  We investigate fourth order Paneitz equations of critical growth in the case
of $n$-dimensional closed conformally flat manifolds, $n \ge 5$. Such equations
arise from conformal geometry and are modelized on the Einstein case of the
geometric equation describing the effects of conformal changes of metrics on
the $Q$-curvature. We obtain sharp asymptotics for arbitrary bounded energy
sequences of solutions of our equations from which we derive stability and
compactness properties. In doing so we establish the criticality of the
geometric equation with respect to the trace of its second order terms.
