  We prove the existence of positive and of nodal solutions for $-\Delta u =
|u|^{p-2}u+\mu |u|^{q-2}u$, $u\in {\rm H_0^1}(\Omega)$, where $\mu >0$ and
$2<q<p=2N(N-2)$, for a class of open subsets $\Omega$ of $\mathbb{R}^N$ lying
between two infinite cylinders.
