Existence of solutions to a generalized self-dual Chern-Simons equation on graphs

Let $ G=(V,E) $ be a connected finite graph and $ \Delta $ the usual graph Laplacian. In this paper, we consider a generalized self-dual Chern-Simons equation on the graph $G$ \begin{eqnarray}\label{one1} \Delta{u}=-\lambda{e^{F(u)}[e^{F(u)}-1]^2}+4\pi\sum_{i=1}^{M}{\delta_{p_{j}}}, \end{eqnarray} where \begin{equation} F(u)=\left\{\begin{array}{l} \widetilde{F}(u), \ \quad u\leq0, 0, \quad \quad \quad u>0, \end{array} \right. \end{equation} $ \widetilde{F}(u) $ satisfies $ u=1+{\widetilde {F}(u)}-e^{\widetilde {F}(u)} $, $ \lambda>0 $, $M$ is any fixed positive integer, $ \delta_{p_{j}} $ is the Dirac delta mass at the vertex $p_j$, and $p_1$, $p_2$, $\cdots$, $p_M$ are arbitrarily chosen distinct vertices on the graph. We first prove that there is a critical value ${\lambda}_c$ such that if $\lambda \geq{\lambda}_c$, then the generalized self-dual Chern-Simons equation has a solution $u_{\lambda}$. Applying the existence result, we develop a new method to construct a solution of the equation which is monotonic with respect to $\lambda$ when $\lambda \geq{\lambda}_c$. Then we establish that there exist at least two solutions of the equation via the variational method for $\lambda>{\lambda}_c$. Furthermore, we give a fine estimate of the monotone solution which can be applied to other related problems.