The Brezis-Nirenberg problem in 4D

Abstract

The problem \begin{equation} \label{bn} -Δu=|u|^{4\over n-2}u+λV u\ \hbox{in}\ Ω,\ u=0\ \hbox{on}\ \partialΩ \end{equation} where $\Omega$ is a bounded regular domain in ${\mathbb{R}}^n$, $\lambda \in \mathbb{R}$ and $V\in C^0(\overline{\Omega }),$ that was introduced by Brezis and Nirenberg in their famous paper, where they address the existence of positive solutions in the autonomous case, i.e. the potential $V$ is constant. Since then, a huge amount of work has been done. In the following we will make a brief history highlighting the results which are much closer to the problem we wish to study in the present paper.