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

TL;DR

This paper explores concentration-compactness phenomena related to Q-curvature in higher dimensions, revealing conditions under which concentration occurs and establishing compactness results on certain manifolds.

Contribution

It extends the analysis of concentration-compactness phenomena to higher order Liouville equations across various geometric settings, including open domains, closed manifolds, and spheres.

Findings

Concentration phenomena depend on the sign of Q-curvature.

On closed manifolds, concentration occurs only at points with positive Q-curvature.

Compactness is guaranteed on certain flat manifolds with non-positive Euler characteristic.

Abstract

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.