Existence of Weak Solutions for a Nonlocal Problem Involving the p(x)-Laplace Operator

Mustafa Avci

doi:10.13189/ujam.2013.010306

Abstract

This paper deals with the existence of weak solutions for some nonlocal problem involving the p (x)- Laplace operator. Using a direct variational method and the theory of the variable exponent Sobolev spaces, we set some conditions that ensures the existence of nontrivial weak solutions. p(x) p(x) dx)∆p(x)u = λa (x) |u| q(x)−2 u in Ω, u = 0 on ∂Ω, (P)

Highlights

In the present paper, we are concerned with the Dirichlet boundary value problem ∫ −M (|∇u|p(x) p(x) dx)∆p(x)u = λa (x) |u|q(x)−2 u Ω in Ω, (P) u = 0 on ∂Ω,() where Ω ⊂ RN, N ≥ 3, is a smooth bounded domain, λ > 0; p, q ∈ C Ω and a is a non-negative measurable real-valued function, M is a continuous function which obey some specific conditions
The purpose of the present paper is to find a nontrivial weak solution for a p (x)-Kirchhoff-type equation (P) in the variable exponent Sobolev spaces
By Proposition 2.3, we get un → u in W01,p(x) (Ω), so we conclude that u is a nontrivial weak solution for problem (P) [21]

Summary

Introduction

The purpose of the present paper is to find a nontrivial weak solution for a p (x)-Kirchhoff-type equation (P) in the variable exponent Sobolev spaces. By help of the well-known theorems mountain pass theorem and Ekeland variational principle, the existence result is obtained. The importance of problem (P) arises mainly from the existence of the p (x)-Laplacian △p(x)u = div |∇u|p(x)−2 ∇u . () For any p ∈ C+ Ω , denote 1 < p− := inf p (x) ≤ p (x) ≤ p+ := supp (x) < ∞, and define the variable exponent x∈Ω

Objectives

Results

Conclusion