Existence of Solutions for Inclusion Problems in Musielak-Orlicz-Sobolev Space Setting

Abstract

In this paper, we mainly prove the existence of (weak) solutions of an inclusion problem with the Dirichlet boundary condition of the following form: $L\in A\left(x,u,Du\right)+F\left(x,u,Du\right),\text{in}\Omega$ , and $u=0,\text{on}\partial \Omega ,$ in Musielak-Orlicz-Sobolev spaces $W_0^1{L}_{\Phi }\left(\Omega \right)$ by using the surjective theorem, where $\Omega \subset {ℝ}^N$ is a bounded Lipschitz domain, $L$ belongs to the dual space ${\left(W_0^1{L}_{\Phi }\left(\Omega \right)\right)}^{\ast }$ of $W_0^1{L}_{\Phi }\left(\Omega \right)$ , $A$ is a multivalued maximal monotone operator, and $F$ is a multivalued convection term. Some examples for $A$ and $F$ are given in the paper. Then, we give some properties of the solution set of the inclusion problem. We also show the existence of (weak) solutions of the inclusion problem with an obstacle effect.