  In this paper, we consider the nonlinear Schr\"odinger equation with a
repulsive inverse power potential. First, we show that some global
well-posedness results and "blow-up or grow-up" results below the ground state
without the potential. Then, we prove equivalence of the conditions on the
initial data below the ground state without potential. We note that recently,
we established existence of a radial ground state and characterized it by the
virial functional for NLS with a general potential in two or higher space
dimensions in [8]. Then, we also prove a global well-posedness result and a
"blow-up or grow-up" result below the radial ground state with a repulsive
inverse power potential obtained in [8].
