Uniqueness and nondegeneracy of ground states for nonlinear Schr\"odinger equations with attractive inverse-power potential

TL;DR

This paper investigates the uniqueness and nondegeneracy of ground states in nonlinear Schrödinger equations with inverse-power potentials, refining previous results and exploring stability properties of solutions.

Contribution

It provides new proofs of uniqueness and nondegeneracy for ground states and discusses their orbital instability, extending prior work in the field.

Findings

Proved uniqueness of ground states under certain conditions

Established nondegeneracy of these ground states

Analyzed orbital instability of standing waves

Abstract

We study uniqueness and nondegeneracy of ground states for stationary nonlinear Schr\"odinger equations with a focusing power-type nonlinearity and an attractive inverse-power potential. We refine the results of Shioji and Watanabe (2016) and apply it to prove the uniqueness and nondegeneracy of ground states for our equations. We also discuss the orbital instability of ground state-standing waves.