On Brezis-Nirenberg problems: open questions and new results in dimension six

Abstract: In this paper, we consider the Brezis-Nirenberg problem \begin{equation*} \left{\begin{aligned} &-\Delta u = \lambda u+|u|^{2^*-2}u, \quad &\mbox{in}\,\Omega,\ &u=0,\quad &\mbox{on}\, \partial\Omega, \end{aligned}\right. \end{equation*} where $\Omega$ is a smoothly bounded domain of $\mathbb R<sup>N$ with $N\geq 3$, $\lambda&gt;0$ is a parameter and $2<sup>*=\frac{2N}{N-2}$ is the critical Sobolev exponent. We first recall the history of the Brezis-Nirenberg problem and then provide new results of it in dimension six. Finally, we also list some open questions on the Brezis-Nirenberg problem.