Multiple solutions of supercritical elliptic problems in perturbed domains

We are concerned with existence and multiplicity of nontrivial solutions for the Dirichlet problem Delta u + vertical bar u vertical bar(p-2)u = 0 in Omega, u = 0 on delta Omega, where Omega is a bounded domain of R-n, n >= 3, and p > 2n/n-2. We show that suitable perturbations of the domain, which modify its topological properties, give rise to a number of solutions which tends to infinity as the size of the perturbation tends to zero (some examples show that the perturbed domains may be even contractible). More precisely, we prove that for all k is an element of N, if the size of the perturbation is small enough (depending on k), there exist at least k pairs of nontrivial solutions, which concentrate near the perturbation as the size of the perturbation tends to zero. The method we use, which is completely variational, gives also further informations on the qualitative properties of the solutions; in particular. these solutions (which may change sign) do not have more than k nodal regions and at least two solutions (which minimize the corresponding energy functional) have constant sign.