Infinitely Many Solutions for a Non-homogeneous Differential Inclusion with Lack of Compactness

In this paper, we consider the following class of differential inclusion problems in {\mathbb{R}^{N}} involving the {p(x)} -Laplacian: -\Delta_{p(x)}u+V(x)\lvert u\rvert^{p(x)-2}u\in a(x)\partial F(x,u)\quad\text{% in}\ \mathbb{R}^{N}. We are concerned with a multiplicity property, and our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue–Sobolev space. Applying the nonsmooth symmetric mountain pass lemma and the fountain theorem, we establish conditions such that the associated energy functional possesses infinitely many critical points, and then we obtain infinitely many solutions.