Multiple positive bound states for the subcritical NLS equation on metric graphs

TL;DR

This paper proves the existence of positive bound states for the subcritical nonlinear Schrödinger equation on metric graphs, showing that for large enough mass, each finite edge hosts a stable bound state.

Contribution

It establishes the existence and stability of positive bound states on metric graphs for the subcritical NLS, with a novel variational characterization involving additional constraints.

Findings

Positive bound states exist on each finite edge beyond a mass threshold.

These bound states are orbitally stable due to their minimization properties.

The results apply to noncompact metric graphs with subcritical focusing nonlinearity.

Abstract

We consider the Schroedinger equation with a subcritical focusing power nonlinearity on a noncompact metric graph, and prove that for every finite edge there exists a threshold value of the mass, beyond which there exists a positive bound state achieving its maximum on that edge only. This bound state is characterized as a minimizer of the energy functional associated to the NLS equation, with an additional constraint (besides the mass prescription): this requires particular care in proving that the minimizer satisfies the Euler--Lagrange equation. As a consequence, for a sufficiently large mass every finite edge of the graph hosts at least one positive bound state that, owing to its minimality property, is orbitally stable.