### 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 |
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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.