Morse inequalities at infinity for a resonant mean field equation

TL;DR

This paper develops Morse theory at infinity for a resonant mean field equation on surfaces, establishing inequalities that relate critical points, their indices at infinity, and topological invariants, leading to new existence and multiplicity results.

Contribution

It introduces Morse inequalities at infinity for the mean field equation in the resonant case, linking critical points, their indices, and topological data under generic conditions.

Findings

Established Morse inequalities at infinity for the equation.

Derived new existence results for solutions.

Proved multiplicity results based on topological methods.

Abstract

In this paper we study the following mean field type equation \begin{equation*} (MF) \qquad -\D_g u \, = \varrho ( \frac{K e^{u}}{\int_{\Sig} K e^{u} dV_g} \, - \, 1) \, \mbox{ in } \Sigma, \end{equation*} where $(\Sigma, g)$ is a closed oriented surface of unit volume $Vol_g(\Sigma)$ = 1, $K$ positive smooth function and $\varrho= 8 \pi m$, $ m \in \N$. Building on the critical points at infinity approach initiated in \cite{ABL17} 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 $B_m(\Sigma)$.\\ We derive from these \emph{Morse inequalities at infinity} various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. $\varrho \in 8 \pi \N$.