# NLS ground states on metric trees: existence results and open questions

## TL;DR

This paper investigates the existence and characteristics of ground states for the nonlinear Schrödinger equation on metric trees, revealing new phenomena like positive energy minimizers and threshold effects, and explores symmetry constraints with novel inequalities.

## Contribution

It introduces new existence results, threshold phenomena, and a Poincaré inequality with remainder specific to NLS on metric trees, advancing understanding beyond traditional graph settings.

## Key findings

- Existence of minimizers with positive energy
- Identification of unexpected threshold phenomena
- Development of a new Poincaré inequality with remainder

## Abstract

We consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new features appear, such as the existence of minimizers with positive energy, and the emergence of unexpected threshold phenomena. We also study the problem with a radial symmetry constraint that is in principle different from the free problem due to the failure of the P\'olya-Szeg\H{o} inequality for radial rearrangements. A key role is played by a new Poincar\'e inequality with remainder.