Existence of solutions for p(x)-Laplacian nonhomogeneous Neumann problems with indefinite weight

In this paper we study the nonhomogeneous Neumann problem of p(x)p(x)-Laplacian equations, where the weighted function V(x)V(x) is indefinite and the potential is only locally Lipschitz. By the theory of the generalized Lebesgue–Sobolev space W1,p(x)(Ω)W1,p(x)(Ω), a nonsmooth Mountain Pass theorem and the Weierstrass theorem, we obtain the existence results of the problem.