Minimal mass blow-up solutions for the $L^2$ critical NLS with inverse-square potential

TL;DR

This paper investigates the minimal mass blow-up solutions for the focusing $L^2$ critical nonlinear Schrödinger equation with inverse-square potential, establishing global well-posedness below ground state mass and classifying blow-up solutions at the threshold.

Contribution

It proves global well-posedness for solutions below ground state mass and classifies all minimal mass blow-up solutions as pseudo-conformal transformations of ground states.

Findings

Solutions with mass below ground state are global.

All minimal mass blow-up solutions are pseudo-conformal transforms of ground states.

Ground states have the same minimal mass despite potential non-uniqueness.

Abstract

We study minimal mass blow-up solutions of the focusing $L^2$ critical nonlinear Schr\"odinger equation with inverse-square potential, \[ i\partial_t u + \Delta u + \frac{c}{|x|^2}u+|u|^{\frac{4}{N}}u = 0, \] with $N\geqslant 3$ and $0<c<\frac{(N-2)^2}{4}$. We first prove a sharp global well-posedness result: all $H^1$ solutions with a mass (i.e. $L^2$ norm) strictly below that of the ground states are global. Note that, unlike the equation in free space, we do not know if the ground state is unique in the presence of the inverse-square potential. Nevertheless, all ground states have the same, minimal, mass. We then construct and classify finite time blow-up solutions at the minimal mass threshold. Up to the symmetries of the equation, every such solution is a pseudo-conformal transformation of a ground state solution.