Negative energy ground states for the $L^2$-critical NLSE on metric graphs

TL;DR

This paper studies the existence of negative energy ground states with prescribed mass for the $L^2$-critical nonlinear Schrödinger equation on noncompact metric graphs, revealing new phenomena compared to the real line case.

Contribution

It introduces novel techniques for analyzing variational problems on metric graphs, demonstrating the existence of negative energy ground states for certain graph classes.

Findings

Existence of negative energy ground states on specific graphs.

Comparison with the real line case showing new phenomena.

Development of new analytical methods for variational problems.

Abstract

We investigate the existence of ground states with prescribed mass for the focusing nonlinear Schr\"odinger equation with $L^2$-critical power nonlinearity on noncompact quantum graphs. We prove that, unlike the case of the real line, for certain classes of graphs there exist ground states with negative energy for a whole interval of masses. A key role is played by a thorough analysis of Gagliardo-Nirenberg inequalities and on estimates of the optimal constants. Most of the techniques are new and suited to the investigation of variational problems on metric graphs.