Morse theory and the resonant Q-curvature problem

Abstract

In this paper, we study the prescribed $Q$-curvature problem on closed four-dimensional Riemannian manifolds when the total integral of the $Q$-curvature is a positive integer multiple of the one of the four-dimensional round sphere. This problem has a variational structure with a lack of compactness. Using some topological tools of the theory of "critical points at infinity" combined with a refined blow-up analysis and some dynamical arguments, we identify the accumulations points of all noncompact flow lines of a pseudogradient flow, the so called critical points at infinity of the associated variational problem, and associate to them a natural Morse index. We then prove strong Morse type inequalities, extending the full Morse theory to this noncompact variational problem. Finally, we derive from our results Poincaré-Hopf index type criteria for existence, extending known results in the literature and deriving new ones.

References in corpus (4)

• Concentration-compactness phenomena in the higher order Liouville's equation
• The profile of bubbling solutions of a class of fourth order geometric equations on 4-manifolds
• Conformal Metrics with Constant Q-Curvature
• An Upper Bound of the Total Q-Curvature and Its Isoperimetric Deficit for Higher-dimensional Conformal Euclidean Metrics

Cited by in corpus (1)

• Variational theory for the resonant T-curvature equation