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

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 ${\mathbb{R}}^N$ with homogeneous Dirichlet boundary conditions on $\partial \Omega$. We assume that $\{q(x)=p^{\ast }(x)\}\ne \varnothing$, where $p^{\ast }(x)=Np(x)/(N-p(x))$ 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.