Three solutions for a class of quasilinear elliptic systems involving the p(x)-Laplace operator

Honghui Yin,Zuodong Yang

doi:10.1186/1687-2770-2012-30

Abstract

The existence of at least three weak solutions is established for a class of quasilinear elliptic systems involving the p(x)-Laplace operator with Neumann boundary condition. The technical approach is mainly based on a three critical points theorem due to Ricceri. MSC: 35D05; 35J60; 58E05.

Highlights

We study problem (1) by using the three critical points theorem by Ricceri [26] too
W1,p(x)( ) = u ∈ Lp(x)( )| |∇u| ∈ Lp(x)( ), and it can be equipped with the norm u p(x) = |u|p(x) + |∇u|p(x), ∀u ∈ W1,p(x)( ), and we call it variable exponent Sobolev space
Doi:10.1186/1687-2770-2012-30 Cite this article as: Yin and Yang: Three solutions for a class of quasilinear elliptic systems involving the p(x)Laplace operator

Summary

Introduction

The study of differential equations and variational problems with nonstandard p(x)-growth conditions has been attracting attention of many authors in the last two decades. It arises from nonlinear elasticity theory, electro-rheological fluids, etc. We cite papers [20,21,22,23], where the authors established the existence of at least three weak solutions to the problems with Dirichlet or Neumann boundary value conditions. Li and Tang in [24] obtained the existence of at least three weak solutions to problem (1) when p(x) ≡ p with Dirichlet boundary value conditions.

We denote

For any u

Sobolev spaces