Existence of at least five solutions for a differential inclusion problem involving the p(x)p(x)-Laplacian

We studied a nonlinear Dirichlet problem driven by the p(x)p(x)-Laplacian and having a nonsmooth potential (hemivariational inequalities). Using a variational method combined with suitable truncation techniques based on nonsmooth critical point theory for locally Lipschitz function, we proved the existence of at least five solutions under the suitable conditions.