Harnack's inequality and Green functions on locally finite graphs

TL;DR

This paper investigates gradient estimates, Harnack's inequality, and Green functions for Schrödinger equations on locally finite graphs, revealing key properties and extending classical analysis to graph structures.

Contribution

It introduces new gradient estimates and Harnack inequalities for Schrödinger equations on graphs, and establishes properties of Green functions in this setting.

Findings

Derived gradient estimates for positive solutions.

Established Harnack's inequality on graphs.

Analyzed properties of Green functions on graphs.

Abstract

In this paper we study the gradient estimate for positive solutions of Schrodinger equations on locally finite graph. Then we derive Harnack's inequality for positive solutions of the Schrodinger equations. We also set up some results about Green functions of the Laplacian equation on locally finite graph. Interesting properties of Schrodinger equation are derived.