  We prove the compactness of solutions to general fourth order elliptic
equations which are L^1-perturbations of the Q-curvature equation on compact
Riemannian 4-maniods. Consequently, we prove the global existence and
convergence of the Q-curvature flow on a generic class of Riemannian
4-manifolds. As a by product, we give a positive answer to an open question by
A. Malchiodi on the existence of bounded Palais-Smale sequences for the
Q-curvature problem when the Paneitz operator is positive with trivial kernel.
