Conformal scalar curvature equation on Sⁿ: functions with two close critical points (twin pseudo-peaks)

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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 greater or equal to 3) when the prescribed function (after being projected to Rⁿ) 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.