Existence results for a Neumann problem involving the p(x)-Laplacian with discontinuous nonlinearities

Abstract

In this paper the existence of a nontrivial solution to a parametric Neumann problem for a class of nonlinear elliptic equations involving the $p\left(x\right)$-Laplacian and a discontinuous nonlinear term is established. Under a suitable condition on the behavior of the potential at $0^{+}$, we obtain an interval $]0,{\lambda }^{\ast }]$, such that, for any $\lambda \in ]0,{\lambda }^{\ast }]$ our problem admits at least one nontrivial weak solution. The solution is obtained as a critical point of a locally Lipschitz functional. In addition to providing a new conclusion on the existence of a solution even for $\lambda ={\lambda }^{\ast }$, our theorem also includes other results in the literature for regular problems.