Sign-changing blowing-up solutions for the Brezis--Nirenberg problem in dimensions four and five

TL;DR

This paper constructs sign-changing solutions to the Brezis-Nirenberg problem in four and five dimensions, where the positive part concentrates and blows up at the domain's center as the parameter approaches the first eigenvalue.

Contribution

It proves the existence of symmetric, sign-changing solutions with specific blow-up behavior in dimensions four and five, extending understanding of solution structures in critical elliptic problems.

Findings

Positive part concentrates and blows up at the domain's center

Negative part vanishes as parameter approaches the first eigenvalue

Solutions exist for symmetric domains in dimensions four and five

Abstract

We consider the Brezis-Nirenberg problem: $$-\Delta u =\lambda u + |u|^{p-1}u\qquad \mbox{in}\,\, \Omega,\quad u=0\,\, \mbox{on}\,\,\ \partial\Omega,$$ where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$, $p=\frac{N+2}{N-2}$ and $\lambda>0$. In this paper we prove that, if $\Omega$ is symmetric and $N=4,5$, there exists a sign-changing solution whose positive part concentrates and blows-up at the center of symmetry of the domain, while the negative part vanishes, as $\lambda\rightarrow \lambda_1$, where $\lambda_1=\lambda_1(\Omega)$ denotes the first eigenvalue of $-\Delta$ on $\Omega$, with zero Dirichlet boundary condition.