We study a natural counterpart of the Nirenberg problem,
namely to prescribe the Q-curvature of a conformal metric on the
standard S4 as a given function f. Our approach uses a geometric
flow within the conformal class, which either leads to a solution
of our problem as, in particular, in the case when f ≡ const,
or otherwise induces a blow-up of the metric near some point of
S4. Under suitable assumptions on f, also in the latter case the
asymptotic behavior of the flow gives rise to existence results via
Morse theory.