Boundary Value Problems
Volume 2010 (2010), Article ID 856932, 18 pages
doi:10.1155/2010/856932
Research Article

Multiple Positive Solutions for a Class of Concave-Convex Semilinear Elliptic Equations in Unbounded Domains with Sign-Changing Weights

Tsing-San Hsu

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

Received 8 September 2010; Accepted 18 October 2010

Academic Editor: Julio Rossi

Copyright © 2010 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

We study the existence and multiplicity of positive solutions for the following Dirichlet equations: − Δ 𝑢 + 𝑢 = 𝜆 𝑎 ( 𝑥 ) | 𝑢 | 𝑞 − 2 𝑢 + 𝑏 ( 𝑥 ) | 𝑢 | 𝑝 − 2 𝑢 in Ω , 𝑢 = 0 on 𝜕 Ω , where 𝜆 > 0 , 1 < 𝑞 < 2 < 𝑝 < 2 ∗ ( 2 ∗ = 2 𝑁 / ( 𝑁 − 2 ) if 𝑁 ≥ 3 ; 2 ∗ = ∞ if 𝑁 = 1 , 2 ), Ω is a smooth unbounded domain in ℝ 𝑁 , 𝑎 ( 𝑥 ) , and 𝑏 ( 𝑥 ) satisfy suitable conditions, and 𝑎 ( 𝑥 ) maybe change sign in Ω .