Abstract and Applied Analysis
Volume 2011 (2011), Article ID 864296, 16 pages
http://dx.doi.org/10.1155/2011/864296
Research Article

Multiple Positive Solutions for Semilinear Elliptic Equations with Sign-Changing Weight Functions in ℝ 𝑁

Tsing-San Hsu

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

Received 9 March 2011; Accepted 3 May 2011

Academic Editor: Nobuyuki Kenmochi

Copyright © 2011 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

Existence and multiplicity of positive solutions for the following semilinear elliptic equation: − Δ 𝑢 + 𝑢 = 𝑎 ( 𝑥 ) | 𝑢 | 𝑝 − 2 𝑢 + 𝜆 𝑏 ( 𝑥 ) | 𝑢 | 𝑞 − 2 𝑢 in ℝ 𝑁 , 𝑢 ∈ 𝐻 1 ( ℝ 𝑁 ) , are established, where 𝜆 > 0 , 1 < 𝑞 < 2 < 𝑝 < 2 ∗ ( 2 ∗ = 2 𝑁 / ( 𝑁 − 2 ) if 𝑁 ≥ 3 , 2 ∗ = ∞ if 𝑁 = 1 , 2 ) , 𝑎 , 𝑏 satisfy suitable conditions, and 𝑏 maybe changes sign in ℝ 𝑁 . The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.