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Open Access Research

Multiple solutions for p-Laplacian systems with critical homogeneous nonlinearity

Dengfeng Lü

Author Affiliations

School of Mathematics and Statistics, Hubei Engineering University, Hubei 432000, P. R. China

Boundary Value Problems 2012, 2012:27  doi:10.1186/1687-2770-2012-27

Published: 28 February 2012

Abstract

In this article, we deal with existence and multiplicity of solutions to the p-Laplacian system of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M1">View MathML</a>

where Ω ⊂ ℝN is a bounded domain with smooth boundary ∂Ω, Δpu = div(|∇u|p-2u) is the p-Laplacian operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M2">View MathML</a> denotes the Sobolev critical exponent, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M3">View MathML</a> is a homogeneous function of degree p*. By using the variational method and Ljusternik-Schnirelmann theory, we prove that the system has at least cat(Ω) distinct nonnegative solutions.

AMS 2010 Mathematics Subject Classifications: 35J50; 35B33.

Keywords:
p-Laplacian system; Ljusternik-Schnirelmann theory; critical exponent; multiple solutions