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<p>Let <i>H</i> = (<i>V, E</i>) be a connected finite hypergraph, which is an extension of the graph theory in which the edges may connect more than two vertices and form hyperedges. We study the Kazdan–Warner equation</p><p><InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Delta \phi =c-h{e}^{\phi }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mo>=</mo> <mi>c</mi> <mo>-</mo> <mi>h</mi> <msup> <mrow> <mi>e</mi> </mrow> <mi>ϕ</mi> </msup> </mrow> </math></EquationSource> </InlineEquation></p><p>on <i>H,</i> where <i>c</i> is a constant and <i>h</i> is a known function defined on <i>H</i> . Based on the work by Grigor’yan, Lin, and Yang [A. Grigor’yan, Y. Lin, and Y. Yang, <i>Calc. Var. Partial Differen. Equat.,</i> <b>55</b>, No. 4, Article 92 (2016)], we employ the variational calculus to extend the main results concerning the solutions to the Kazdan–Warner equation from finite graphs to hypergraphs. We obtain similar results for the cases where <i>c &gt;</i> 0 and <i>c &lt;</i> 0 provided that <i>h</i> satisfies certain conditions on hypergraphs. However, for the case where <i>c</i> = 0<i>,</i> we cannot get the same results.</p>

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Kazdan–Warner Equation on Hypergraphs

  • 期刊论文

  • 发表日期：

    2026年3月11日

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  • Haigang Zhang,
  • Juan Zhao

摘要

Let H = (V, E) be a connected finite hypergraph, which is an extension of the graph theory in which the edges may connect more than two vertices and form hyperedges. We study the Kazdan–Warner equation

\(\Delta \phi =c-h{e}^{\phi }\) Δ ϕ = c - h e ϕ

on H, where c is a constant and h is a known function defined on H . Based on the work by Grigor’yan, Lin, and Yang [A. Grigor’yan, Y. Lin, and Y. Yang, Calc. Var. Partial Differen. Equat., 55, No. 4, Article 92 (2016)], we employ the variational calculus to extend the main results concerning the solutions to the Kazdan–Warner equation from finite graphs to hypergraphs. We obtain similar results for the cases where c > 0 and c < 0 provided that h satisfies certain conditions on hypergraphs. However, for the case where c = 0, we cannot get the same results.