Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz-Sobolev spaces

TL;DR

This paper investigates the existence and multiple solutions of a class of quasilinear problems involving the $ riangle_ ext{ extPhi}$-Laplacian in Orlicz-Sobolev spaces, using variational methods and topological category theory.

Contribution

It establishes new results on solution multiplicity for quasilinear problems in Orlicz-Sobolev spaces via variational techniques and Lusternik-Schnirelmann category.

Findings

Proves existence of multiple solutions depending on domain topology.

Utilizes variational methods tailored to Orlicz-Sobolev spaces.

Shows solution multiplicity linked to the Lusternik-Schnirelmann category.

Abstract

This work is concerned with the existence and multiplicity of solutions for the following class of quasilinear problems $$ -\Delta_{\Phi}u+\phi(|u|)u=f(u)~\text{in} ~\Omega_{\lambda}, u(x)>0 ~\text{in}~\Omega_{\lambda}, u=0~ \mbox{on} ~\partial\Omega_{\lambda}, $$ where $\Phi(t)=\int_0^{|t|} \phi(s) s \, ds $ is an $N-$function, $\Delta_{\Phi}$ is the $\Phi-$Laplacian operator, \linebreak $\Omega_{\lambda}=\lambda \Omega,$ $\Omega$ is a smooth bounded domain in $\mathbb{R}^N,$ $N \geq 2$, $\lambda$ is a positive parameter and $f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Here, we use variational methods to get multiplicity of solutions by using of Lusternik-Schnirelmann category of ${\Omega}$ in itself.