A refined blow-up analysis of the Brezis-Nirenberg equation and its application: The one-bubble case for $N\ge 4$

In this paper, we consider the famous Brezis-Nirenberg equation \begin{eqnarray*} \left{ \aligned &-Δu=λu+|u|^{\frac{4}{N-2}}u,\quad&\mbox{in},, Ω,\ &u=0,\quad&\mbox{on},, \partialΩ, \endaligned \right. \end{eqnarray*} where $N\ge 3$ is the dimension, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$ 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 $N\ge 4$. As applications, we prove that the $4d$ Brezis-Nirenberg equation has a nontrivial solution (least energy solution) for $\lambda \in \sigma (-\mathrm{\Delta })$ in general bounded domains, where $\sigma (-\mathrm{\Delta })$ is the spectrum of $-\mathrm{\Delta }$ in $H_0^1(\mathrm{\Omega })$. Our result completes the existence theory of the Brezis-Nirenberg equation for $N\ge 4$ in \cite{AP2025,CFP1985,CFS,CSS1986,CW2005,CSZ2012,SWW2009,TYZ2022} since 1984.