Concentration-compactness phenomena in the higher order Liouville's equation

We investigate different concentration-compactness and phenomena related to the Q-curvature in arbitrary even dimension, We first treat the case of an open domain in R(2m), then that of a closed manifold and, finally, the particular case of the sphere S(2m). In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in R(2m), blow-up phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have compactness. (C) 2009 Elsevier Inc. All rights reserved.