Minimal mass blow-up solutions for a inhomogeneous NLS equation

We consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^N$ \begin{align}\label{inls} i \partial_t u +\Delta u +V(x)|u|^{\frac{4-2b}{N}}u = 0, \end{align} where $V(x) = k(x)|x|^{-b}$, with $b>0$. Under suitable assumptions on $k(x)$, we established the threshold for global existence and blow-up and then study the existence and non-existence of minimal mass blow-up solutions.