The aim of this paper is to study a nonlinear scalar field equation on a surface $\Sigma$ via a Morse-theoretical approach, based on some of the methods in . Employing these ingredients, we derive an alternative and direct proof (plus a clear interpretation) of a degree formula obtained in , which used refined blow-up estimates from  and . Related results are derived for the prescribed $Q$-curvature equation on four manifolds.
"Morse theory and a scalar field equation on compact surfaces." Adv. Differential Equations 13 (11-12) 1109 - 1129, 2008.