Existence of solutions for a Neumann problem involving the p(x)-Laplacian

We study the existence and multiplicity of weak solutions for a parametric Neumann problem driven by the p(x)-Laplacian. Under a suitable condition on the behavior of the potential at $0^+$, we obtain an interval such that when a parameter $lambda$ is in this interval, our problem admits at least one nontrivial weak solution. We show the multiplicity of solutions for potentials satisfying also the Ambrosetti-Rabinowitz condition. Moreover, if the right-hand side f satisfies the Ambrosetti-Rabinowitz condition, then we obtain the existence of two nontrivial weak solutions.