Existence and Multiplicity of Solutions for a Quasilinear Elliptic Inclusion with a Nonsmooth Potential

Abstract

This paper is concerned with a nonlinear elliptic inclusion driven by a multivalued subdifferential of nonsmooth potential and a nonlinear inhomogeneous differential operator. We obtain two multiplicity theorems in the Orlicz-Sobolev space. In the first multiplicity theorem, we produce three nontrivial smooth solutions. Two of these solutions have constant sign (one is positive, the other is negative). In the second multiplicity theorem, we derive an unbounded sequence of critical points for the problem. Our approach is variational, based on the nonsmooth critical point theory. We also show that $C^1$-local minimizers are also local minimizers in the Orlicz-Sobolev space for a large class of locally Lipschitz functions.