Infinitely many small solutions for the p(x)-Laplacian operator with nonlinear boundary conditions

In this paper, we prove the existence of infinitely many small solutions to the following quasilinear elliptic equation ?Δ _p(x) u +  |u| ^p(x)-2 u =  f (x, u) in a smooth bounded domain Ω of ${\mathbb{R}^N}$ with nonlinear boundary conditions ${|\nabla u|^{p-2}\frac{\partial u}{\partial\nu} = |u|^{{q(x)-2}}u}$ . We also assume that ${\{q(x) = p^\ast(x)\}\neq \emptyset}$ , where p*(x) =  Np(x)/(N ? p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountain-pass lemma due to Kajikiya, and property of these solutions is also obtained.