Open Access Research

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

Erlin Guo* and Peihao Zhao

Full TextView PDF (347KB)

Author affiliations

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

For all author emails, please log on.

Citation and License

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