Existence of normalized solutions to nonlinear Schrödinger 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 $I:H^1\left({\mathbb{Z}}^N\right)\to \mathbb{R}$, $$I(u)=\frac{1}{2}{\int }_{{\mathbb{Z}}^N}|\nabla u{|}^{2}d\hat{I}¼-{\int }_{{\mathbb{Z}}^N}F(u)d\hat{I}¼,$$ constrained on $S_m:=\left\{u\in H^1\left({\mathbb{Z}}^N\right)\mid \Vert u{\Vert }_{{\ell }^2\left({\mathbb{Z}}^N\right)}^2=m\right\}$, where $N\ge 2$, $m>0$ is prescribed, $f\in C(\mathbb{R},\mathbb{R})$ satisfying some technical assumptions and $F(t):={\int }_0^{t}f(\ddot{I}„)d\ddot{I}„$. We prove the following minimization problem $$\underset{u\in S_m}{\inf }I(u)$$ has an excitation threshold $m^{\ast }\in [0,+\infty ]$ 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 $m^{\ast }\in (0,+\infty )$ or $m^{\ast }=0$, 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 $m^{\ast }$ also separates the existence and nonexistence of global minimizers for $I(u)$ constrained on $S_m$.

15 pages