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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

Citation and License

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

Published: 28 February 2012


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.

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