A Relativistic Abelian Chern–Simons Model on Graph

In this paper, we consider a relativistic Abelian Chern–Simons equation {Δu = λ (a(b − a)eu − b(b − a)ev + a2e2u − abe2v + b(b − a)eu+v + 4π N1 j=1 ∑ δp j , Δv = λ (−b(b − a)eu + a(b − a)ev − abe2u + a2e2v + b(b − a)eu+v + 4π N2 j=1 ∑ δq j , on a connected finite graph G = (V, E), where λ 0 is a constant; a b 0; N1 and N2 are positive integers; p1, p2,..., pN1 and q1, q2,..., qN2 denote distinct vertices of V. Additionally, δp j and δq 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 λ0, such that the above equation admits a solution when λ ≥ λ0. Furthermore, we employ the mountain pass theorem developed by Ambrosetti–Rabinowitz to establish that the equation has at least two solutions when λ λ0.