# Estimates of the mean field equations with integer singular sources: Non-simple blowup

Ting Jung Kuo, Chang Shou Lin

12 引文 斯高帕斯（Scopus）

## 摘要

Let M be a compact Riemann surface, αj > -1, and h (x) a positive C2 function of M. In this paper, we consider the following mean field equation: Δu (x) + ρ (h (x) eu(x)/∫M h (x) eu(x) - 1/|M|) = 4π ∑j=1dαjqj - 1/|M|) in M. We prove that for αj ∈ ℕ and any ρ > ρ0, the equation has one solution at least if the Euler characteristic χ (M) ≤ 0, where ρ0 = maxM(2K - Δln h + N∗), K is the Gaussian curvature, and N∗ = 4π ∑j=1d αj. This result was proved in [10] when αj = 0. Our proof relies on the bubbling analysis if one of the blowup points is at the vortex qj. In the case where αj ∉ ℕ, the sharp estimate of solutions near qj has been obtained in [11]. However, if αj ∈ ℕ, then the phenomena of non-simple blowup might occur. One of our contributions in part 1 is to obtain the sharp estimate for the non-simple blowup phenomena.

原文 英語 377-424 48 Journal of Differential Geometry 103 3 https://doi.org/10.4310/jdg/1468517500 已發佈 - 2016 7月 是

• 分析
• 代數與數理論
• 幾何和拓撲