ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITION

Abstract

Abstract. This paper deals with the superlinear elliptic problem with-out Ambrosetti and Rabinowitz type growth condition of the form:−div(1 + |∇u| p(x) √ 1+|∇u| 2p(x) )|∇u| p(x)−2 ∇u= λf(x,u), a.e. in Ω,u = 0, on ∂Ω,where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, λ > 0 isa parameter. The purpose of this paper is to obtain the existence resultsof nontrivial solutions for every parameter λ. Firstly, by using the moun-tain pass theorem a nontrivial solution is constructed for almost everyparameter λ > 0. Then we consider the continuation of the solutions.Our results are a generalization of that of Manuela Rodrigues. 1. IntroductionDuring the last fifteen years, variational problems and partial differentialequations with various types of nonstandard growth conditions have becomeincreasingly popular. This is partly due to their frequent appearance in ap-plications such as the modeling of electrorheological fluids [1, 12] and imageprocessing [2], but these problems are very interesting from a purely mathe-matical point of view as well.In this paper, we consider the following nonlinear eigenvalue problems forp(x)-Laplacian-like operators originated from a capillary phenomena of thefollowing form:(P)−div(1+