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

Ting Jung Kuo, Chang Shou Lin

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)377-424
Number of pages48
JournalJournal of Differential Geometry
Volume103
Issue number3
DOIs
Publication statusPublished - 2016 Jul

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ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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