Abstract
Let \(G=(V,E)\) be a connected 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 and Warner (Ann. Math. 99(1):14–47, 1974).
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Acknowledgments
The authors appreciate the referees for good comments and valuable suggestions which improve the representation of this paper. The argument of proving the solvability of \(\Delta v=\overline{h}-h\) in the proof of Theorem 3 is provided by a referee. A. Grigor’yan is partly supported by SFB 701 of the German Research Council. Y. Lin is supported by the National Science Foundation of China (Grant No. 11271011). Y. Yang is supported by the National Science Foundation of China (Grant No. 11171347).
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Communicated by J. Jost.
