A Multiplicity Result for an Elliptic Anisotropic Differential Inclusion Involving Variable Exponents

Abstract

In this paper we are concerned with the study of a class of quasilinear elliptic differential inclusions involving the anisotropic \(\overrightarrow {p}(\cdot)\)-Laplace operator, on a bounded open subset of \({\mathbb R}^n\) which has a smooth boundary. The abstract framework required to study this kind of differential inclusions lies at the interface of three important branches in analysis: nonsmooth analysis, the variable exponent Lebesgue–Sobolev spaces theory and the anisotropic Sobolev spaces theory. Using the concept of nonsmooth critical point we are able to prove that our problem admits at least two non-trivial weak solutions.