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

Multiplicity Results for p-Laplacian with Critical Nonlinearity of Concave-Convex Type and Sign-Changing Weight Functions

Tsing-San Hsu

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

Received 4 December 2008; Revised 24 June 2009; Accepted 7 September 2009

Academic Editor: Pavel Drábek

Copyright © 2009 Tsing-San Hsu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.