A refined blow-up analysis of the Brezis-Nirenberg equation and its application: The one-bubble case for $N\geq4$
In this paper, we consider the famous Brezis-Nirenberg equation \begin{eqnarray*} \left\{ \aligned &-\Delta u=\lambda u+|u|^{\frac{4}{N-2}}u,\quad&\mbox{in}\,\, \Omega,\\ &u=0,\quad&\mbox{on}\,\, \partial\Omega, \endaligned \right. \end{eqnarray*} where $N\geq3$ is the dimension, $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$ and $\lambda>0$ is a parameter. By developing a refined blow-up analysis based on the inverse reduction argument developed in \cite{WW2019,WW2019-2}, we classify, for the fist time, the Struwe decomposition of the Brezis-Nirenberg equation in the one-bubble case as the parameter $\lambda$ varies for $N\geq4$. As applications, we prove that the $4d$ Brezis-Nirenberg equation has a nontrivial solution (least energy solution) for $\lambda\in\sigma(-\Delta)$ in general bounded domains, where $\sigma(-\Delta)$ is the spectrum of $-\Delta$ in $H^1_0(\Omega)$. Our result completes the existence theory of the Brezis-Nirenberg equation for $N\geq4$ in \cite{AP2025,CFP1985,CFS,CSS1986,CW2005,CSZ2012,SWW2009,TYZ2022} since 1984.