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arXiv · 2307.04426

The Brezis-Nirenberg problem in 4D

Angela Pistoia·Serena Rocci
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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.

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Angela Pistoia, Serena Rocci. 2023-07-10. The Brezis-Nirenberg problem in 4D. https://arxiv.org/abs/2307.04426

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