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

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 ${b}_{\sup }={\sup }_{x\in {\Omega }}b(x)$.