Morse inequalities at infinity for a resonant mean field equation

Abstract

In this paper, we study the following mean field type equation: [Formula: see text] where [Formula: see text] is a closed oriented surface of unit volume [Formula: see text] = 1, [Formula: see text] positive smooth function and [Formula: see text], [Formula: see text]. 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 [Formula: see text]. 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. [Formula: see text].