Estimates of the mean field equations with integer singular sources

Non-simple blowup

Ting Jung Kuo, Chang Shou Lin

Research output: Contribution to journalArticle

4 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=1 dαjqj - 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=1 d 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
Publication statusPublished - 2016 Jul 1
Externally publishedYes

Fingerprint

Mean Field Equation
Blow-up
Integer
Estimate
Total curvature
Euler Characteristic
Riemann Surface
Vortex

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

Cite this

Estimates of the mean field equations with integer singular sources : Non-simple blowup. / Kuo, Ting Jung; Lin, Chang Shou.

In: Journal of Differential Geometry, Vol. 103, No. 3, 01.07.2016, p. 377-424.

Research output: Contribution to journalArticle

Kuo, Ting Jung ; Lin, Chang Shou. / Estimates of the mean field equations with integer singular sources : Non-simple blowup. In: Journal of Differential Geometry. 2016 ; Vol. 103, No. 3. pp. 377-424.
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