## Abstract and Applied Analysis

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

Tsung-Fang Wu

#### Abstract

We consider the elliptic problem $-\Delta u+u=b(x){|u|}^{p-2}u+h(x)$ in $\Omega$, $u\in {H}_{0}^{1}(\Omega )$, where $2\lt p \lt (2N/(N-2)) (N\geq 3), 2 \lt p \lt \infty (N=2), \Omega$ is a smooth unbounded domain in ${\mathbb{R}}^{N}, b(x)\in C(\Omega)$, and $h(x)\in {H}^{-1} (\Omega )$. We use the shape of domain $\Omega$ to prove that the above elliptic problem has a ground-state solution if the coefficient $b(x)$ satisfies $b(x)\rightarrow {b}^{\infty} > 0$ as $|x|\rightarrow \infty$ and $b(x)\geq c$ for some suitable constants $c\in (0,{b}^{\infty})$, and $h(x)\equiv 0$. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficient $b(x)$ also satisfies the above conditions, $h(x)\geq 0$ and $0\lt \Vert h\Vert_{{H}^{-1}} \lt (p-2){(1/(p-1))}^{(p-1)/(p-2)}{[{b}_{\sup}{S}^{p}({\Omega})]}^{1/(2-p)}$, where $S({\Omega })$ is the best Sobolev constant of subcritical operator in ${H}_{0}^{1}({\Omega })$ and ${b}_{\sup }={\sup }_{x\in {\Omega }}b(x)$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2007 (2007), Article ID 18187, 25 pages.

Dates
First available in Project Euclid: 27 February 2008

https://projecteuclid.org/euclid.aaa/1204126588

Digital Object Identifier
doi:10.1155/2007/18187

Mathematical Reviews number (MathSciNet)
MR2302187

Zentralblatt MATH identifier
1157.35377

#### Citation

Wu, Tsung-Fang. Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains. Abstr. Appl. Anal. 2007 (2007), Article ID 18187, 25 pages. doi:10.1155/2007/18187. https://projecteuclid.org/euclid.aaa/1204126588

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