Existence of Solutions to Mean Field Equations on Graphs

Abstract

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

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

and

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

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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References

  1. 1.

    Caglioti, E., Lions, P.L., Marchioro, C., Pulvirenfi, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143, 501–525 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Caffarelli, L.A., Yang, Y.: Vortex condensation in the Chern–Simons Higgs model: an existence theorem. Commun. Math. Phys. 168, 321–336 (1995)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Chen, C.C., Lin, C.S.: Topological degree for a mean field equation on Riemann surfaces. Commun. Pure Appl. Math. LVI, 1667–1727 (2003)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chen, C.C., Lin, C.S., Wang, G.: Concentration phenomenon of two-vortex solutions in a Chern–Simons model. Ann. Scuola Norm. Sup. Pisa CI. Sci. (5) III, 367–379 (2004)

    MATH  Google Scholar 

  5. 5.

    Grigor’yan, A., Lin, Y., Yang, Y.: Kazdan–Warner equation on graph. Calc. Var. Part. Differ. Equ. 55(4), 1–13 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ge, H.: Kazdan–Warner equation on graph in the negative case. J. Math. Anal. Appl. 453, 1022–1027 (2017)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Ge, H., Jiang, W.: Kazdan–Warner equation on infinite graphs. J. Korean Math. Soc. 55(5), 1091–1101 (2017)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Keller, M., Schwarz, M.: The Kazdan–Warner equation on canonically compactifiable graphs. Calc. Var. Part. Differ. Equ. 57(2), 1–20 (2018)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Kazdan, J., Warner, F.: Curvature functions for compact 2-manifolds. Ann. Math. 99(1), 14–47 (1974)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Lin, C.-S., Wang, C.-L.: Elliptic functions, Green functions and the mean field equations on tori. Ann. Math. 172(2), 911–954 (2010)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Lin, C.-S., Yan, S.: Existence of bubbling solutions for Chern–Simons model on a torus. Arch. Ration. Mech. Anal. 207(2), 353–392 (2013)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

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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Communicated by H. T. Yau.

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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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