Multiple solutions for the p(x)��laplace operator with critical growth.

ANALÍA SILVA

ADVANCED NONLINEAR STUDIES

ADVANCED NONLINEAR STUDIES, INC

Año: 2011 vol. 11 p. 63 - 75

1536-1365

 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 quasilinear elliptic equation $-\Delta_{p(x)} u = |u|^{q(x)-2}u +\lambda f(x,u)$ in asmooth bounded domain $\Omega$ of $\mathbb{R}^N$ 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\indent critical Sobolev exponent for variable exponents and $\Delta_{p(x)} u = \mbox{div}(|\nabla u|^{p(x)-2}\nabla u)$ is the $p(x)-$laplacian. The proof is based on variational arguments and the extension of  concentration compactness method for variable exponent spaces.