Existence of Solutions to Mean Field Equations on Graphs

An Huang Department of Mathematics, Brandies University, Waltham, MA, USA Yong Lin Yau Mathematical Sciences Center, Tsinghua University, Beijing, China Shing-Tung Yau Department of Mathematics, Harvard University, Cambridge, MA, USA

Analysis of PDEs mathscidoc:2207.03007

Communications in Mathematical Physics, 377, 613–621, 2020.2
In this paper, we prove two existence results of solutions to mean field equations Δu+e^u=ρδ_0 and Δu=λe^u(e^u−1)+4π\sum_{j=1}^M δ_{p_j} on an arbitrary connected finite graph, where ρ>0 and λ>0 are constants, M is a positive integer, and p_1,…,p_M are arbitrarily chosen distinct vertices on the graph.
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  title={Existence of Solutions to Mean Field Equations on Graphs},
  author={An Huang, Yong Lin, and Shing-Tung Yau},
  booktitle={Communications in Mathematical Physics},
An Huang, Yong Lin, and Shing-Tung Yau. Existence of Solutions to Mean Field Equations on Graphs. 2020. Vol. 377. In Communications in Mathematical Physics. pp.613–621. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707153951790910567.
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