Sign-changing blowing-up solutions for supercritical Bahri-Coron's problem

TL;DR

This paper constructs the first sign-changing bubbling solutions for a supercritical elliptic problem with a small hole in the domain, using a nodal solution to the Yamabe problem with many kernels.

Contribution

It provides the first example of sign-changing bubbling solutions for a supercritical problem with a small domain hole, extending previous results.

Findings

First sign-changing bubbling solutions for supercritical problem

Uses a nodal Yamabe solution with many kernels

Addresses domain with small hole in elliptic PDE context

Abstract

Let $\Omega$ be a bounded domain in $\R^n$, $n\ge 3$ with smooth boundary $\partial\Omega$ and a small hole. We give the first example of sign-changing {\it bubbling} solutions to the nonlinear elliptic problem $$ -\Delta u=|u|^{{n+2\over n-2} +\ve -1 } u \, \, \mbox{ in } \Omega , \quad \quad u=0 \mbox{ on } \partial \Omega, $$ where $\ve$ is a small positive parameter. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem $$ -\Delta w = |w|^{\frac{4}{n-2}} w, \ \ w \in {\mathcal D}^{1,2} (\R^n) $$ which has large number ($3n$) of kernels.