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

Repovš, Dušan (avtor), Vetro, Calogero (avtor)

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.

• Jezik: Angleški jezik
• Ključne besede: weighted (p, q)-Laplacian, resonant Carathéodory function, parametric power term, positive and negative solutions, nodal solutions
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: PEF - Pedagoška fakulteta; FMF - Fakulteta za matematiko in fiziko
• Različica publikacije: Objavljena publikacija
• Leto izida: 2022
• Št. strani: Str. 499-517
• Številčenje: Vol. 7, Iss. 1
• PID: 20.500.12556/RUL-132481
• UDK: 517.956
• ISSN pri članku: 2473-6988
• DOI: 10.3934/math.2022032
• COBISS.SI-ID: 81064195
• Datum objave v RUL: 27.10.2021

Gradivo je del revije

• Naslov: AIMS mathematics
• Skrajšan naslov: AIMS math.
• Založnik: AIMS Press
• ISSN: 2473-6988
• COBISS.SI-ID: 18357081

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.
• Začetek licenciranja: 13.10.2021