Topological degree for Chern–Simons Higgs models on finite graphs

Let (V, E) be a finite connected graph. We are concerned about the Chern–Simons Higgs model 0.1 \(\begin{aligned} \Delta u=\lambda e^u(e^u-1)+f, \end{aligned}\) $$\begin{array}{c}\hfill \mathrm{\Delta }u=\lambda e^u(e^u-1)+f,\end{array}$$ where \(\Delta \) $\mathrm{\Delta }$ is the graph Laplacian, \(\lambda \) $\lambda$ is a real number and f is a function on V. When \(\lambda >0\) $\lambda >0$ and \(f=4\pi \sum _{i=1}^N\delta _{p_i}\) $f=4\pi {\sum }_{i=1}^N{\delta }_{p_i}$ , \(N\in {\mathbb {N}}\) $N\in \mathbb{N}$ , \(p_1,\cdots ,p_N\in V\) $p_1,\cdots ,p_N\in V$ , 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 \(\lambda \) $\lambda$ and a general function f, whose integral mean is denoted by \({\overline{f}}\) $\overline{f}$ , and prove that when \(\lambda {\overline{f}}<0\) $\lambda \overline{f}<0$ , the equation (0.1) has a solution; when \(\lambda {\overline{f}}>0\) $\lambda \overline{f}>0$ , there exist two critical numbers \(\Lambda ^*>0\) ${\mathrm{\Lambda }}^{\ast }>0$ and \(\Lambda _*<0\) ${\mathrm{\Lambda }}_{\ast }<0$ such that if \(\lambda \in (\Lambda ^*,+\infty )\cup (-\infty ,\Lambda _*)\) $\lambda \in ({\mathrm{\Lambda }}^{\ast },+\infty )\cup (-\infty ,{\mathrm{\Lambda }}_{\ast })$ , then (0.1) has at least two solutions, including one local minimum solution; if \(\lambda \in (0,\Lambda ^*)\cup (\Lambda _*,0)\) $\lambda \in (0,{\mathrm{\Lambda }}^{\ast })\cup ({\mathrm{\Lambda }}_{\ast },0)$ , then (0.1) has no solution; while if \(\lambda =\Lambda ^*\) $\lambda ={\mathrm{\Lambda }}^{\ast }$ or \(\Lambda _*\) ${\mathrm{\Lambda }}_{\ast }$ , 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.