Existence and multiplicity of solutions for nonlocal p(x)-Laplacian equations with nonlinear Neumann boundary conditions

Erlin Guo,Peihao Zhao

doi:10.1186/1687-2770-2012-1

Erlin Guo, Peihao Zhao

Open Access

https://doi.org/10.1186/1687-2770-2012-1

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Abstract

In this article, we study the nonlocal p(x)-Laplacian problem of the following form a ( ∫ Ω 1 p ( x ) ( | ∇ u | p ( x ) + | u | p ( x ) ) d x ) ( - d i v ( | ∇ u | p ( x ) - 2 ∇ u ) + | u | p ( x ) - 2 u ) = b ( ∫ Ω F ( x , u ) d x ) f ( x , u ) in Ω a ∫ Ω 1 p ( x ) ( | ∇ u | p ( x ) + | u | p ( x ) ) d x | ∇ u | p ( x ) - 2 ∂ u ∂ ν = g ( x , u ) on ∂ Ω , where Ω is a smooth bounded domain and ν is the outward normal vector on the boundary ∂Ω, and F ( x , u ) = ∫ 0 u f ( x , t ) d t . By using the variational method and the theory of the variable exponent Sobolev space, under appropriate assumptions on f, g, a and b, we obtain some results on existence and multiplicity of solutions of the problem.Mathematics Subject Classification (2000): 35B38; 35D05; 35J20.

Highlights

In this article, we consider the following problem ⎪⎪⎪⎪⎨ a1 (|∇u|p(x) + |u|p(x))dx (−div(|∇u|p(x)−2∇u) + |u|p(x)−2u) p(x) (P) ⎪⎪⎪⎪⎩ a= b F(x, u)dx f (x, u) in ∂ ∂ u ν, where Ω is a smooth bounded domain in RN, p ∈ Cwith 1 < p- := infΩ p(x) ≤ p (x) ≤ p+ := supΩ p(x) < N, a(t) is a continuous real-valued function, f : Ω × R ® R, g : u∂Ω × R ® R satisfy the Caratheodory condition, and F(x, u) = f (x, t)dt
Since the equation contains an integral related to the unknown u over Ω, it is no longer an identity pointwise, and is often called nonlocal problem
The p(x)-Kirchhoff type equations with Dirichlet boundary value problems have been studied by Dai and Hao [24], and much weaker conditions have been given by Fan [25]

Summary

Introduction

Where Ω is a smooth bounded domain in RN, p ∈ Cwith 1 < p- := infΩ p(x) ≤ p (x) ≤ p+ := supΩ p(x) < N, a(t) is a continuous real-valued function, f : Ω × R ® R, g : u. Corrêa and Figueiredo [13] presented several sufficient conditions for the existence of positive solutions to a class of nonlocal boundary value problems of the p-Kirchhoff type equation. The p(x)-Kirchhoff type equations with Dirichlet boundary value problems have been studied by Dai and Hao [24], and much weaker conditions have been given by Fan [25]. We focus the case of nonlocal p(x)-Laplacian problems with nonlinear Neumann boundary conditions This is a new topics even when p(x) ≡ p is a constant. Since E is sequentially weakly lower semi-continuous and X is reflexive, E attains its infimum in X at some u0 Î X In this case E is differentiable at u0, u0 is a solution of (P).

There exist an open subset

Note that in this case the inequalities