On the effect of the domain geometru on the existence of sign changing solutions to elliptic problems with critical and supercritical growth

This paper deals with the existence of sign changing solutions of the problem \[ \begin{eqnarray*}\left\{\eqalign{-\Delta u=|u|^{p-1}u+{\varepsilon} w(x)|u|^{q-1}u \tqs {\rm in} \ \Omega,\\ u=0 \tqs {\rm on} \ \partial\Omega,}\right.\end{eqnarray*} \] where Ω is a bounded regular domain in {\Bbb R}^N , N ≥ 4, ε > 0, p = (N + 2)/(N − 2), q ≥ 1, q ≠ p and w\in{\rm C}^1(\bar \Omega) .