Standing waves for defocusing nonlinear Schr\"odinger equations with point interaction

TL;DR

This paper studies standing wave solutions of the defocusing nonlinear Schrödinger equation with point interaction in 2D and 3D, establishing existence, uniqueness, symmetry, positivity, stability, and decay properties.

Contribution

It introduces a framework for analyzing standing waves with point interactions, proving their existence, uniqueness, and qualitative features in the defocusing case.

Findings

Existence and uniqueness of standing waves with point interaction

Radial symmetry, positivity, and stability of solutions

Sharp decay estimates in the zero-mass case

Abstract

We consider standing waves of the nonlinear Schr\"odinger equation $i\partial_t u = -\Delta_\alpha u + |u|^{p-1}u$ in the defocusing case in dimensions $N=2$ and $N=3$. Here, $-\Delta_\alpha$ denotes the Laplacian with a point interaction. This operator is bounded from below by a negative constant; consequently, unlike in the free case, the associated energy functional admits non-trivial minimizers. We establish existence and uniqueness of standing waves, and prove further qualitative properties, including radial symmetry, positivity, and stability. Moreover, we build an appropriate functional space for the zero-mass case and establish sharp decay estimates in this case.