Local minimizers in absence of ground states for the critical NLS energy on metric graphs

TL;DR

This paper investigates the existence of local minimizers for the critical nonlinear Schrödinger equation on metric graphs, especially in cases where ground states do not exist, highlighting conditions for stable solutions.

Contribution

It demonstrates the existence of constrained local minimizers in scenarios lacking ground states, extending understanding of stable solutions on metric graphs.

Findings

Local minimizers exist without ground states under certain conditions

Ground state existence depends on mass and graph topology

Stable solutions can be achieved without ground states

Abstract

We consider the mass-critical nonlinear Schr\"odinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in arXiv:1605.07666, where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.