Uniqueness and nondegeneracy of ground states for 2d-nonlinear scalar field equations with point interaction

TL;DR

This paper proves the uniqueness and nondegeneracy of ground states for 2D nonlinear scalar field equations with point interaction, using Pohozaev identities and variational methods, providing key insights into their structure.

Contribution

It establishes the uniqueness of positive radial ground states and their nondegeneracy in the energy space for 2D nonlinear scalar field equations with point interaction.

Findings

Unique positive radial ground states are proven.

Ground states are nondegenerate critical points.

Method combines Pohozaev identities and variational techniques.

Abstract

We study uniqueness and nondegeneracy of ground states for nonlinear scalar field equations in two dimensions with a point interaction at the origin. It is known that the all ground states are radial, positive, and decreasing functions. In this paper we prove the uniqueness of positive radial solutions by a method of Poho\v{z}aev identities. As a corollary, we obtain the uniqueness of ground states. Moreover, by a variational and ODE technique, we show that the ground state is a nondegenerate critical point of the action in the energy space.