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We investigate different concentration-compactness phenomena related to theQ-curvature in arbitrary even dimension. We first treat the case of an opendomain in $R^{2m}$, then that of a closed manifold and, finally, the particularcase of the sphere $S^{2m}$. In all cases we allow the sign of the Q-curvatureto vary, and show that in the case of a closed manifold, contrary to the caseof open domains in $R^{2m}$, concentration phenomena can occur only at pointsof positive Q-curvature. As a consequence, on a locally conformally flatmanifold of non-positive Euler characteristic we always have compactness.

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