The Brezis-Nirenberg problem in 4D

TL;DR

This paper investigates the Brezis-Nirenberg problem specifically in four-dimensional domains, exploring the existence and properties of solutions to a critical elliptic PDE with potential, building on extensive prior research.

Contribution

The paper provides new insights into the existence of solutions for the Brezis-Nirenberg problem in 4D, extending previous results to this specific dimension and potential configurations.

Findings

Analysis of solution existence in 4D domain

Extension of classical results to non-constant potentials

Identification of conditions for positive solutions

Abstract

The problem \begin{equation} \label{bn} -\Delta u=|u|^{4\over n-2}u+\lambda V u\ \hbox{in}\ \Omega,\ u=0\ \hbox{on}\ \partial\Omega \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.