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Solutions to a generalized Chern-Simons Higgs model on finite graphs by topological degree

Songbo Hou, Wenjie Qiao

Abstract

Consider a finite connected graph denoted as $G=(V, E)$. This study explores a generalized Chern-Simons Higgs model, characterized by the equation: $$ Δu = λe^u (e^u - 1)^{2p+1} + f,$$ where $Δ$ denotes the graph Laplacian, $λ$ is a real number, $p$ is a non-negative integer, and $f$ is a function on $V$. Through the computation of the topological degree, this paper demonstrates the existence of a single solution for the model. Further analysis of the interplay between the topological degree and the critical group of an associated functional reveals the presence of multiple solutions. These findings extend the work of Li, Sun, Yang (arXiv:2309.12024) and Chao, Hou (J. Math. Anal. Appl. (2023) 126787).

Solutions to a generalized Chern-Simons Higgs model on finite graphs by topological degree

Abstract

Consider a finite connected graph denoted as . This study explores a generalized Chern-Simons Higgs model, characterized by the equation: where denotes the graph Laplacian, is a real number, is a non-negative integer, and is a function on . Through the computation of the topological degree, this paper demonstrates the existence of a single solution for the model. Further analysis of the interplay between the topological degree and the critical group of an associated functional reveals the presence of multiple solutions. These findings extend the work of Li, Sun, Yang (arXiv:2309.12024) and Chao, Hou (J. Math. Anal. Appl. (2023) 126787).
Paper Structure (4 sections, 6 theorems, 84 equations)

This paper contains 4 sections, 6 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th1']}
  3. Proof of Theorem \ref{['th2']}
  4. Proof of Theorem \ref{['th3']}

Key Result

Theorem 1.1

Let $(V, E)$ represent a connected finite graph with symmetric weights, i.e., $w_{xy}=w_{yx}$ for all $xy \in E$. Suppose $\sigma \in [0,1]$, $\lambda$, and $f$ satisfy for some real number $\Lambda>0$. If $u$ is a solution of then there exists a constant $C$, depending only on $\Lambda$ and the graph $V$, such that $|u(x)| \leq C$ for all $x \in V$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof