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

Let M be a compact Riemann surface, α_j > -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) e^u(x)/∫_M h (x) e^u(x) - 1/|M|) = 4π ∑_j=1 ^dα_j (δ_qj - 1/|M|) in M. We prove that for α_j ∈ ℕ and any ρ > ρ₀, the equation has one solution at least if the Euler characteristic χ (M) ≤ 0, where ρ₀ = 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.