High energy sign-changing solutions for Coron's problem

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