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arXiv · 1903.07891

Existence of Solutions to Mean Field Equations on Graphs

An Huang·Yong Lin·Shing-Tung Yau
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Abstract

In this paper, we prove two existence results of solutions to mean field equations $$Δu+e^u=ρδ_0$$ and $$Δu=λe^u(e^u-1)+4 π\sum_{j=1}^{M}{δ_{p_j}}$$ 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 vertices on the graph.

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An Huang, Yong Lin, Shing-Tung Yau. 2019-03-19. Existence of Solutions to Mean Field Equations on Graphs. https://doi.org/10.1007/s00220-020-03708-1

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