Construction of Blow-up Sequence for the Conformal Scalar Curvature Equation on S^n. I, II, and Appendix

TL;DR

This paper develops a method using annular domains and Lyapunov-Schmidt reduction to construct scalar curvature functions on high-dimensional spheres that induce solutions to the conformal scalar curvature equation which blow up, with prescribed smoothness.

Contribution

It introduces a novel construction technique for scalar curvature functions on S^n that produce blow-up solutions to the conformal scalar curvature equation.

Findings

Constructed scalar curvature functions on S^n for n > 5.

Demonstrated existence of blowing-up positive solutions.

Ensured prescribed functions have C^{n-1, β} smoothness.

Abstract

Using the Lyapunov-Schmidt reduction method, we describe how to use annular domains to construct (scalar curvature) functions on S^n, (n > 5), so that each one of them enables the conformal scalar curvature equation to have a blowing-up sequence of positive solutions. The prescribed scalar curvature function is shown to have C^{n - 1, \beta} smoothness.