Abstract and Applied Analysis
Volume 2007 (2007), Article ID 18187, 21 pages
Research Article

Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains

Tsung-Fang Wu

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

Received 30 September 2006; Revised 26 December 2006; Accepted 3 January 2007

Academic Editor: Vy Khoi Le

Copyright © 2007 Tsung-Fang Wu. 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.


We consider the elliptic problem Δu+u=b(x)|u|p2u+h(x) in Ω, uH01(Ω), where 2<p<(2N/(N2)) (N3), 2<p< (N=2), Ω is a smooth unbounded domain in N, b(x)C(Ω), and h(x)H1(Ω). We use the shape of domain Ω to prove that the above elliptic problem has a ground-state solution if the coefficient b(x) satisfies b(x)b>0 as |x| and b(x)c for some suitable constants c(0,b), and h(x)0. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficient b(x) also satisfies the above conditions, h(x)0 and 0<hH1<(p2)(1/(p1))(p1)/(p2)[bsupSp(Ω)]1/(2p), where S(Ω) is the best Sobolev constant of subcritical operator in H01(Ω) and bsup=supxΩb(x).