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

Bobo Hua, Linlin Sun, Jiaxuan Wang

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.

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

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.

Paper Structure

This paper contains 15 sections, 26 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The setting of graphs and some lemmas
  4. Topological degree
  5. A Priori Estimates
  6. Topological degree
  7. Existence results
  8. Simple cases
  9. Some cases for $\mathbf{c\neq0}$
  10. $\mathbf{c=0}$
  11. The complete discussion for existence results
  12. General exponential nonlinearity
  13. A Priori estimates
  14. Topological degree
  15. Existence results

Key Result

Theorem 1.1

Let $G=(V,E,\omega,m)$ be a finite connected weighted graph. For $m\in\mathbb{N},$ let $u_m$ be a solution to the following equation on $G$ and Then there exists a subsequence $\{u_{m_k}\}$, which satisfies the following alternative:

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1: Elliptic estimate
  • Lemma 2.2: the method of sub- and supersolutions
  • Lemma 2.3
  • proof
  • ...and 47 more