Compact embeddings in the generalized Sobolev space image and existence of solutions for nonlinear elliptic problems Original Research Article

Our aim in this paper is to deal with the compact embedding in the generalized Sobolev space View the MathML sourceW01,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 View the MathML source−div(|∇u(x)|p(x)−2∇u(x))=f(x,u(x))in G,u(x)=0on ∂G, Turn MathJax on 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].