Fractional mean field equations on finite graphs

TL;DR

This paper investigates fractional mean field equations on finite graphs, establishing existence results using variational methods, topological degree, and heat flows, extending previous work from the classical case to fractional orders.

Contribution

The paper extends the analysis of mean field equations from the classical case to fractional orders on finite graphs, providing new existence results under various conditions.

Findings

Existence of solutions established for different cases based on the sign of h.

Application of variational methods, topological degree, and heat flows.

Extension of previous results from s=1 to fractional s in (0,1).

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 $s=1$.