Compact embeddings for sobolev spaces of variable exponents and existence of solutions for nonlinear elliptic problems involving the p(x)-laplacian and its critical exponent

Abstract

We give a sufficient condition for the compact embedding from W0k,p(·) (Ω) to Lq(·) Ω in case ess infxεΩ(Np(x)/(N - kp(x)) - q(x)) = 0, where Ω is a bounded open set in RN. As an application, we find a nontrivial nonnegative weak solution of the nonlinear elliptic equation We also consider the existence of a weak solution to the problem above even if the embedding is not compact.