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

TL;DR

This paper investigates the nonlinear Schrödinger equation with a repulsive inverse power potential, establishing well-posedness, blow-up criteria, and the equivalence of initial data conditions relative to the ground state.

Contribution

It extends previous results by proving the equivalence of initial data conditions below the ground state for NLS with a repulsive inverse power potential.

Findings

Global well-posedness below the ground state

Blow-up or grow-up behavior characterized

Equivalence of initial data conditions established

Abstract

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].