Infinitely many solutions for equations of p(x)-Laplace type with the nonlinear Neumann boundary condition

We investigate the following nonlinear Neumann boundary-value problem with associated p(x)-Laplace-type operator where the function φ(x, v) is of type |v|p(x)−2 v with continuous function p: → (1,∞) and both f : Ω × ℝ → ℝ and g : ∂Ω × ℝ → ℝ satisfy a Carathéodory condition. We first show the existence of infinitely many weak solutions for the Neumann problems using the Fountain theorem with the Cerami condition but without the Ambrosetti and Rabinowitz condition. Next, we give a result on the existence of a sequence of weak solutions for problem (P) converging to 0 in L ∞ -norm by employing De Giorgi's iteration and the localization method under suitable conditions.