布雷齐斯 - 尼伦伯格方程的精细爆破分析及其应用：\(N\geq4\)时的单泡情形
A refined blow-up analysis of the Brezis-Nirenberg equation and its application: The one-bubble case for $N\geq4$
AI总结：
研究\(N\geq4\)时布雷齐斯 - 尼伦伯格方程，基于反约化论证发展精细爆破分析，对单泡情形下随参数\(\lambda\)变化的施特鲁韦分解进行分类，并证明\(4d\)方程在一般有界区域对特定\(\lambda\)有非平凡解，完善相关存在性理论。
AI中文摘要：
本文考虑著名的布雷齐斯 - 尼伦伯格方程，其中\(N\geq3\)为维度，\(\Omega\subset\mathbb{R}^N\)是有光滑边界的有界区域，\(\lambda>0\)为参数。通过基于文献[WW2019,WW2019 - 2]中反约化论证发展的精细爆破分析，首次对\(N\geq4\)时参数\(\lambda\)变化的单泡情形下布雷齐斯 - 尼伦伯格方程的施特鲁韦分解进行分类。作为应用，证明了\(4d\)布雷齐斯 - 尼伦伯格方程在一般有界区域中对于\(\lambda\in\sigma(-\Delta)\)有非平凡解（最小能量解），完善了自1984年以来关于\(N\geq4\)时布雷齐斯 - 尼伦伯格方程的存在性理论。
英文摘要：
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 $N\geq3$ is the dimension, $Ω\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partialΩ$ and $λ>0$ 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 $λ$ varies for $N\geq4$. As applications, we prove that the $4d$ Brezis-Nirenberg equation has a nontrivial solution (least energy solution) for $λ\inσ(-Δ)$ in general bounded domains, where $σ(-Δ)$ is the spectrum of $-Δ$ in $H^1_0(Ω)$. 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.
