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.