### Abstract

Let M be a compact Riemann surface, α_{j} > -1, and h (x) a positive C^{2} function of M. In this paper, we consider the following mean field equation: Δu (x) + ρ (h (x) e^{u(x)}/∫_{M} h (x) e^{u(x)} - 1/|M|) = 4π ∑_{j=1} ^{d}α_{j} (δ_{qj} - 1/|M|) in M. We prove that for α_{j} ∈ ℕ and any ρ > ρ_{0}, the equation has one solution at least if the Euler characteristic χ (M) ≤ 0, where ρ_{0} = max_{M}(2K - Δln h + N∗), K is the Gaussian curvature, and N∗ = 4π ∑_{j=1} ^{d} α_{j}. This result was proved in [10] when α_{j} = 0. Our proof relies on the bubbling analysis if one of the blowup points is at the vortex q_{j}. In the case where α_{j} ∉ ℕ, the sharp estimate of solutions near q_{j} has been obtained in [11]. However, if α_{j} ∈ 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 language | English |
---|---|

Pages (from-to) | 377-424 |

Number of pages | 48 |

Journal | Journal of Differential Geometry |

Volume | 103 |

Issue number | 3 |

Publication status | Published - 2016 Jul 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Journal of Differential Geometry*,

*103*(3), 377-424.

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

Research output: Contribution to journal › Article

*Journal of Differential Geometry*, vol. 103, no. 3, pp. 377-424.

}

TY - JOUR

T1 - Estimates of the mean field equations with integer singular sources

T2 - Non-simple blowup

AU - Kuo, Ting Jung

AU - Lin, Chang Shou

PY - 2016/7/1

Y1 - 2016/7/1

N2 - 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 dαj (δqj - 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.

AB - 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 dαj (δqj - 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.

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M3 - Article

VL - 103

SP - 377

EP - 424

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 3

ER -