Topological degree for Chern–Simons Higgs models on finite graphs.

Let (V, E) be a finite connected graph. We are concerned about the Chern–Simons Higgs model 0.1 Δ u = λ e u (e u - 1) + f , where Δ is the graph Laplacian, λ is a real number and f is a function on V. When λ > 0 and f = 4 π ∑ i = 1 N δ p i , N ∈ N , p 1 , ⋯ , p N ∈ V , the equation (0.1) was investigated by Huang et al. (Commun Math Phys 377:613–621, 2020) and Hou and Sun (Calc Var 61:139, 2022) via the upper and lower solutions principle. We now consider an arbitrary real number λ and a general function f, whose integral mean is denoted by f ¯ , and prove that when λ f ¯ < 0 , the equation (0.1) has a solution; when λ f ¯ > 0 , there exist two critical numbers Λ ∗ > 0 and Λ ∗ < 0 such that if λ ∈ (Λ ∗ , + ∞) ∪ (- ∞ , Λ ∗) , then (0.1) has at least two solutions, including one local minimum solution; if λ ∈ (0 , Λ ∗) ∪ (Λ ∗ , 0) , then (0.1) has no solution; while if λ = Λ ∗ or Λ ∗ , then (0.1) has at least one solution. Our method is calculating the topological degree and using the relation between the degree and the critical group of a related functional. Similar method is also applied to the Chern–Simons Higgs system, and a partial result for the multiple solutions of the system is obtained. [ABSTRACT FROM AUTHOR]