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The behavior of solutions of a parametric weighted (p,q)-Laplacian equation
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Repovš, Dušan
(
Author
),
ID
Vetro, Calogero
(
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)
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https://www.aimspress.com/article/doi/10.3934/math.2022032
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Abstract
We study the behavior of solutions for the parametric equation ▫$$-\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z) = \lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda >0,$$▫ under Dirichlet condition, where ▫$\Omega \subseteq \mathbb{R}^N$▫ is a bounded domain with a ▫$C^2$▫-boundary ▫$\partial \Omega$▫, ▫$a_1, a_2 \in L^\infty(\Omega)$▫ with ▫$a_1(z), a_2(z) > 0$▫ for a.a. ▫$z \in \Omega$▫, ▫$p, q \in (1, \infty)$▫ and ▫$\Delta_{p}^{a_1}$▫, ▫$\Delta_{q}^{a_2}$▫ are weighted versions of ▫$p$▫-Laplacian and ▫$q$▫-Laplacian. We prove existence and nonexistence of nontrivial solutions, when ▫$f(z,x)$▫ asymptotically as ▫$x \to \pm \infty$▫ can be resonant. In the studied cases, we adopt a variational approach and use truncation and comparison techniques. When ▫$\lambda$▫ is large, we establish the existence of at least three nontrivial smooth solutions with sign information and ordered. Moreover, the critical parameter value is determined in terms of the spectrum of one of the differential operators.
Language:
English
Keywords:
weighted (p
,
q)-Laplacian
,
resonant Carathéodory function
,
parametric power term
,
positive and negative solutions
,
nodal solutions
Work type:
Article
Typology:
1.01 - Original Scientific Article
Organization:
PEF - Faculty of Education
FMF - Faculty of Mathematics and Physics
Publication version:
Version of Record
Year:
2022
Number of pages:
Str. 499-517
Numbering:
Vol. 7, Iss. 1
PID:
20.500.12556/RUL-132481
UDC:
517.956
ISSN on article:
2473-6988
DOI:
10.3934/math.2022032
COBISS.SI-ID:
81064195
Publication date in RUL:
27.10.2021
Views:
3019
Downloads:
142
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Record is a part of a journal
Title:
AIMS mathematics
Shortened title:
AIMS math.
Publisher:
AIMS Press
ISSN:
2473-6988
COBISS.SI-ID:
18357081
Licences
License:
CC BY 4.0, Creative Commons Attribution 4.0 International
Link:
http://creativecommons.org/licenses/by/4.0/
Description:
This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.
Licensing start date:
13.10.2021
Projects
Funder:
ARRS - Slovenian Research Agency
Project number:
P1-0292
Name:
Topologija, geometrija in nelinearna analiza
Funder:
ARRS - Slovenian Research Agency
Project number:
N1-0114
Name:
Algebrajski odtisi geometrijskih značilnosti v homologiji
Funder:
ARRS - Slovenian Research Agency
Project number:
N1-0083
Name:
Forsing, fuzija in kombinatorika odprtih pokritij
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