Paper detail

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

We consider the supercritical problem \[ -Δv=|v|^{p-2}v in Θ_ε, v=0 on \partialΘ_ε, \] where $Θ$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3$, $p>2^{\ast}:=2N/(N-2)$, and $Θ_ε$ is obtained by deleting the $ε$-neighborhood of some sphere which is embedded in $Θ$. In some particular situations we show that, for $ε>0$ small enough, this problem has a positive solution $v_ε$ and that these solutions concentrate and blow up along the sphere as $ε$ 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 $Ω_ε:=\{x\inΩ:|x-ξ_{0}|>ε\}$ of lower dimension, by means of some Hopf map. We show that, if $Ω$ is a bounded smooth domain in $\mathbb{R}^{n}$, $n\geq3$, $ξ_{0} is inΩ,$ $Q is in C^{2}(\b{\Oarmega})$ is positive and $\nabla Q(ξ_{0})\neq0$ then, for $ε>0$ small enough, this problem has a positive solution $u_ε$, and that these solutions concentrate and blow up at $ξ_{0}$ as $ε$ goes to 0.