On some nonhomogeneous elliptic problems in unbounded domains

In this paper we consider the problem \[\left\{\begin{array}{ll} - \Delta u + u\ =\ |u|^{p-2}u + f(x)&\quad\mbox{in $\Omega$,}\\ u = u_0 &\quad\mbox{on $\partial\Omega$,} \end{array}\right.\] where $\Omega = \R^N\setminus \bar\omega$, $\omega$ being a nonempty open bounded domain of $\R^N$ having smooth boundary $\partial\omega = \partial\Omega$, $N \ge 3$, $2 < p < \frac{2 N}{N - 2}$, $u_0 \in H^{1/2}(\partial\Omega)$, $f \in L^2(\Omega)$. We prove that, if $u_0$ and $f$ are nonnegative and satisfy suitable conditions, there exist at least two positive solutions. Moreover, the existence of a solution is shown to hold even when no condition on the sign of $u_0$ and $f$ is assumed