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

In this paper the existence of a nontrivial solution to a parametric Neumann problem for a class of nonlinear elliptic equations involving the p(x)-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,λ∗], such that, for any λε]0,λ∗] 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 λ=λ∗, our theorem also includes other results in the literature for regular problems.