Compactness properties for geometric fourth order elliptic equations with application to the Q-curvature flow

We prove the compactness of solutions of general fourth order elliptic equations which areL^{1}-perturbations of theQ-curvature equation on compact Riemannian 4-manifolds. Consequently, we prove the global existence and convergence of theQ-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 [J. reine angew. Math. 594 (2006), 137–174] on the existence of bounded Palais–Smale sequences for theQ-curvature problem when the Paneitz operator is positive with trivial kernel.