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Open Access Research

Existence and multiplicity of positive solutions for a class of p(x)-Kirchhoff type equations

Ruyun Ma, Guowei Dai* and Chenghua Gao

Author Affiliations

Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China

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Boundary Value Problems 2012, 2012:16  doi:10.1186/1687-2770-2012-16

Published: 13 February 2012

Abstract

In this article, we study the existence and multiplicity of positive solutions for the Neumann boundary value problems involving the p(x)-Kirchhoff of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/16/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/16/mathml/M1">View MathML</a>

Using the sub-supersolution method and the variational method, under appropriate assumptions on f and M, we prove that there exists λ* > 0 such that the problem has at least two positive solutions if λ > λ*, at least one positive solution if λ = λ* and no positive solution if λ < λ*. To prove these results we establish a special strong comparison principle for the Neumann problem.

2000 Mathematical Subject Classification: 35D05; 35D10; 35J60.

Keywords:
p(x)-Kirchhoff; positive solution; sub-supersolution method; comparison principle