Multiple solutions for the $p(x)-$laplace operator with critical growth

TL;DR

This paper proves the existence of at least three solutions for a variable exponent p(x)-laplace elliptic problem with critical growth, extending previous results to more general variable exponent settings using variational and concentration compactness methods.

Contribution

It extends multiplicity results for quasilinear elliptic problems to the variable exponent case with critical growth, employing advanced variational techniques.

Findings

At least three nontrivial solutions exist for the problem.

The methods extend concentration compactness to variable exponent spaces.

Results generalize previous fixed exponent cases.

Abstract

The aim of this paper is to extend previous results regarding the multiplicity of solutions for quasilinear elliptic problems with critical growth to the variable exponent case. We prove, in the spirit of \cite{DPFBS}, the existence of at least three nontrivial solutions to the following quasilinear elliptic equation $-\Delta_{p(x)} u = |u|^{q(x)-2}u +\lambda f(x,u)$ in a smooth bounded domain $\Omega$ of $\R^N$ with homogeneous Dirichlet boundary conditions on $\partial\Omega$. We assume that $\{q(x)=p^*(x)\}\not=\emptyset$, where $p^*(x)=Np(x)/(N-p(x))$ is the critical Sobolev exponent for variable exponents and $\Delta_{p(x)} u = {div}(|\nabla u|^{p(x)-2}\nabla u)$ is the $p(x)-$laplacian. The proof is based on variational arguments and the extension of concentration compactness method for variable exponent spaces.