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Kazdan-Warner equation on graph

Alexander Grigor’yan Department of Mathematics, University of Bielefeld, 33501, Bielefeld, Germany Yong Lin Department of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of China Yunyan Yang Department of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of China

Analysis of PDEs mathscidoc:2207.03001

Calculus of Variations and Partial Differential Equations, 55, (92), 2016.7
Let G=(V,E) be a connected finite graph and Δ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan–Warner equation Δu=c−he^u has a solution on V, where c is a constant, and h:V→R is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan and Warner (Ann. Math. 99(1):14–47, 1974).
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@inproceedings{alexander2016kazdan-warner,
  title={Kazdan-Warner equation on graph},
  author={Alexander Grigor’yan, Yong Lin, and Yunyan Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707120946978948556},
  booktitle={Calculus of Variations and Partial Differential Equations},
  volume={55},
  number={92},
  year={2016},
}
Alexander Grigor’yan, Yong Lin, and Yunyan Yang. Kazdan-Warner equation on graph. 2016. Vol. 55. In Calculus of Variations and Partial Differential Equations. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707120946978948556.
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