# Abstract

In this paper we are concerned with the study of a class of quasilinear elliptic diff erential inclusions involving the anisotropic $\overrightarrow{p}(\cdot)$-Laplace operator, on a bounded open subset of $IR^n$ which has a smooth boundary. The abstract framework required to study this kind of di erential 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.