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

Consider a finite connected graph denoted as $G=(V,E)$. This study explores a generalized Chern-Simons Higgs model, characterized by the equation: $$\mathrm{\Delta }u=\lambda e^u(e^u-1{)}^{2p+1}+f,$$ where $\mathrm{\Delta }$ denotes the graph Laplacian, $\lambda$ 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).