Abstract and Applied Analysis
Volume 2009 (2009), Article ID 652109, 24 pages
doi:10.1155/2009/652109

Multiplicity results for $p$-Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions

Abstract

The multiple results of positive solutions for the following quasilinear elliptic equation: −Δpu=λf(x)|u|q−2u+g(x)|u|p∗−2u in Ω,u=0 on ∂Ω, are established. Here, 0∈Ω is a bounded smooth domain in ℝN,Δp denotes the p-Laplacian operator, 1≤q<p<N,p∗=Np/(N−p),λ is a positive real parameter, and f,g are continuous functions on Ω¯ which are somewhere positive but which may change sign on Ω. The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.