Global dynamics below the ground state for the focusing Schr\"odinger equation with a potential

TL;DR

This paper investigates the long-term behavior of solutions to the focusing nonlinear Schrödinger equation with a potential, establishing conditions for scattering and blow-up phenomena in non-radial and radial cases.

Contribution

It extends the analysis of the Schrödinger equation by proving scattering and blow-up results without radial symmetry assumptions, under certain potential conditions.

Findings

Proves scattering for solutions below the ground state.

Establishes blow-up results in mass-supercritical and energy-subcritical regimes.

Provides methods to exclude blow-up in symmetric cases.

Abstract

In this paper, we consider the nonlinear Schr\"odinger equation with a real valued potential V=V(x). We study global behavior of solutions to the equation with a data below the ground state under some conditions for the potential V and prove a scattering result and a blowing-up result in mass-supercritical and energy-subcritical. Our proof of the blowing-up or growing-up result without radially symmetric assumption is based on the argument by Du-Wu-Zhang in [6]. We can exclude the possibility of the growing-up result by the argument in [23], [15], and [10] if "the data and the potential are radially symmetric" or "the data has finite variance".