Morse inequalities at infinity for a resonant mean field equation.

In this paper, we study the following mean field type equation: (M F) − Δ g u = ϱ K e u ∫ Σ K e u d V g − 1 in  Σ , where (Σ , g) is a closed oriented surface of unit volume V o l g (Σ) = 1, K positive smooth function and ϱ = 8 π m , m ∈ 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 (Σ). 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. ϱ ∈ 8 π ℕ. [ABSTRACT FROM AUTHOR]