Existence and orbital stability of standing waves to a nonlinear Schr\"odinger equation with inverse square potential on the half-line

TL;DR

This paper studies the existence and stability of standing waves in a nonlinear Schrödinger equation with inverse-square potential on the half-line, using variational methods and profile decomposition to handle non-compactness.

Contribution

It proves the existence and orbital stability of ground state standing waves for sub-critical nonlinearities and orbital instability for super-critical cases.

Findings

Existence of standing waves established.

Orbital stability for mass sub-critical nonlinearity.

Orbital instability for mass super-critical nonlinearity.

Abstract

We investigate the properties of standing waves to a nonlinear Schr\"odinger equation with inverse-square potential on the half-line. We first establish the existence of standing waves. Then, by a variational characterization of the ground states, we establish the orbital stability of standing waves for mass sub-critical nonlinearity. Owing to the non-compactness and the absence of translational invariance of the problem, we apply a profile decomposition argument. We obtain convergent minimizing sequences by comparing the problem to the problem at "infinity" (i.e., the equation without inverse square potential). Finally, we establish orbital instability by a blow-up argument for mass super-critical nonlinearity.