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 ∫ V κ d μ < 0 . We consider the following Kazdan–Warner equation on G : Δ u + κ - K λ e 2 u = 0 , where K λ = K + λ and K : V → R is a non-constant function satisfying max x ∈ V K ( 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.