Abstract and Applied Analysis
Volume 2007 (2007), Article ID 43018, 19 pages
doi:10.1155/2007/43018
Research Article
Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains
Center for General Education, Chang Gung University, Kwei-Shan, Tao-Yuan 333, Taiwan
Received 7 July 2006; Revised 25 December 2006; Accepted 25 December 2006
Academic Editor: Martin J. Bohner
Copyright © 2007 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 will show that under suitable conditions on f and h, there exists a positive number λ∗ such that the nonhomogeneous elliptic equation
−Δu+u=λ(f(x,u)+h(x)) in Ω, u∈H01(Ω), N≥2, has at least two positive solutions if λ∈(0,λ∗), a unique positive solution if λ=λ∗, and no positive solution if λ>λ∗, where Ω is the entire space or an exterior domain or an unbounded cylinder domain
or the complement in a strip domain of a bounded domain.
We also obtain some properties of the set of solutions.