# Conformal scalar curvature equation on S^n: functions with two close critical points (twin pseudo-peaks)

## TL;DR

This paper investigates the existence of solutions to the conformal scalar curvature equation on spheres when the prescribed function has two nearby critical points with similar properties, using a non-perturbative Lyapunov-Schmidt reduction approach.

## Contribution

It introduces a novel application of Lyapunov-Schmidt reduction to handle functions with twin pseudo-peaks on S^n, providing new existence results for the scalar curvature problem.

## Key findings

- Existence of solutions with twin pseudo-peaks under certain flatness conditions
- Balance between critical point contributions and bubble interactions is key
- Method extends previous approaches to functions with closely spaced critical points

## Abstract

By using the Lyapunov-Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on S^n (n greater or equal to 3) when the prescribed function (after being projected to R^n) 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.