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

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

2008-09-12

arxiv.org/abs/0809.2172v2

J. Funct. Anal. 256 (2009), 3743-3771

Mathematics

Analysis of PDEs

26 pages, revised version

Scientific paper

10.1016/j.jfa.2009.02.017

We investigate different concentration-compactness 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 $R^{2m}$, concentration 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.

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