  We consider the supercritical problem \[ -\Delta v=|v|^{p-2}v in
\Theta_{\epsilon}, v=0 on \partial\Theta_{\epsilon}, \] where $\Theta$ is a
bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3$, $p>2^{\ast}:=2N/(N-2)$,
and $\Theta_{\epsilon}$ is obtained by deleting the $\epsilon$-neighborhood of
some sphere which is embedded in $\Theta$. In some particular situations we
show that, for $\epsilon>0$ small enough, this problem has a positive solution
$v_{\epsilon}$ and that these solutions concentrate and blow up along the
sphere as $\epsilon$ tends to 0.
  Our approach is to reduce this problem to a critical problem of the form \[
-\Delta u=Q(x)|u|^{4/(n-2)}u in \Omega_{\epsilon}, u=0 on
\partial\Omega_{\epsilon}, \] in a punctured domain
$\Omega_{\epsilon}:=\{x\in\Omega:|x-\xi_{0}|>\epsilon\}$ of lower dimension, by
means of some Hopf map. We show that, if $\Omega$ is a bounded smooth domain in
$\mathbb{R}^{n}$, $n\geq3$, $\xi_{0} is in\Omega,$ $Q is in
C^{2}(\b{\Oarmega})$ is positive and $\nabla Q(\xi_{0})\neq0$ then, for
$\epsilon>0$ small enough, this problem has a positive solution $u_{\epsilon}$,
and that these solutions concentrate and blow up at $\xi_{0}$ as $\epsilon$
goes to 0.
