# Prescribing Morse scalar curvatures: blow-up analysis

## TL;DR

This paper analyzes the blow-up behavior of solutions to prescribed Morse scalar curvature problems, establishing precise blow-up rates in the subcritical case and excluding tower bubbles across all dimensions.

## Contribution

It provides a detailed blow-up rate analysis for subcritical solutions and rules out tower bubbles, advancing understanding of scalar curvature prescription problems.

## Key findings

- Precise blow-up rates for subcritical solutions
- Exclusion of tower bubbles in all dimensions
- Foundation for future existence results

## Abstract

We study finite-energy blow-ups for prescribed Morse scalar curvatures in both the subcritical and the critical regime. After general considerations on Palais-Smale sequences we determine precise blow up rates for subcritical solutions: in particular the possibility of tower bubbles is excluded in all dimensions. In subsequent papers we aim to establish the sharpness of this result, proving a converse existence statement, together with a one to one correspondence of blowing-up subcritical solutions and {\em critical points at infinity}. This analysis will be then applied to deduce new existence results for the geometric problem.