In this paper, we study the following mean field type equation: $$(MF)-{\mathrm{\Delta }}_{g}u=\varrho \left(\frac{K{e}^u}{{\int }_{\mathrm{\Sigma }}K{e}^{u}d{V}_g}-1\right)\text{in }\mathrm{\Sigma },$$ where $(\mathrm{\Sigma },g)$ is a closed oriented surface of unit volume ${Vol}_g(\mathrm{\Sigma })$ = 1, $K$ positive smooth function and $\varrho =8\pi m$, $m\in \mathbb{N}$. Building on the critical points at infinity approach initiated in [M. Ahmedou, M. Ben Ayed and M. Lucia, On a resonant mean field type equation: A “critical point at infinity” approach, Discrete Contin. Dyn. Syst. 37(4) (2017) 1789–1818] we develop, under generic condition on the function K and the metric g, a full Morse theory by proving Morse inequalities relating the Morse indices of the critical points, the indices of the critical points at infinity, and the Betti numbers of the space of formal barycenters $B_m(\mathrm{\Sigma })$. We derive from these Morse inequalities at infinity various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. $\varrho \in 8\pi \mathbb{N}$.

在本文中，我们研究了以下平均场类型方程：$$(米F)-{\mathrm{△}}_G你=\varrho \left(\frac{钾{\mathrm{电子}}^{你}}{{\int }_{\mathrm{\Sigma }}钾{\mathrm{电子}}^{你}d{V}_G}-\mathrm{1个}\right)\text{在 }\mathrm{\Sigma },$$在哪里$(\mathrm{\Sigma },G)$是单位体积的封闭定向曲面${Vo升}_G(\mathrm{\Sigma })$= 1,$钾$正平滑函数和$\varrho =\mathrm{8个}\pi 米$,$米\in \mathbb{N}$. 建立在 [M. Ahmedou、M. Ben Ayed 和 M. Lucia，关于共振平均场类型方程：“无穷远临界点”方法，Discrete Contin。动态。系统。 37 (4) (2017) 1789–1818] 我们在函数K和度量g的一般条件下，通过证明与临界点的摩尔斯指数相关的摩尔斯不等式，开发了一个完整的摩尔斯理论，临界点的指数在无穷大，以及正式重心空间的 Betti 数${乙}_{米}(\mathrm{\Sigma })$. 我们从无穷远处的这些莫尔斯不等式推导出各种新的存在性以及共振情况下平均场方程的多重性结果，即$\varrho \in \mathrm{8个}\pi \mathbb{N}$.