  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.
