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 vetex $p_s$, and $p_1, p_2,\dots, p_N$ are arbitrarily chosen distinct vertices on the graph. We show that there exists a critial 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