Compact embeddings in the generalized Sobolev space W01,p(⋅)(G) and existence of solutions for nonlinear elliptic problems

Our aim in this paper is to deal with the compact embedding in the generalized Sobolev space W01,p(⋅)(G) with a variable exponent satisfying the log-Hölder condition. As an application, we find a nontrivial weak solution of the nonlinear elliptic problem −div(|∇u(x)|p(x)−2∇u(x))=f(x,u(x))in G,u(x)=0on ∂G, which is an extension of Fan–Zhang [X. Fan, G.-H. Zhang, Existence of solutions for p(x)p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52, 2003, pp. 1843–1852, Theorem 4.7].