Solutions to discrete fractional Sch\"{o}dinger equations

TL;DR

This paper investigates the existence and multiplicity of solutions for a discrete fractional Schrödinger equation using variational methods, highlighting differences from the continuous case due to the nonlocal operator.

Contribution

It introduces a novel approach to analyze discrete fractional Schrödinger equations with a nonlocal operator defined via discrete Fourier transform.

Findings

Proved existence of solutions under certain conditions.

Established multiplicity results for solutions.

Differentiated the discrete case from the continuous fractional Schrödinger equation.

Abstract

In this paper, we study the discrete fractional Schr\"{o}dinger equation $$ (-\Delta)^\alpha u+h(x) u=f(x,u),\quad x\in \mathbb{Z}^d,$$ where $d\in\mathbb{N}^*,\,\alpha \in(0, 1)$ and the nonlocal operator $(-\Delta)^\alpha $ is defined by discrete Fourier transform, which differs from the continuous case. Under suitable assumptions on $h$ and $f$, we prove the existence and multiplicity of solutions to this equation by variational method.