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By using the Lyapunov–Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on Sn (n ≥ 3) when the prescribed function (after being projected to IRn) has two close critical points, which have the same value (positive), equal “flatness” (“twin”; flatness < n − 2), 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.

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