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

In this paper, we consider the Brezis-Nirenberg problem \begin{equation*} \left{\begin{aligned} &-Δu = λu+|u|^{2^-2}u, \quad &\mbox{in},Ω,\ &u=0,\quad &\mbox{on}, \partialΩ, \end{aligned}\right. \end{equation} where $\mathrm{\Omega }$ is a smoothly bounded domain of ${\mathbb{R}}^N$ with $N\ge 3$, $\lambda >0$ is a parameter and $2^{\ast }=\frac{2N}{N-2}$ 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.