Conformal scalar curvature equation on Sn: Functions with two close critical points (Twin Pseudo-Peaks)

Abstract

By using the Lyapunov–Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on [Formula: see text] ([Formula: see text]) when the prescribed function (after being projected to [Formula: see text]) has two close critical points, which have the same value (positive), equal “flatness” (“twin”; flatness [Formula: see text]), and exhibit maximal behavior in certain directions (“pseudo-peaks”). The proof relies on a balance between the two main contributions to the reduced functional — one from the critical points and the other from the interaction of the two bubbles.