SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Multiple positive solutions for a class of quasi-linear elliptic equations involving concave-convex nonlinearities and Hardy terms

Tsing-San Hsu

Author affiliations

Center for General Education, Chang Gung University, Kwei-San, Tao-Yuan 333, Taiwan ROC

Citation and License

Boundary Value Problems 2011, 2011:37  doi:10.1186/1687-2770-2011-37

Published: 19 October 2011

Abstract

In this paper, we are concerned with the following quasilinear elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M1">View MathML</a>

where Ω ⊂ ℝN is a smooth domain with smooth boundary ∂Ω such that 0 ∈ Ω, Δpu = div(|∇u|p-2u), 1 < p < N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M2">View MathML</a>, λ >0, 1 < q < p, sign-changing weight functions f and g are continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M4">View MathML</a> is the best Hardy constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M5">View MathML</a> is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold, the multiplicity of positive solutions to this equation is verified.

Keywords:
Multiple positive solutions; critical Sobolev exponent; concave-convex; Hardy terms; sign-changing weights