Existence results for some nonlinear elliptic systems on graphs

TL;DR

This paper investigates nonlinear elliptic systems on graphs, establishing new embedding theorems and maximum principles, and proves the existence of solutions using variational methods, extending classical results from continuous spaces to discrete graph structures.

Contribution

It introduces new Sobolev embedding and maximum principle results for elliptic systems on graphs, extending existence theories from Riemann surfaces and Euclidean spaces to discrete graph settings.

Findings

Established a Sobolev embedding theorem on graphs

Proved a new strong maximum principle for elliptic systems on graphs

Confirmed the existence of solutions using variational methods

Abstract

In this paper, several nonlinear elliptic systems are investigated on graphs. One type of the sobolev embedding theorem and a new version of the strong maximum principle are established. Then, by using the variational method, the existence of different types of solutions to some elliptic systems is confirmed. Such problems extend the existence results on closed Riemann surface to graphs and extend the existence results for one single equation on graphs [A. Grigor'yan, Y. Lin, Y. Yang, J. Differential Equations, 2016] to nonlinear elliptic systems on graphs. Such problems can also be viewed as one type of discrete version of the elliptic systems on Euclidean space and Riemannian manifold.