Existence of multiple solutions for a p(x)-Laplace equation

This article shows the existence of at least three nontrivial solutions to the quasilinear elliptic equation $$ -Delta_{p(x)}u+|u|^{p(x)-2}u=f(x,u) $$ in a smooth bounded domain $Omegasubsetmathbb{R}^{n}$, with the nonlinear boundary condition $| abla u|^{p(x)-2}frac{partial u}{partial u}=g(x,u)$ or the Dirichlet boundary condition $u=0$ on $partialOmega$. In addition, this paper proves that one solution is positive, one is negative, and the last one is a sign-changing solution. The method used here is based on Nehari results, on three sub-manifolds of the space $W^{1,p(x)}(Omega)$.