Chern-Simons-Higgs type equations on canonically compactifiable graphs

Abstract

In this paper, we prove existence results of solutions to three kinds of Chern-Simons-Higgs type equations, including mean field equations and Chern-Simons-Higgs equations as well as the generalized Chern-Simons-Higgs equations on canonically compactifiable graphs, which is a special infinite graphs giving inclusive relationship between Banach spaces on graphs. The paper mainly employs variational principles in Banach spaces as well as upper and lower solutions method, with the main challenge being the lack of finite bound of number of vertices and other certain properties, leading to difficulties of estimates of bound for functionals. We choose suitable restrict spaces in Lagrange multiplier theorem and use Moser-Trudinger inequalities to overcome these difficulties.