The fourth order Paneitz equations of critical growth is investigated in the case of -dimensional closed conformally flat manifolds, . Such equations arise from conformal geometry, they are modelled on the Einstein case of a geometric equation describing the effects of conformal changes on the -curvature. Sharp asymptotics is obtained for arbitrary bounded energy sequences of solutions and stability and compactness are achieved, so establishing the criticality of the geometric equation with respect to the trace of its second order terms. For each positive numbers , satisfying , a smooth compact Riemannian manifold of dimension , the fourth order Paneitz equation of critical Sobolev growth is
where is the Laplace-Beltrami operator and is the critical Sobolev exponent. Let and denote the Ricci and scalar curvature, respectively, of and consider , the smooth -tensor field given by
If the conformal equation relating the -curvatures and is
When is Einstein, so that , for some , the last equation becomes
where , and . Given , consider the set ( is the Sobolev space of functions in with two derivatives in ). The equation (1) is said to be compact if for any , is compact in the -topology, and it is stable if it is compact and for any , there exists such that for any and satisfying , it holds that
The main theorems proved in the paper are the following:
Theorem 1. Let be a smooth compact conformally flat Riemannian manifold of dimension and positive real numbers such that . Let and be sequences of real numbers converging to and , respectively, and be a bounded sequence in as of smooth positive solutions of (2) such that weakly in as . Then either strongly in any -topology, or .
Theorem 2. Under the same hypothesis of the theorem above, assume that one of the following conditions holds true:
and for al , where is the mass at of the operator .
and , where if are the minimum and the maximum eigenvalues of , resp., and .
or and in M.
Then for any sequences and , and any bounded sequence of positive solutions of (2) there holds that, up to a subsequence , in for some smooth positive solution of (1). In particular, equation (1) is stable.