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 $\mathbb{S}^5$ these a priori estimates, combined with the perturbation result in the first part of the present work, allow us to obtain some existence result using a continuity method. On $\mathbb{S}^6$ we prove the existence of at least one solution when an index formula associated to this conformal invariant is different from zero