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On the defocusing stationary nonlinear Schrödinger equation on metric graphs

Élio Durand-Simonnet, Damien Galant, Boris Shakarov

Abstract

We study the defocusing nonlinear Schrödinger 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 $δ$-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.

On the defocusing stationary nonlinear Schrödinger equation on metric graphs

Abstract

We study the defocusing nonlinear Schrödinger 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 -subcritical regime. For -type vertex conditions, we provide more precise results: ground states exist for all masses in the -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.
Paper Structure (12 sections, 30 theorems, 163 equations)

This paper contains 12 sections, 30 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric graphs
  4. Hamiltonian operator
  5. Properties of solutions
  6. Energy ground states
  7. General case
  8. Delta type vertex conditions
  9. Bifurcation from the linear ground state
  10. Multiplicity results
  11. Solutions with a prescribed frequency
  12. Normalized solutions with a prescribed mass

Key Result

Theorem 1.2

Let $\mathcal{G}$ be noncompact. Assume that $l_{\operatorname{H}}<0$. Then, there exists $\mu_1 >0$ such that $\tau_\mu$ admits a minimizer for any $\mu \in [0,\mu_1]$. Moreover, if $p\in(1,5)$, then there exists $\mu_2 \geq \mu_1$ such that $\tau_\mu$ does not admit any minimizer for $\mu > \mu_2$

Theorems & Definitions (62)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: Gårding's Inequality
  • ...and 52 more