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.

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Coron 问题的高能变号解决方案

我们研究以下问题的符号变化解的存在性 $$ (P) \quad \quad \quad \left\{ \begin{array}{ll} \Delta u+|u|^{p-1}u=0 \quad & {\rm in} \quad \Omega_\epsilon; u=0 \quad & {\rm on} \quad\partial \Omega_\epsilon, \end{array} \right. $$ 其中 $p=\frac{n+2}{n-2}$ 是临界 Sobolev 指数，$\Omega_\epsilon$ 是 ${\mathcal R}^n$, $n\ 中的有界光滑域geq 3$，形式为 $\Omega_\epsilon=\Omega\backslash B(0,\epsilon)$ 和 $\Omega$ 一个包含原点 $0$ 和 $B(0,\epsilon)$ 的平滑有界域球以原点为中心，半径 $\epsilon >0$。当参数 $\epsilon$ 足够小时，我们构造了一种新的具有高能量的符号变化解决方案来解决问题 $(P)$。