Let G = (V,E) be a connected finite graph and Δ be the usual graph Laplacian. In this paper, the authors consider a generalized self-dual Chern-Simons equation on the graph G
(0.1) \(\Delta u = - \lambda {{\rm{e}}^{F( u )}}{[ {{{\rm{e}}^{F( u )}} - 1} ]^2} + 4\pi \sum\limits_{j = 1}^M {{\delta _{{p_j}}}},\)
where
\(F(u) =\begin{cases}{\tilde F\left( u \right)}, & u \le 0,\\0, & u > 0,\end{cases}\)
\(\tilde F(u)\) satisfies \(u=1+\tilde F(u)-\rm{{e}}^{\tilde F(u)}\) , λ > 0, M is any fixed positive integer, \(\delta_{p_{j}}\) is the Dirac delta mass at the vertex pj, and p1, p2, …, pM are arbitrarily chosen distinct vertices on the graph. They first prove that there is a critical value λc such that if λ ≥ λc, then the generalized self-dual Chern-Simons equation has a solution uλ. Applying the existence result, they develop a new method to construct a solution of (0.1) which is monotonic with respect to λ when λ ≥ λc. Then they establish that there exist at least two solutions of the equation for λ > λc via the variational method. Furthermore, they give a fine estimate of the monotone solution, which can be applied to other related problems.
