On the superlinear problem involving thep(x)-Laplacian

Abstract

This paper deals with the superlinear elliptic problem without Ambrosetti and Rabinowitz type growth condition of the form: ( − div(|∇u| p(x)−2 ∇u) = �f(x,u) in , u = 0 on@, where ⊂ R N (N ≥ 2) is a bounded domain with smooth boundary @, � > 0 is a parameter. Existence of nontrivial solution is established for arbitrary � > 0. Firstly, by using the mountain pass theorem a nontrivial solution is constructed for almost every parameter � > 0. Then, it is considered the continuation of the solutions. Our results are a generalization of Miyagaki and Souto.