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

Abstract

We consider the supercritical problem \[ -Δv=|v|^{p-2}v in Θ_ε, v=0 on \partialΘ_ε, \] where $\Theta$ is a bounded smooth domain in ${\mathbb{R}}^N$, $N\ge 3$, $p>2^{\ast }:=2N/(N-2)$, 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 \[ -Δu=Q(x)|u|^{4/(n-2)}u in Ω_ε, u=0 on \partialΩ_ε, \] 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 ${\mathbb{R}}^n$, $n\ge 3$, ${\xi }_{0}isin\Omega ,$ $Q is in C^{2}(\b{\Oarmega})$ is positive and $\nabla Q({\xi }_0)\ne 0$ 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.

References in corpus (1)

• Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents