A Relativistic Abelian Chern-Simons Model on Graph

In this paper, we consider a relativistic Abelian Chern-Simons equation { Delta u=lambda(a(b-a)e(u)-b(b-a)e(v)+a(2)e(2u)-abe2v+b(b-a)e(u+v))+4 pi(N1)& sum;(j=1)delta(pj)Delta v=lambda(-b(b-a)e(u)+a(b-a)e(v)-abe(2u)+a(2)e(2v)+b(b-a)e(u+v))+4 pi(N2)& sum;(j=1)delta(qj), on a connected finite graph G=(V,E), where lambda>0 is a constant;a>b>0;N1andN2are positive integers;p1,p2,...,pN1andq1,q2,...,qN2denote distinctvertices ofV. Additionally,delta pjand delta qjrepresent the Dirac delta masses located atverticespjandqj. By employing the method of constrained minimization, we prove that there exists a critical value lambda 0, such that the above equation admits a solutionwhen lambda >=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.