Constant Sign Solutions for Variable Exponent System Neumann Boundary Value Problems with Singular Coefficient

Xianbin Wu

doi:10.1155/2014/396704

Abstract

We deal with the existence of constant sign solutions for the following variable exponent system Neumann boundary value problem:-div(|∇u|p(x)-2∇u)+λ|u|p(x)-2u=Fu(x,u,v) in Ω, -div(|∇v|q(x)-2∇v)+λ|v|q(x)-2v=Fv(x,u,v)inΩ, ∂u/∂γ=0=∂v/∂γ on ∂Ω. We give several sufficient conditions for the existence of the constant sign solutions, whenF(x,·,·)satisfies neither sub-(p(x),q(x)) growth condition, nor Ambrosetti-Rabinowitz condition (subcritical). In particular, we obtain the existence of eight constant sign solutions.

Highlights

In recent years, there is a lot of interest in the study of various mathematical problems with variable exponent
Throughout this paper, the letters c, ci, Ci, i = 1, 2, . . ., denote positive constants which may vary from line to line but are independent of the terms which will take part in any limit process
In order to discuss the problem (P), we need some theories on space W1,p(⋅)(Ω) which we call variable exponent Sobolev space

Summary

Introduction

There is a lot of interest in the study of various mathematical problems with variable exponent (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]). We consider the existence of constant sign solutions for the following problem:. On the existence of constant sign solutions of p-Laplacian problems, we refer to [35,36,37,38,39,40]. The corresponding functional of (P) is coercive; if F(x, ⋅, ⋅) satisfies the super-(p+, q+) growth condition (subcritical), that is, the following Ambrosetti-Rabinowitz condition:. We deal with the existence of constant sign solutions of the problem (P), when the corresponding functional neither is coercive nor satisfies Ambrosetti-Rabinowitz condition. We discuss the existence of solutions of (P), when F satisfies sub-(p(x), q(x)) growth condition near the origin in local; that is, the following condition.

Preliminary Results and Notations

Properties of Operators

Existence and Multiplicity of Solutions