Existence theorems for a generalized Chern–Simons equation on finite graphs

Consider G = (V, E) as a finite graph, where V and E correspond to the vertices and edges, respectively. We study a generalized Chern–Simons equation $Δu=λeu(ebu−1)+4π∑j=1Nδpj$ on G, where λ and b are positive constants; N is a positive integer; p₁, p₂, …, p_N are distinct vertices of V; and $δpj$ is the Dirac delta mass at p_j. We prove that there exists a critical value λ_c such that the equation has a solution if λ ≥ λ_c and the equation has no solution if λ < λ_c. We also prove that if λ > λ_c, the equation has at least two solutions that include a local minimizer for the corresponding functional and a mountain-pass type solution. Our results extend and complete those of Huang et al. [Commun. Math. Phys. 377(1), 613–621 (2020)] and Hou and Sun [Calculus Var. Partial Differ. Equations 61(4), 139 (2022)].