Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of stationary solutions of supercritical nonlinear Schr{ö}dinger equations on some metric graphs

Nabile Boussaïd, Jack Borthwick

TL;DR

We study stationary solutions to the supercritical nonlinear Schrödinger equation on noncompact metric graphs under Kirchhoff conditions, with mass constraint and p>6. The authors develop a mountain-pass variational framework using an auxiliary family $E_{\rho}$ and leverage a recent abstract theory for bounded Palais–Smale sequences to overcome noncompactness, obtaining positive constrained critical points for almost every $\rho\in[\tfrac{1}{2},1]$ and any mass $\mu>0$. The work clarifies spectral assumptions and a no-zero-positive-state condition to guarantee convergence to a positive limit, delivering an existence result that extends prior compact-graph and finite-edge results to more general noncompact graphs with distributed nonlinearity. This contributes to the understanding of nonlinear waves on quantum graphs, with implications for wave propagation on network-like structures in physics and engineering.

Abstract

We consider the existence of stationary wave solutions with prescribed mass to a supercritical nonlinear Schr{ö}dinger equation on a noncompact connected metric graph without a small mass assumption.

Existence of stationary solutions of supercritical nonlinear Schr{ö}dinger equations on some metric graphs

TL;DR

We study stationary solutions to the supercritical nonlinear Schrödinger equation on noncompact metric graphs under Kirchhoff conditions, with mass constraint and p>6. The authors develop a mountain-pass variational framework using an auxiliary family and leverage a recent abstract theory for bounded Palais–Smale sequences to overcome noncompactness, obtaining positive constrained critical points for almost every and any mass . The work clarifies spectral assumptions and a no-zero-positive-state condition to guarantee convergence to a positive limit, delivering an existence result that extends prior compact-graph and finite-edge results to more general noncompact graphs with distributed nonlinearity. This contributes to the understanding of nonlinear waves on quantum graphs, with implications for wave propagation on network-like structures in physics and engineering.

Abstract

We consider the existence of stationary wave solutions with prescribed mass to a supercritical nonlinear Schr{ö}dinger equation on a noncompact connected metric graph without a small mass assumption.

Paper Structure

This paper contains 19 sections, 9 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Framework
  3. Notations.
  4. Properties of quantum graphs and examples
  5. Spectral properties of the Kirchoff laplacian
  6. Concave functions
  7. Aknowledgements
  8. Some preliminary definitions and results
  9. Mountain pass solutions
  10. Gagliardo-Nirenberg inequalities on graphs.
  11. Mountain pass geometry.
  12. Properties of the essential spectrum.
  13. An abstract compactness lemma.
  14. Main statements
  15. Local positive solutions to stationary NLS
  16. ...and 4 more sections

Key Result

Theorem 4

Let $\mathcal{G}$ be noncompact connected metric graph such that Assumptions Assump:EssSpec and Assump:NoZeroPostiveZerostate hold then for any fixed $\mu>0$, there exists $(u, \lambda)\in H^1_\mu(\mathcal{G})\times \mathbb{R}^+$, $u>0$, which solves Eq:Kirchhoff.

Theorems & Definitions (22)

  • Remark 3
  • Theorem 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 8: Theorem 1 in BCJS-2022
  • Remark 9
  • Remark 10
  • Lemma 11
  • proof
  • ...and 12 more