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.
Categories: math.AP