The aim of this paper is to establish the existence of at least three solutions for the nonlinear Neumann boundary-value problem involving the p(x)-Laplacian of the form $-{\Delta }_{p\left(x\right)}u+a\left(x\right)|u^{\left|p\right(x)-2}u=\mu g(x,u)$ in Ω, ${\left|\nabla u\right|}^{p\left(x\right)-2}\partial u/\partial \nu =\lambda f(x,u)$ on ∂Ω. Our technical approach is based on the three critical points theorem due to Ricceri.

We establish the existence of a T-p(x)-solution for the p(x)-elliptic problem $-div\left(a\right(x,u,\nabla u\left)\right)+g(x,u)=f-divF$ in Ω, where Ω is a bounded open domain of ${ℝ}^N$, N ≥ 2 and $a:\Omega \times ℝ\times {ℝ}^N\to {ℝ}^N$ is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but with only a weak monotonicity condition. The right hand side f lies in L¹(Ω) and F belongs to ${\prod }_{i=1}^N{L}^{p^{\text{'}}\left(·\right)}\left(\Omega \right)$.