NLS ground states on metric graphs with localized nonlinearities

TL;DR

This paper studies the existence of ground states for nonlinear Schrödinger equations on metric graphs with localized nonlinearities, identifying thresholds that determine their existence or nonexistence using variational methods.

Contribution

It introduces thresholds based on the measure of the nonlinear region that predict ground state existence on metric graphs, adapting classical calculus of variations techniques.

Findings

Two measure thresholds determine ground state existence.

Existence or nonexistence of ground states depends on these thresholds.

Classical variational techniques are adapted to metric graphs.

Abstract

We investigate the existence of ground states for the focusing subcritical NLS energy on metric graphs with localized nonlinearities. In particular, we find two thresholds on the measure of the region where the nonlinearity is localized that imply, respectively, existence or nonexistence of ground states. In order to obtain these results we adapt to the context of metric graphs some classical techniques from the Calculus of Variations.