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Ground states for the NLS equation with combined nonlinearity on periodic metric graphs

Nicola Soave, Lorenzo Villata

TL;DR

This work analyzes the existence of ground states for the nonlinear Schrödinger energy with combined nonlinearities on noncompact $1$- and $2$-periodic metric graphs under a fixed mass. It blends variational methods with sharp Gagliardo–Nirenberg inequalities on graphs to characterize the impact of periodic geometry and dimensional crossover on existence and energy levels, distinguishing focusing and defocusing regimes. The authors establish mass-threshold phenomena, derive critical masses $\mu_{p,\mathcal{G}}$ and $\mu_{6,\mathcal{G}}$, and reveal how the interplay between $p$, $q$, and the perturbation parameter $\alpha$ yields rich bifurcation-like behavior, including regime diagrams and explicit thresholds that depend on the underlying graph topology. By extending homogeneous $2$-periodic results to general graphs and refining inhomogeneous results on finite noncompact graphs, the paper advances the understanding of NLS on periodic networks with nonlinearities in tandem and provides a framework for analyzing dimensional crossover effects in graph-based models.

Abstract

We investigate the existence of ground states with prescribed mass for the Non-Linear Schrödinger energy with combined nonlinearities on $1$ and $2$-periodic metric graphs. This is the natural prosecution of previous studies concerning on the one hand the homogeneous NLS equation on periodic graphs, and on the other hand the NLS equation with combined nonlinearity on noncompact metric graphs with finitely many vertexes and edges. As in the latter case, it turns out that the interplay between different nonlinearities creates new phenomena with respect to the homogenous setting, but, due to the periodicity, in a quite different way; in particular, for $2$-periodic graphs, the so called dimensional crossover occurs. As a by-product, we extend existing results for the homogeneous NLS on the square and honeycomb grids to general $2$-periodic graphs. Furthermore, we also improve previous results obtained for the inhomogeneous NLS on noncompact graphs with finitely many vertexes and edges.

Ground states for the NLS equation with combined nonlinearity on periodic metric graphs

TL;DR

This work analyzes the existence of ground states for the nonlinear Schrödinger energy with combined nonlinearities on noncompact - and -periodic metric graphs under a fixed mass. It blends variational methods with sharp Gagliardo–Nirenberg inequalities on graphs to characterize the impact of periodic geometry and dimensional crossover on existence and energy levels, distinguishing focusing and defocusing regimes. The authors establish mass-threshold phenomena, derive critical masses and , and reveal how the interplay between , , and the perturbation parameter yields rich bifurcation-like behavior, including regime diagrams and explicit thresholds that depend on the underlying graph topology. By extending homogeneous -periodic results to general graphs and refining inhomogeneous results on finite noncompact graphs, the paper advances the understanding of NLS on periodic networks with nonlinearities in tandem and provides a framework for analyzing dimensional crossover effects in graph-based models.

Abstract

We investigate the existence of ground states with prescribed mass for the Non-Linear Schrödinger energy with combined nonlinearities on and -periodic metric graphs. This is the natural prosecution of previous studies concerning on the one hand the homogeneous NLS equation on periodic graphs, and on the other hand the NLS equation with combined nonlinearity on noncompact metric graphs with finitely many vertexes and edges. As in the latter case, it turns out that the interplay between different nonlinearities creates new phenomena with respect to the homogenous setting, but, due to the periodicity, in a quite different way; in particular, for -periodic graphs, the so called dimensional crossover occurs. As a by-product, we extend existing results for the homogeneous NLS on the square and honeycomb grids to general -periodic graphs. Furthermore, we also improve previous results obtained for the inhomogeneous NLS on noncompact graphs with finitely many vertexes and edges.
Paper Structure (10 sections, 27 theorems, 115 equations, 1 figure)

This paper contains 10 sections, 27 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gagliardo-Nirenberg inequality and critical mass
  4. The homogeneous problem on periodic graphs
  5. $1$-periodic case
  6. $2$-periodic case
  7. The focusing case
  8. On 1-periodic graphs
  9. On 2-periodic graphs
  10. The defocusing case

Key Result

Proposition 1.1

Let $\mathcal{G}$ be a $1$ or $2$-periodic graph, and $\alpha>0$. If either $p<6$ and $\mu>0$, or $p=6$ and $\mu \in (0, \tilde{\mu}_{\mathcal{G}})$, then $E_{p,q,\alpha}(\cdot\,,\mathcal{G})$ is bounded from below and coercive in $H^1_{\mu}(\mathcal{G})$. Otherwise, if $p=6$ and $\mu\geq \tilde{\mu

Figures (1)

  • Figure 1: On the left, the fundamental domain $K$, and on the right the new fundamental domain $\tilde{K}$.

Theorems & Definitions (52)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 42 more