Existence of Solutions to Mean Field Equations on Graphs


In this paper, we prove two existence results of solutions to mean field equations

$$\begin{aligned} \Delta u+e^u=\rho \delta _0 \end{aligned}$$


$$\begin{aligned} \Delta u=\lambda e^u(e^u-1)+4 \pi \sum _{j=1}^{M}{\delta _{p_j}} \end{aligned}$$

on an arbitrary connected finite graph, where \(\rho >0\) and \(\lambda >0\) are constants, M is a positive integer, and \(p_1,\ldots ,p_M\) are arbitrarily chosen distinct vertices on the graph.

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Y. Lin is supported by the National Science Foundation of China (Grant No. 11271011 and 11761131002), S.-T. Yau is supported by the NSF DMS-0804. Part of the work was done when Y. Lin visited the Harvard CMSA in 2018.

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Huang, A., Lin, Y. & Yau, S. Existence of Solutions to Mean Field Equations on Graphs. Commun. Math. Phys. 377, 613–621 (2020). https://doi.org/10.1007/s00220-020-03708-1

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