Multiple solutions of Kazdan–Warner equation on graphs in the negative case

Let G= (V, E) be a finite connected graph, and let κ: V→ R be a function such that ∫ _Vκd μ< 0. We consider the following Kazdan–Warner equation on G: Δu+κ-Kλe2u=0,where K_λ= K+ λ and K: V→ R is a non-constant function satisfying max _x∈VK(x) = 0 and λ∈ R. By a variational method, we prove that there exists a λ^∗> 0 such that when λ∈ (- ∞, λ^∗] the above equation has solutions, and has no solution when λ≥ λ^∗. In particular, it has only one solution if λ≤ 0 ; at least two distinct solutions if 0 < λ< λ^∗; at least one solution if λ= λ^∗. This result complements earlier work of Grigor’yan et al. (Calc Var Partial Diff Equ, 55(4):13 2016), and is viewed as a discrete analog of that of Ding and Liu (Trans Am Math Soc, 347:1059–1066 1995) and Yang and Zhu (Ann Acad Sci Fenn Math, 44:167–181 2019) on manifolds.