Equivalence of conditions on initial data below the ground state to NLS with a repulsive inverse power potential

Abstract

In this paper, we consider the nonlinear Schrödinger equation (NLS) with a repulsive inverse power potential. First, we show 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. Recently, we established the 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 obtained by Hamano and Ikeda [“Characterization of the ground state to the intercritical NLS with a linear potential by the virial functional,” in Advances in Harmonic Analysis and Partial Differential Equations, Trends in Mathematics (Birkhäuser/Springer, Cham, 2020), pp. 279–307]. 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 by Hamano and Ikeda [“Characterization of the ground state to the intercritical NLS with a linear potential by the virial functional,” in Advances in Harmonic Analysis and Partial Differential Equations, Trends in Mathematics (Birkhäuser/Springer, Cham, 2020), pp. 279–307].