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

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 $S_{m}$.