1	https://doi.org/10	https://doi.org/10	PROPN
ma-257	420	2	.	.	PUNCT
ma-257	421	1	1090	1090	NUM
ma-257	421	2	/	/	SYM
ma-257	421	3	gsm/019.[28	gsm/019.[28	PROPN
ma-257	421	4	]	]	X
ma-257	421	5	h.	h.	NOUN
ma-257	421	6	kielhöfer	kielhöfer	PROPN
ma-257	421	7	,	,	PUNCT
ma-257	421	8	stability	stability	NOUN
ma-257	421	9	and	and	CCONJ
ma-257	421	10	semilinear	semilinear	NOUN
ma-257	421	11	evolution	evolution	NOUN
ma-257	421	12	equations	equation	NOUN
ma-257	421	13	in	in	ADP
ma-257	421	14	hilbert	hilbert	PROPN
ma-257	421	15	space	space	NOUN
ma-257	421	16	,	,	PUNCT
ma-257	421	17	arch	arch	NOUN
ma-257	421	18	.	.	PUNCT
ma-257	422	1	ration	ration	NOUN
ma-257	422	2	.	.	PUNCT
ma-257	423	1	mech	mech	PROPN
ma-257	423	2	.	.	PUNCT
ma-257	424	1	anal	anal	PROPN
ma-257	424	2	.	.	PUNCT
ma-257	425	1	57	57	NUM
ma-257	425	2	(	(	PUNCT
ma-257	425	3	2	2	NUM
ma-257	425	4	)	)	PUNCT
ma-257	425	5	(	(	PUNCT
ma-257	425	6	1974)150–165	1974)150–165	NUM
ma-257	425	7	.	.	PUNCT
ma-257	425	8	https://doi.org/10.1007/bf00248417.[29	https://doi.org/10.1007/bf00248417.[29	PROPN
ma-257	425	9	]	]	PUNCT
ma-257	425	10	d.	d.	PROPN
ma-257	425	11	h.	h.	PROPN
ma-257	425	12	sattinger	sattinger	PROPN
ma-257	425	13	,	,	PUNCT
ma-257	425	14	monotone	monotone	ADJ
ma-257	425	15	methods	method	NOUN
ma-257	425	16	in	in	ADP
ma-257	425	17	nonlinear	nonlinear	ADJ
ma-257	425	18	elliptic	elliptic	ADJ
ma-257	425	19	and	and	CCONJ
ma-257	425	20	parabolic	parabolic	ADJ
ma-257	425	21	boundary	boundary	ADJ
ma-257	425	22	value	value	NOUN
ma-257	425	23	problems	problem	NOUN
ma-257	425	24	,	,	PUNCT
ma-257	425	25	indiana	indiana	PROPN
ma-257	425	26	univ	univ	PROPN
ma-257	425	27	.	.	PUNCT
ma-257	426	1	math.j	math.j	PROPN
ma-257	426	2	.	.	PROPN
ma-257	426	3	21	21	NUM
ma-257	426	4	(	(	PUNCT
ma-257	426	5	11	11	NUM
ma-257	426	6	)	)	PUNCT
ma-257	426	7	(	(	PUNCT
ma-257	426	8	1972	1972	NUM
ma-257	426	9	)	)	PUNCT
ma-257	426	10	979–1000	979–1000	NUM
ma-257	426	11	.	.	PUNCT
ma-257	427	1	https://doi.org/10.1512/iumj.1972.21.21079.[30	https://doi.org/10.1512/iumj.1972.21.21079.[30	X
ma-257	427	2	]	]	X
ma-257	427	3	p.	p.	PROPN
ma-257	427	4	drábek	drábek	PROPN
ma-257	427	5	,	,	PUNCT
ma-257	427	6	p.	p.	PROPN
ma-257	427	7	krejcí	krejcí	PROPN
ma-257	427	8	,	,	PUNCT
ma-257	427	9	p.	p.	PROPN
ma-257	427	10	takác	takác	PROPN
ma-257	427	11	,	,	PUNCT
ma-257	427	12	nonlinear	nonlinear	ADJ
ma-257	427	13	differential	differential	ADJ
ma-257	427	14	equations	equation	NOUN
ma-257	427	15	,	,	PUNCT
ma-257	427	16	vol	vol	NOUN
ma-257	427	17	.	.	PUNCT
ma-257	428	1	404	404	NUM
ma-257	428	2	,	,	PUNCT
ma-257	428	3	crc	crc	NOUN
ma-257	428	4	press	press	NOUN
ma-257	428	5	,	,	PUNCT
ma-257	428	6	1999	1999	NUM
ma-257	428	7	.	.	PUNCT
ma-257	429	1	https://doi.org/10	https://doi.org/10	PROPN
ma-257	429	2	.	.	PUNCT
ma-257	430	1	1201/9780429332555	1201/9780429332555	NUM
ma-257	430	2	.	.	PUNCT
ma-257	431	1	https://doi.org/10.28924/ada/ma.5.1	https://doi.org/10.28924/ada/ma.5.1	PROPN
ma-257	431	2	https://doi.org/10.1016/j.chaos.2005.08.165	https://doi.org/10.1016/j.chaos.2005.08.165	PROPN
ma-257	431	3	https://doi.org/10.1016/s0362-546x(01)00564-8	https://doi.org/10.1016/s0362-546x(01)00564-8	NOUN
ma-257	431	4	https://doi.org/10.1016/s0362-546x(01)00564-8	https://doi.org/10.1016/s0362-546x(01)00564-8	NOUN
ma-257	431	5	https://doi.org/10.14317/jami.2018.173	https://doi.org/10.14317/jami.2018.173	PROPN
ma-257	431	6	https://doi.org/10.1090/gsm/019	https://doi.org/10.1090/gsm/019	X
ma-257	431	7	https://doi.org/10.1090/gsm/019	https://doi.org/10.1090/gsm/019	PRON
ma-257	431	8	https://doi.org/10.1007/bf00248417	https://doi.org/10.1007/bf00248417	ADP
ma-257	431	9	https://doi.org/10.1512/iumj.1972.21.21079	https://doi.org/10.1512/iumj.1972.21.21079	NOUN
ma-257	431	10	https://doi.org/10.1201/9780429332555	https://doi.org/10.1201/9780429332555	NOUN
ma-257	431	11	https://doi.org/10.1201/9780429332555	https://doi.org/10.1201/9780429332555	NOUN
ma-257	431	12	1	1	NUM
ma-257	431	13	.	.	PUNCT
ma-257	432	1	introduction	introduction	NOUN
ma-257	432	2	2	2	NUM
ma-257	432	3	.	.	PUNCT
ma-257	432	4	existence	existence	NOUN
ma-257	432	5	and	and	CCONJ
ma-257	432	6	non	non	ADJ
ma-257	432	7	-	-	NOUN
ma-257	432	8	existence	existence	NOUN
ma-257	432	9	results	result	VERB
ma-257	432	10	3	3	NUM
ma-257	432	11	.	.	PUNCT
ma-257	432	12	stability	stability	NOUN
ma-257	432	13	and	and	CCONJ
ma-257	432	14	instability	instability	NOUN
ma-257	432	15	results	result	VERB
ma-257	432	16	references	reference	NOUN