Prescribing a fourth order conformal invariant on the standard sphere, Part II: blow-up analysis and applications

TL;DR

This paper conducts a detailed blow-up analysis for a fourth order elliptic equation on the sphere, deriving estimates and existence results for solutions in dimensions 5 and 6 related to conformal invariants.

Contribution

It provides new blow-up analysis techniques and existence results for conformal invariant equations on the sphere, extending previous work with refined estimates.

Findings

A priori estimates established in dimensions 5 and 6

Existence of solutions on the 5-dimensional sphere

Existence of at least one solution on the 6-dimensional sphere

Abstract

In this paper we perform a fine blow-up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere. We derive from this analysis some a priori estimates in dimension 5 and 6. On the five dimensionl sphere these a priori estimates combined with the perturbation result of Part I allow us to obtain some existence results. On the six dimensional sphere we prove the existence of at least one solution when an associated index is different from zero.