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  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 . 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.
|ジャーナル||Journal of Differential Geometry|
|出版ステータス||Published - 2016 7 1|
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