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

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.

26 pages, revised version

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• Classification of solutions to the higher order Liouville's equation on R^{2m}
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Cited by in corpus (18)

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• Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three
• A threshold phenomenon for embeddings of $H_0^m$ into Orlicz spaces
• Conformal metrics on R^{2m} with constant Q-curvature and large volume
• Conformally Euclidean metrics on ${\mathbb{R}}^n$ with arbitrary total $Q$-curvature
• Asymptotics and quantization for a mean-field equation of higher order
• Quantization for the prescribed Q-curvature equation on open domains
• Morse theory and the resonant Q-curvature problem
• Concentration phenomena for the fractional $Q$-curvature equation in dimension 3 and fractional Poisson formulas
• Bubbling of the prescribed $Q$-curvature equation on $4$-manifolds in the null case
• Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space
• Uniform bounds for higher-order semilinear problems in conformal dimension
• Strong field behavior of Wang-Yau Quasi-local energy
• Existence of solutions to a higher dimensional mean-field equation on manifolds
• "Large" conformal metrics of prescribed Q-curvature in the negative case
• Existence and compactness of conformal metrics on the plane with unbounded and sign-changing Gaussian curvature
• The Prescribed $Q$-Curvature Flow for Arbitrary Even Dimension in a Critical Case
• Isolated Singularities of Polyharmonic Operator in Even Dimension