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

Shuang Liu, Yunyan Yang*

*Corresponding author for this work

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Abstract

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 Kλ= K+ λ and K: V→ R is a non-constant function satisfying max xVK(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.

Original languageEnglish
Article number164
JournalCalculus of Variations and Partial Differential Equations
Volume59
Issue number5
DOIs
Publication statusPublished - 1 Oct 2020
Externally publishedYes

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Liu, S., & Yang, Y. (2020). Multiple solutions of Kazdan–Warner equation on graphs in the negative case. Calculus of Variations and Partial Differential Equations, 59(5), Article 164. https://doi.org/10.1007/s00526-020-01840-3