Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

We consider the supercritical problem -\Delta u = |u|^{p-2}u in \Omega, u=0 on \partial\Omega, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N\geq3,$ and $p\geq2^{*}:= 2N/(N-2).$ 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 $p\geq 2(N-1)/(N-3)$ 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 $1\leq k\leq N-3$ there are bounded smooth domains in $\mathbb{R}^{N}$ 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.