High energy sign-changing solutions for Coron's problem

Abstract

We study the existence of sign changing solutions to the following problem $$ (P) \quad \quad \quad \left\{ \begin{array}{ll} Δu+|u|^{p-1}u=0 \quad & {\rm in} \quad Ω_ε; u=0 \quad & {\rm on} \quad\partial Ω_ε, \end{array} \right. $$ where $p=\frac{n+2}{n-2}$ is the critical Sobolev exponent and $Ω_ε$ is a bounded smooth domain in ${\mathcal R}^n$, $n\geq 3$, with the form $Ω_ε=Ω\backslash B(0,ε)$ with $Ω$ a smooth bounded domain containing the origin $0$ and $B(0,ε)$ the ball centered at the origin with radius $ε>0$. We construct a new type of sign-changing solutions with high energy to problem $(P)$, when the parameter $ε$ is small enough.