Journal of the
Korean Mathematical Society JKMS

Let $G=(V,E)$ be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on $G$ \begin{equation*} \Delta u=\lambda \mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 \pi \sum_{s=1}^{N} \delta_{p_{s}}, \end{equation*} where $\lambda>0$, $\delta_{p_{s}}$ is the Dirac mass at the vertex $p_s$, and $p_1, p_2,\dots, p_N$ are arbitrarily chosen distinct vertices on the graph. We show that there exists a critical value $\hat{\lambda}$ such that when $\lambda > \hat{\lambda}$, the generalized Chern-Simons equation has at least two solutions, when $\lambda = \hat{\lambda}$, the generalized Chern-Simons equation has a solution, and when $\lambda < \hat\lambda$, the generalized Chern-Simons equation has no solution.

Keywords: Chern-Simons equation, finite graph, existence, uniqueness, variational method

MSC numbers: 58E30, 35J91, 05C22

Supported by: This work is financially supported by the Natural Science Foundation of Henan Province (Grant No. 222300 420416), the China Postdoctoral Science Foundation (Grant No. 2022M711045), and the National Natural Science Foundation of China (Grant No. 12201184).