Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

We consider the supercritical problem -Delta u = vertical bar u vertical bar(p-2) u in Omega, u = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N, N >= 3, and p >= 2* := 2N/N-2. Bahri and Coron showed that if Omega has nontrivial homology this problem has a positive solution for p = 2*. However, this is not enough to guarantee existence in the supercritical case. For p >= 2(N - 1)/N-3 Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as p increases. More precisely, we show that for p > 2(N - k)/N-k-2 with 1 <= k <= N - 3 there are bounded smooth domains in R-N whose cup-length is k + 1 in which this problem does not have a nontrivial solution. For N = 4, 8, 16 we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents.