Morse theory and a scalar field equation on compact surfaces

The aim of this paper is to study a nonlinear scalar eld equation on a surface via a Morse-theoretical approach, based on some of the methods in [25]. Employing these ingredients, we derive an alternative and direct proof (plus a clear interpretation) of a degree formula obtained in [18], which used re ned blow-up estimates from [34] and [17]. Related results are derived for the prescribed Q-curvature equation on four manifolds