The behavior of solutions of a parametric weighted (p,q)-Laplacian equation

Repovš, Dušan (author), Vetro, Calogero (author)

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

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