Fractional mean field equations: Theory and application on finite graphs

Abstract

In this paper, the author introduces a nonlocal perspective by incorporating the fractional Laplacian ${(-\Delta )}^s$, and considers the fractional mean field equation on a finite graph $G=(V,E)$, say${(-\Delta )}^{s}u=\rho (\frac{h{e}^u}{{\int }_{V}h{e}^{u}d\mu }-\frac{1}{\left|V\right|}),\forall x\in V,$ 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, using various methods such as 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 $s=1$, and potentially broaden the understanding and application of fractional operators in discrete mathematical structures, emphasizing connections to both continuous and discrete theories.