Morse theory and a scalar field equation on compact surfaces

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