  We consider the problem \[ -\Delta u=|u|^{p-2}u in \Omega, u=0 on
\partial\Omega, \] where
$\Omega:=\{(y,z)\in\mathbb{R}^{m+1}\times\mathbb{R}^{N-m-1}:
0<a<|y|<b<\infty\}$, $0\leq m\leq N-1$ and $N\geq2$. 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 $p\geq2_{N,m}^{\ast}$.
