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

Abstract

Let $ G=(V,E) $ be a connected finite graph and $ Δ$ the usual graph Laplacian. In this paper, we consider a generalized self-dual Chern-Simons equation on the graph $G$ \begin{eqnarray}\label{one1} Δ{u}=-λ{e^{F(u)}[e^{F(u)}-1]^2}+4π\sum_{i=1}^{M}{δ_{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)} $, $ λ>0 $, $M$ is any fixed positive integer, $ δ_{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 $λ_c$ such that if $λ\geqλ_c$, then the generalized self-dual Chern-Simons equation has a solution $u_λ$. Applying the existence result, we develop a new method to construct a solution of the equation which is monotonic with respect to $λ$ when $λ\geqλ_c$. Then we establish that there exist at least two solutions of the equation via the variational method for $λ>λ_c$. Furthermore, we give a fine estimate of the monotone solution which can be applied to other related problems.