Compact imbedding theorems in Musielak spaces

Abstract

We provide compact imbedding results for Musielak–Sobolev spaces built on smooth bounded domains from Musielak functions, on which we impose, among others, natural integral conditions that extend those used in the framework of Orlicz spaces. We obtain the Rellich–Kondrachov theorem for regular Musielak functions. We then apply the results obtained to get a Poincaré-type inequality in Musielak spaces, which we use to solve a class of nonlinear elliptic problems.