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

Abstract

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.