Compact embeddings in the generalized Sobolev space [formula omitted] and existence of solutions for nonlinear elliptic problems

Abstract

Our aim in this paper is to deal with the compact embedding in the generalized Sobolev space W 0 1 , 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 ) = 0 on ∂ G , which is an extension of Fan–Zhang [X. Fan, G.-H. Zhang, Existence of solutions for p ( x ) -Laplacian Dirichlet problem, Nonlinear Anal. 52, 2003, pp. 1843–1852, Theorem 4.7].