2023/01/19 by Xiaobao Zhu, Zhu, Xiaobao
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2301.08309
openalex publication_date 2023/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (M,g) be a compact Riemann surface with unit area, h∈ C∞(M) a function which is positive somewhere, ρ>0, pi∈ M and αi∈(-1,+∞) for i=1,⋯,ℓ, we consider the mean field equation Δv + 4π∑i=1ℓαi(1-δpi) = ρ(1-(hev)/(∫Mhevdμ)), on M, where Δ and dμ are the Laplace-Beltrami operator and the area element of (M,g) respectively. Using variational method and blowup analysis, we prove some existence results in the critical case ρ=8π(1+min\0,min1≤ i≤ℓαi\). These results can be seen as partial generalizations of works of Chen-Li (J. Geom. Anal. 1: 359--372, 1991), Ding-Jost-Li-Wang (Asian J. Math. 1: 230--248, 1997), Mancini (J. Geom. Anal. 26: 1202--1230, 2016), Yang-Zhu (Proc. Amer. Math. Soc. 145: 3953--3959, 2017), Sun-Zhu (arXiv:2012.12840) and Zhu (arXiv:2212.09943). Among other things, we prove that the blowup (if happens) must be at the point where the conical angle is the smallest one and h is positive, this is the most important contribution of our paper.