Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains

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 $-\Delta u+u=\lambda \left(f\left(x,u\right)+h\left(x\right)\right)$ in $\Omega$, $u\in H_0^1\left(\Omega \right)$, $N\ge 2$, has at least two positive solutions if $\lambda \in \left(0,{\lambda }^{\ast }\right)$, 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.