In this paper we consider the global well-posedness (GWP) and finite time blowup problem for the 3D focusing energy-critical inhomogeneous NLS with spatial inhomogeneity the coefficient $g$ such that $g\left(x\right)\sim {\left|x\right|}^{-b}$ for $0\le b<2$. The difficulty of this problem comes from the singularity of $g$. In the previous result (Cho and Lee, 2020; Cho et al., 2020) the authors showed the GWP for $0\le b<\frac{4}{3}$ by Kenig–Merle argument based on the standard Strichartz estimates. Here we extend the GWP to the coefficient with more serious singularity, $\frac{4}{3}\le b<\frac{3}{2}$. For this purpose, we improve the local theory and develop a new profile decomposition based on weighted Strichartz estimates.

在本文中，我们考虑具有空间不均匀性的3D聚焦能量临界不均匀NLS的全局适定性（GWP）和有限时间爆炸问题 $G$ 这样 $G（X）〜{\left|X\right|}^{-b}$ 对于 $0\le b<2$。这个问题的困难来自于奇异性$G$。在之前的结果中（Cho和Lee，2020； Cho等，2020），作者展示了$0\le b<\frac{4}{3}$由Kenig-Merle根据标准的Strichartz估计得出。在这里，我们将GWP扩展为具有更严重的奇点的系数，$\frac{4}{3}\le b<\frac{3}{2}$。为此，我们改进了局部理论，并基于加权的Strichartz估计开发了新的轮廓分解。