Uniqueness of ground states of some coupled nonlinear Schrodinger systems and their application

TL;DR

This paper proves the uniqueness of ground states in certain coupled nonlinear Schrödinger systems, using symmetrization and radial symmetry, and applies these results to derive a sharp inequality.

Contribution

It introduces a novel approach combining symmetrization and ODE analysis to establish ground state uniqueness and derives a new sharp inequality as an application.

Findings

Proved uniqueness of ground states for specific coupled Schrödinger systems.

Established existence of ground states via Schwartz symmetrization.

Derived a sharp vector-valued Gagliardo-Nirenberg inequality.

Abstract

We establish the uniqueness of ground states of some coupled nonlinear Schrodinger systems in the whole space. We firstly use Schwartz symmetrization to obtain the existence of ground states for a more general case. To prove the uniqueness of ground states, we use the radial symmetry of the ground states to transform the systems into an ordinary differential system, and then we use the integral forms of the system. More interestingly, as an application of our uniqueness results, we derive a sharp vector-valued Gagliardo-Nirenberg inequality.