The existence of topological solutions to theChern–Simons model on lattice graphs

We prove the existence of topological solutions to the self-dual Chern-Simons model and the abelian Higgs system on the lattice graphs n for n 2. This extends results of Huang, Lin and Yau (2020) from finite graphs to lattice graphs.M j=1 n j p j(2)with positive integers n 1 , . . ., n M and distinct vortices p 1 , . . ., p M 2 .Here > 0, and p j is the Dirac mass at p j .A solution of (1) or ( 2) is called topological if u(x) 0 as |x| +, and called nontopological if u(x) - as |x| +.For the abelian Higgs system (2), Jaffe and Taubes [1980] proved the existence and uniqueness of general finite energy multivortex solutions to the Bogomol'nyi equations, and there have been many studies on this model since then, such as [Jacobs and Rebbi 1979;Jaffe and Taubes 1980;Wang and Yang 1992].The self-dual Chern-Simons system (1) is the minimal self-dual model containing the Chern-Simons term.The Chern-Simons vortices were discovered in [Jackiw and Weinberg 1990;Hong et al. 1990], which attracted people to investigate the existence problem.The existence of topological solutions in 2 was established in [Wang 1991;Spruck and Yang 1995] by the variational method and iteration argument, and the existence of self-dual doubly periodic vortex solutions was proved in [Caffarelli and Yang 1995].