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

Abstract

Let $M$ be a compact Riemann surface, $\alpha j \gt -1$, and $h(x)$ a positive $C^2$ function of $M$. In this paper, we consider the following mean field equation: \[ \Delta u (x) + \rho \left( \frac{h(x) e^{u(x)}}{\int_{M} h (x) e^{u(x)}} - \frac{1}{\lvert M \rvert} \right) = 4\pi \sum^{d}_{j=1} \alpha_j \left( \delta_{q_j} - \frac{1}{\lvert M \rvert} \right) \textrm{ in } M \textrm{.} \] We prove that for $\alpha_j \in \mathbb{N}$ and any $\rho \gt \rho_0$, the equation has one solution at least if the Euler characteristic $\chi (M) \leq 0$, where $\rho_0 = \underset{M}{\max} ( 2K - \Delta \mathrm{ln} h + N^*)$, $K$ is the Gaussian curvature, and $N^* = 4\pi \sum^{d}_{j=1} \alpha_j$. This result was proved in [10] when $\alpha_j = 0$. Our proof relies on the bubbling analysis if one of the blowup points is at the vortex $q_j$. In the case where $\alpha_j \notin \mathbb{N}$, the sharp estimate of solutions near $q_j$ has been obtained in [11]. However, if $\alpha_j \in \mathbb{N}$, 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.