Existence theory for elliptic equations of general exponential nonlinearity on finite graphs

TL;DR

This paper establishes existence results for semilinear elliptic equations with general exponential nonlinearities on finite graphs, extending classical equations and developing new a priori estimate techniques.

Contribution

It introduces novel methods for a priori estimates and degree computation, enabling existence proofs for a broad class of nonlinear equations on finite graphs.

Findings

Explicit Brouwer degree computation for reduced graphs

Existence of solutions when degree is nonzero

Existence of solutions via sub- and supersolutions when degree vanishes

Abstract

We study semilinear elliptic equations on finite graphs with fully general exponential nonlinearities, thereby extending classical equations such as the Kazdan-Warner and Chern-Simons equations. A key contribution of this work is the development of new techniques for deriving a priori estimates in this generalized setting, which reduce the original finite graph to a graph with only two vertices. This reduction enables us to explicitly compute the Brouwer degree and to establish the existence of solutions when the degree is nonzero. Furthermore, using the method of sub- and supersolutions, we also prove the existence of solutions in cases where the Brouwer degree vanishes.