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 $\mathbb R^N$ with $N\geq 3$, $\lambda>0$ is a parameter and $2^*=\frac{ 2N}{ N-2}$ 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.
Categories: math.AP