Fractional mean field equations on finite graphs

Abstract

In this paper, the author considers the fractional mean field equation on a finite graph $G=(V,E)$, say \begin{equation*} (-\Delta)^s u=\rho\left(\dfrac{he^u}{\int_V he^ud\mu}-\dfrac{1}{|V|}\right),\quad\forall\,x\in V, \end{equation*} where $s\in(0,\,1)$, $\rho\in(-\infty,\,0)\cup(0,\,+\infty)$ are some fixed parameters, $h$ denotes a given real value function on $V$. Based on the sign of the prescribed function $h$, via the variational method, topological degree and two mean field type heat flows, the author obtains the existence of solutions for the above problem in three cases respectively. These results extend the relevant research of Lin-Yang (Calc. Var., 2021), Sun-Wang (Adv. Math., 2022) and Liu-Zhang (J. Math. Anal. Appl., 2023) in the case of $s=1$.