Nodal solutions of the Brezis-Nirenberg problem in dimension 6

TL;DR

This paper demonstrates the existence of sign-changing solutions that blow up at a point in a 6-dimensional domain for the Brezis-Nirenberg problem as a parameter approaches a critical value.

Contribution

It establishes the existence of nodal solutions in dimension six that blow up at a point, extending understanding of solution behavior in critical dimensions.

Findings

Existence of sign-changing solutions in dimension six

Solutions blow up at a point as parameter approaches a critical value

Extends previous results to higher dimensions

Abstract

We show that the classical Brezis-Nirenberg problem $$ -\Delta u=u|u| + \lambda u\ \hbox{in}\ \Omega, u=0\ \hbox{on}\ \partial\Omega, $$ when $\Omega$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a point in $\Omega$ as $\lambda$ approaches a suitable value $\lambda_0>0.$