Open Access Research

Existence and multiplicity of solutions for nonlocal p ( x ) -Laplacian problems in R N

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, 2012:79  doi:10.1186/1687-2770-2012-79

Published: 26 July 2012

Abstract

In this paper, we study the nonlocal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a>-Laplacian problem of the following form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M4">View MathML</a>

By using the method of weight function and the theory of the variable exponent Sobolev space, under appropriate assumptions on f and M, we obtain some results on the existence and multiplicity of solutions of this problem. Moreover, we get much better results with f in a special form.

MSC: 35B38, 35D05, 35J20.

Keywords:
critical points; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a>-Laplacian; nonlocal problem; variable exponent Sobolev spaces