Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

TL;DR

This paper studies the existence and properties of ground states for the 2D focusing nonlinear Schrödinger equation with a point defect, revealing their symmetry, positivity, and singularity characteristics.

Contribution

It establishes the existence of ground states for all positive masses and introduces a novel approach using the Nehari manifold to handle the complex energy space structure.

Findings

Ground states exist for every positive mass.

Ground states exhibit a logarithmic singularity at the defect.

Ground states are positive, radially symmetric, and decreasing radially.

Abstract

We investigate the ground states for the focusing, subcritical nonlinear Schr\"odinger equation with a point defect in dimension two, defined as the minimizers of the energy functional at fixed mass. We prove that ground states exist for every positive mass and show a logarithmic singularity at the defect. Moreover, up to a multiplication by a constant phase, they are positive, radially symmetric, and decreasing along the radial direction. In order to overcome the obstacles arising from the uncommon structure of the energy space, that complicates the application of standard rearrangement theory, we move to the study of the minimizers of the action functional on the Nehari manifold and then establish a connection with the original problem. An ad hoc result on rearrangements is given to prove qualitative features of the ground states.