  This paper is concerned with the global well-posedness and finite time blowup
problem for the 3D focusing energy-critical inhomogeneous NLS. In the previous
results \cite{chkl2, chkl3} the authors considered the same problems with the
spatial inhomogeneity coefficient $g$ such that $g(x) \sim |x|^{-b}$ for $0 \le
b < \frac43$. Here we extend the inhomogeneous index $b$ up to $\frac32$. For
this purpose, we improve the local theory and develop a new profile
decomposition based on weighted space.
