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Doubly nonlinear Schrödinger normalized ground states on 2D grids: existence results and singular limits

Daniele Barbera, Filippo Boni, Simone Dovetta, Lorenzo Tentarelli

TL;DR

This work establishes existence and asymptotic convergence results for normalized ground states of a doubly nonlinear Schrödinger energy on 2D grids with both standard and concentrated nonlinearities. By developing an extension framework from grid to plane and leveraging grid-Gagliardo–Nirenberg inequalities, it proves ground-state existence in finite and periodic vertex configurations and derives sharp singular limits as the grid spacing vanishes. In the $\mathbb{Z}^2$-periodic case, the grid problem converges to the classical 2D plane problem with two bulk nonlinearities, while in the $\mathbb{Z}$-periodic case the limit reduces to line- or strip-concentrated nonlinearities, including singular delta-type conditions along a line. The results provide a rigorous bridge between nonlinear Schrödinger equations on metric graphs and their high-dimensional Euclidean limits, offering precise energy convergence and strong $H^1$-type convergence for ground states.

Abstract

We investigate the existence and the singular limit of normalized ground states for focusing doubly nonlinear Schrödinger equations with both standard and concentrated nonlinearities on two-dimensional square grids. First, we provide existence and non-existence results for such ground states depending on the values of the nonlinearity powers and on the structure of the set of vertices where the concentrated nonlinearities are located. Second, we prove that suitable piecewise-affine extensions of such states converge strongly in $H^1(\R^2)$ to ground states of corresponding doubly nonlinear models defined on the whole plane as the length of the edges in the grid tends to zero. This convergence is proved both for limit models with standard nonlinearities only and for models combining standard and singular nonlinearities concentrated on a line or on a strip.

Doubly nonlinear Schrödinger normalized ground states on 2D grids: existence results and singular limits

TL;DR

This work establishes existence and asymptotic convergence results for normalized ground states of a doubly nonlinear Schrödinger energy on 2D grids with both standard and concentrated nonlinearities. By developing an extension framework from grid to plane and leveraging grid-Gagliardo–Nirenberg inequalities, it proves ground-state existence in finite and periodic vertex configurations and derives sharp singular limits as the grid spacing vanishes. In the -periodic case, the grid problem converges to the classical 2D plane problem with two bulk nonlinearities, while in the -periodic case the limit reduces to line- or strip-concentrated nonlinearities, including singular delta-type conditions along a line. The results provide a rigorous bridge between nonlinear Schrödinger equations on metric graphs and their high-dimensional Euclidean limits, offering precise energy convergence and strong -type convergence for ground states.

Abstract

We investigate the existence and the singular limit of normalized ground states for focusing doubly nonlinear Schrödinger equations with both standard and concentrated nonlinearities on two-dimensional square grids. First, we provide existence and non-existence results for such ground states depending on the values of the nonlinearity powers and on the structure of the set of vertices where the concentrated nonlinearities are located. Second, we prove that suitable piecewise-affine extensions of such states converge strongly in to ground states of corresponding doubly nonlinear models defined on the whole plane as the length of the edges in the grid tends to zero. This convergence is proved both for limit models with standard nonlinearities only and for models combining standard and singular nonlinearities concentrated on a line or on a strip.

Paper Structure

This paper contains 8 sections, 19 theorems, 183 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of ground states on grids: proof of Theorems \ref{['thm:GScomp']}--\ref{['thm:GSZ']}--\ref{['thm:GSZ2']}
  4. Ground states in $\mathbb{R}^2$ with standard and concentrated nonlinearities
  5. Singular limit with $\mathbb{Z}^2$-periodic point nonlinearities: proof of Theorem \ref{['thm:limZ2']}
  6. Singular limit with $\mathbb{Z}$-periodic nonlinearities: proof of Theorems \ref{['thm:limZ_theta']}--\ref{['thm:limZ_S']}
  7. The limit problem $\mathcal{E}_{\mathbb{R}^2,\theta}$: proof of Theorem \ref{['thm:limZ_theta']}
  8. The limit problem $\mathcal{E}_{\mathbb{R}^2,\theta,R}$: proof of Theorem \ref{['thm:limZ_S']}

Key Result

Theorem 1.3

Let $p\in(2,6)$, $q\in(2,4)$, $\alpha,\beta>0$ and $V\subset\mathbb{V}_{\mathcal{G}_\varepsilon}$ be such that $\# V<+\infty$. There results that:

Figures (2)

  • Figure 1: The grid $\mathcal{G}_\varepsilon$.
  • Figure 2: Examples of a $\mathbb{Z}$-periodic (A) and a $\mathbb{Z}^2$-periodic (B) subset $V$ of vertices in a two-dimensional grid (the vertices in $V$ are denoted by red crosses).

Theorems & Definitions (42)

  • Definition 1.1: $\mathbb{Z}$-periodic set $V$
  • Definition 1.2: $\mathbb{Z}^2$-periodic set $V$
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 2.1
  • Lemma 2.2
  • ...and 32 more