Fractional mean field equations on finite graphs

2024/04/02 by Yang Liu, Liu, Yang
Mathematics · #35R02 #39A12 #46E39 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2404.01610

openalex publication_date 2024/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, the author considers the fractional mean field equation on a finite graph G=(V,E), say (-Δ)^s u=ρ(\dfrache^u∫_V he^udμ-\dfrac1|V|), ∀ x∈ V, where s∈(0, 1), ρ∈(-∞, 0)∪(0, +∞) 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.

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