On the defocusing stationary nonlinear Schr\"odinger equation on metric graphs

TL;DR

This paper investigates the existence, stability, and bifurcation of ground states for the defocusing nonlinear Schrödinger equation on metric graphs, revealing how these properties depend on mass, vertex conditions, and graph structure.

Contribution

It provides new existence and stability results for ground states on metric graphs, especially under $ ext{delta}$-type conditions, and analyzes bifurcation and multiplicity phenomena.

Findings

Ground states exist for small masses in the subcritical regime.

Existence of ground states for all masses in critical and supercritical cases with $ ext{delta}$-type conditions.

Bifurcation of ground states from the zero solution at the spectrum's bottom.

Abstract

We study the defocusing nonlinear Schr\"odinger equation on noncompact metric graphs under general self-adjoint vertex conditions ensuring the existence of a negative eigenvalue of the Hamiltonian operator. First, we focus on the existence of energy ground states with prescribed mass. We show that existence and stability always hold for small masses and fail for large masses in the $L^2$-subcritical regime. For $\delta$-type vertex conditions, we provide more precise results: ground states exist for all masses in the $L^2$-critical and supercritical cases, while in the subcritical case, for one vertex graphs, there exists a sharp mass threshold such that ground states exist below it and do not exist above it. Moreover, we show that the ground state bifurcates from the vanishing solution at the bottom of the Hamiltonian spectrum. Finally, we present multiplicity results for stationary solutions, both in the fixed-frequency and fixed-mass settings.