Topological degree for Chern–Simons Higgs models on finite graphs

Abstract

Let (V, E) be a finite connected graph. We are concerned about the Chern–Simons Higgs model

$$\begin{aligned} \Delta u=\lambda e^u(e^u-1)+f, \end{aligned}$$

(0.1)

where $\Delta$ is the graph Laplacian, $\lambda$ is a real number and f is a function on V. When $\lambda >0$ and $f=4\pi \sum _{i=1}^N\delta _{p_i}$, $N\in {\mathbb {N}}$, $p_1,\cdots ,p_N\in V$, the equation (0.1) was investigated by Huang et al. (Commun Math Phys 377:613–621, 2020) and Hou and Sun (Calc Var 61:139, 2022) via the upper and lower solutions principle. We now consider an arbitrary real number $\lambda$ and a general function f, whose integral mean is denoted by ${\overline{f}}$, and prove that when $\lambda {\overline{f}}<0$, the equation (0.1) has a solution; when $\lambda {\overline{f}}>0$, there exist two critical numbers $\Lambda ^*>0$ and $\Lambda _*<0$ such that if $\lambda \in (\Lambda ^*,+\infty )\cup (-\infty ,\Lambda _*)$, then (0.1) has at least two solutions, including one local minimum solution; if $\lambda \in (0,\Lambda ^*)\cup (\Lambda _*,0)$, then (0.1) has no solution; while if $\lambda =\Lambda ^*$ or $\Lambda _*$, then (0.1) has at least one solution. Our method is calculating the topological degree and using the relation between the degree and the critical group of a related functional. Similar method is also applied to the Chern–Simons Higgs system, and a partial result for the multiple solutions of the system is obtained.