Sign changing solutions for elliptic equations with critical growth in cylinder type domains

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$2<q<p=2N(N-2)$,foraclassofopensubsets$\Omega$of$$\mathbb{R}$^N$ lying between two infinite cylinders.