Existence of normalized solutions to nonlinear Schr\"{o}dinger equations with potential on lattice graphs

TL;DR

This paper investigates the existence of normalized ground state solutions for nonlinear Schrödinger equations on lattice graphs, identifying thresholds based on the potential and nonlinearity conditions.

Contribution

It establishes the existence and non-existence thresholds for normalized solutions depending on the potential type and nonlinearity growth near zero.

Findings

Existence of a threshold  for solution existence.

No solutions for small mass below .

Solutions exist for large mass above .

Abstract

We study the existence of ground state normalized solution of the following Schr\"{o}dinger equation: \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda u=f(x,u), & x\in\mathbb{Z}^d \\ \Vert u\Vert_2^2=a \end{cases} \end{equation*} where $V(x)$ is trapping potential or well potential, $f(x,u)$ satisfies Berestycki-Lions type condition and other suitable conditions. We show that there always exists a threshold $\alpha\in[0,\infty)$ such that there do not exist ground state normalized solutions for $a\in (0,\alpha)$, and there exists a ground state normalized solution for $a\in(\alpha,\infty)$. Furthermore, we prove sufficient conditions for the positivity of $\alpha$ that $\alpha=0$ if $f(x,u)$ is mass-subcritical near 0, and $\alpha>0$ if $f(x,u)$ is mass-critical or mass-supercritical near 0.