Mass-energy threshold dynamics for the focusing NLS with a repulsive inverse-power potential

In this paper we study long time dynamics (i.e., scattering and blow-up) of solutions for the focusing NLS with a repulsive inverse-power potential and with initial data lying exactly at the mass-energy threshold, namely, when $E_V(u_0)M(u_0)=E_0(Q)M(Q)$. Moreover, we prove failure of the uniform space-time bounds at the mass-energy threshold.