Positive solutions to a supercritical elliptic problem which concentrate along a thin spherical hole

Abstract

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.