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Open Access Research

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

Erlin Guo* and Peihao Zhao

Author Affiliations

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China

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Boundary Value Problems 2012, 2012:1  doi:10.1186/1687-2770-2012-1

Published: 4 January 2012

Abstract

In this article, we study the nonlocal p(x)-Laplacian problem of the following form

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

where Ω is a smooth bounded domain and ν is the outward normal vector on the boundary ∂Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M2">View MathML</a>. 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.

Keywords:
critical points; p(x)-Laplacian; nonlocal problem; variable exponent Sobolev spaces; nonlinear Neumann boundary conditions