## Abstract and Applied Analysis

### Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains

Tsing-San Hsu

#### Abstract

We will show that under suitable conditions on $f$ and $h$, there exists a positive number $\lambda^{\ast}$ such that the nonhomogeneous elliptic equation $-{\Delta}u+u={\lambda}(f(x,u)+h(x))$ in $\Omega$, $u\in{}{H}_{0}^{1}({\Omega{}})$, $N\geq 2$, has at least two positive solutions if $\lambda \in 0, \lambda^{\ast)$, a unique positive solution if $\lambda=\lambda^{\ast}$, and no positive solution if $\lambda > \lambda^{\ast}$, where $\Omega$ 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2007 (2007), Article ID 43018, 19 pages.

Dates
First available in Project Euclid: 23 March 2007

https://projecteuclid.org/euclid.aaa/1174668659

Digital Object Identifier
doi:10.1155/2007/43018

Mathematical Reviews number (MathSciNet)
MR2283965

Zentralblatt MATH identifier
1157.35371

#### Citation

Hsu, Tsing-San. Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains. Abstr. Appl. Anal. 2007 (2007), Article ID 43018, 19 pages. doi:10.1155/2007/43018. https://projecteuclid.org/euclid.aaa/1174668659