Existence of at least one nontrivial solution for a class of problems involving both p(x)-Laplacian and p(x)-Biharmonic

Abstract

We investigate the existence of a weak nontrivial solution for the following problem. Our analysis is generally bathed on discussions of variational based on the Mountain Pass theorem and some recent theories one the generalized Lebesgue-Sobolev space. This paper guarantees the existence of at least one weak nontrivial solution for our problem. More precisely, by applying Ambrosetti and Rabinowitz’s mountain pass theorem and under appropriate conditions, we show that there exists a positive number such that our problem has at least one nontrivial weak solution.

Keywords

• p(x)-biharmonic operator
• p(x)-harmonic operator
• palais-smale condition
• mountain pass theorem
• generalized lebesgue-sobolev space.