High energy sign-changing solutions for Coron's problem

TL;DR

This paper constructs high-energy sign-changing solutions for a critical Sobolev exponent problem in a domain with a small hole, extending understanding of solutions' existence and behavior in such singularly perturbed domains.

Contribution

It introduces a new class of high-energy sign-changing solutions for the critical PDE in domains with small holes, advancing the analysis of solution structures in singular perturbation problems.

Findings

Existence of high-energy sign-changing solutions for small epsilon

Construction method applicable to domains with holes

Insights into solution behavior near singular perturbations

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.