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

TL;DR

This paper investigates the long-term behavior of solutions to the focusing nonlinear Schrödinger equation with a repulsive inverse-power potential at the critical mass-energy level, revealing scattering, blow-up, and failure of uniform bounds.

Contribution

It provides a detailed analysis of dynamics at the mass-energy threshold for the focusing NLS with a repulsive inverse-power potential, including new results on scattering, blow-up, and bounds failure.

Findings

Characterization of solution behavior at the mass-energy threshold

Proof of scattering and blow-up phenomena at critical level

Demonstration of failure of uniform space-time bounds

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.