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      Kazdan-Warner equation on graph

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          Abstract

          Let \(G=(V,E)\) be a finite graph and \(\Delta\) 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 \(\Delta u=c-he^u\) has a solution on \(V\), where \(c\) is a constant, and \(h:V\rightarrow\mathbb{R}\) 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-Warner (Ann. Math., 1974).

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          Author and article information

          Journal
          2016-07-15
          Article
          1607.04540

          http://creativecommons.org/licenses/by-nc-sa/4.0/

          Custom metadata
          34B45, 35A15, 58E30
          13 pages
          math.AP

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