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

TL;DR

This paper investigates the existence, nonexistence, and multiplicity of solutions for a parametric weighted (p,q)-Laplacian equation with resonance behavior, using variational methods and spectral analysis.

Contribution

It introduces new results on solution behavior for a weighted (p,q)-Laplacian equation with resonance, including existence of multiple solutions and critical parameter determination.

Findings

Existence of at least three solutions for large λ.

Nonexistence results under certain conditions.

Critical λ value linked to spectral properties.

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.