Blow-up or grow-up for the focusing 3D cubic NLS with a repulsive inverse-power potential at the mass--energy threshold

We consider the focusing cubic nonlinear Schrodinger equation with a repulsive inverse-power potential $V(x)=a\mathrm{\mid }x{\mathrm{\mid }}^{-\mu }$, where $a>0$ and $1<\mu \le 2$. At the mass-energy threshold, Miao, Murphy, and Zheng, as well as Ardila, Hamano, and Ikeda, established sharp scattering results in the positive virial region. In this paper, we continue to study the dynamics in the negative virial region and show that, in each time direction, solutions either blow up in finite time or grow up.