Uniqueness of ground state and minimal-mass blow-up solutions for focusing NLS with Hardy potential

TL;DR

This paper proves the uniqueness of the ground state for a focusing nonlinear Schrödinger equation with Hardy potential, establishes a sharp inequality, and characterizes minimal mass blow-up solutions.

Contribution

It provides the first proof of ground state uniqueness, derives a sharp Hardy-Gagliardo-Nirenberg inequality, and characterizes minimal mass blow-up solutions.

Findings

Unique ground state solution established

Sharp Hardy-Gagliardo-Nirenberg inequality derived

Complete characterization of minimal mass blow-up solutions

Abstract

We consider the focusing nonlinear Schr\"odinger equation with the critical inverse square potential. We give the first proof of the uniqueness of the ground state solution. Consequently, we obtain a sharp Hardy-Gagliardo-Nirenberg interpolation inequality. Moreover, we provide a complete characterization for the minimal mass blow-up solutions to the time dependent problem.