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

Abstract

We consider the elliptic problem $-\Delta u+u=b\left(x\right){\left|u\right|}^{p-2}u+h\left(x\right)$ in $\Omega$, $u\in H_0^1\left(\Omega \right)$, where $2<p<(2N/(N-2\left)\right) (N\ge 3), 2<p<\infty \left(N=2\right), \Omega$ is a smooth unbounded domain in ${ℝ}^N, b\left(x\right)\in C\left(\Omega \right)$, and $h\left(x\right)\in H^{-1}\left(\Omega \right)$. We use the shape of domain $\Omega$ to prove that the above elliptic problem has a ground-state solution if the coefficient $b\left(x\right)$ satisfies $b\left(x\right)\to b^{\infty }>0$ as $\left|x\right|\to \infty$ and $b\left(x\right)\ge c$ for some suitable constants $c\in \left(0,b^{\infty }\right)$, and $h\left(x\right)\equiv 0$. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficient $b\left(x\right)$ also satisfies the above conditions, $h\left(x\right)\ge 0$ and $0<{\Vert h\Vert }_{H^{-1}}<\left(p-2\right){\left(1/(p-1)\right)}^{(p-1)/(p-2)}{\left[b_{sup}{S}^p\left(\Omega \right)\right]}^{1/(2-p)}$, where $S\left(\Omega \right)$ is the best Sobolev constant of subcritical operator in $H_0^1\left(\Omega \right)$ and $b_{sup}={sup}_{x\in \Omega }b\left(x\right)$.