A Relativistic Abelian Chern–Simons Model on Graph

Abstract

In this paper, we consider a relativistic Abelian Chern–Simons equation

$$\begin{aligned} \left\{ \begin{array}{l} \Delta u=\lambda \left( a(b-a)e^{u}-b(b-a)e^{v}+a^{2}e^{2u}-abe^{2v}+b(b-a)e^{u+v}\right) +4\pi \sum \limits _{j=1}^{N_{1}} \delta _{p_{j}},\\ \Delta v=\lambda \left( -b(b-a)e^{u}+a(b-a)e^{v}-abe^{2u} +a^{2}e^{2v}+b(b-a)e^{u+v}\right) +4\pi \sum \limits _{j=1}^{N_{2}} \delta _{q_{j}}, \end{array} \right. \end{aligned}$$

on a connected finite graph \(G=(V, E)\), where \(\lambda >0\) is a constant; \(a>b>0\); \(N_{1}\) and \(N_{2}\) are positive integers; \(p_{1}, p_{2}, \ldots , p_{N_{1}}\) and \(q_{1}, q_{2}, \ldots , q_{N_{2}}\) denote distinct vertices of V. Additionally, \(\delta _{p_{j}}\) and \(\delta _{q_{j}}\) represent the Dirac delta masses located at vertices \(p_{j}\) and \(q_{j}\). By employing the method of constrained minimization, we prove that there exists a critical value \(\lambda _{0}\), such that the above equation admits a solution when \(\lambda \ge \lambda _{0}\). Furthermore, we employ the mountain pass theorem developed by Ambrosetti–Rabinowitz to establish that the equation has at least two solutions when \(\lambda >\lambda _{0}\).