Existence of normalized solutions to nonlinear Schr\"odinger equations on lattice graphs

TL;DR

This paper investigates the existence of normalized solutions to nonlinear Schrödinger equations on lattice graphs, establishing an excitation threshold that determines when global minimizers exist based on the mass parameter.

Contribution

It introduces a discrete Schwarz rearrangement method on lattice graphs and classifies the problem into subcritical, critical, and supercritical cases based on the excitation threshold.

Findings

Existence of a critical mass threshold $m^*$ for minimizers.

Classification into $L^2$-subcritical, critical, and supercritical cases.

Threshold $m^*$ separates existence and nonexistence of minimizers.

Abstract

In this paper, using a discrete Schwarz rearrangement on lattice graphs developed in \cite{DSR}, we study the existence of global minimizers for the following functional $I:H^1\left(\mathbb{Z}^N\right)\to \R$, $$I(u)=\frac{1}{2} \int_{\mathbb{Z}^N}|\nabla u|^2 \,d\mu-\int_{\mathbb{Z}^N} F(u)\, d\mu,$$ constrained on $S_m:=\left\{u \in H^1\left(\mathbb{Z}^N\right) \mid\|u\|_{\ell^2\left(\mathbb{Z}^N\right)}^2=m\right\}$, where $N \geq 2$, $m>0$ is prescribed, $f \in C(\mathbb{R}, \mathbb{R})$ satisfying some technical assumptions and $F(t):=\int_0^t f(\tau) \,d\tau$. We prove the following minimization problem $$ \inf_{u \in S_m} I(u) $$ has an excitation threshold $m^*\in [0,+\infty]$ such that \begin{equation*} \inf_{u \in S_m} I(u)<0 \quad \text{if and only if } m>m^*. \end{equation*} Based primarily on $m^* \in (0,+\infty)$ or $m^*=0$, we classify the problem into three different cases: $L^2$-subcritical, $L^2$-critical and $L^2$-supercritical. Moreover, for all three cases, under assumptions that we believe to be nearly optimal, we show that $m^*$ also separates the existence and nonexistence of global minimizers for $I(u)$ constrained on $S_{m}$.