A supercritical elliptic problem in a cylindrical shell

Abstract

We consider the problem \[ -Δu=|u|^{p-2}u in Ω, u=0 on \partialΩ, \] where $\Omega :=\{(y,z)\in {\mathbb{R}}^{m+1}\times {\mathbb{R}}^{N-m-1}:0<a<|y|<b<\infty \}$, $0\le m\le N-1$ and $N\ge 2$. Let $2_{N,m}^{\ast }:=2(N-m)/(N-m-2)$ if $m<N-2$ and $2_{N,m}^{\ast }:=\infty$ if $m=N-2$ or $N-1$. We show that $2_{N,m}^{\ast }$ is the true critical exponent for this problem, and that there exist nontrivial solutions if $2<p<2_{N,m}^{\ast }$ but there are no such solutions if $p\ge 2_{N,m}^{\ast }$.