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

Djadli, Zindine; Malchiodi, Andrea; Ould Ahmedou, Mohameden

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 2, pp. 387-434.

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 $({𝕊}^n,h)$. We derive from this analysis some a priori estimates in dimension $5$ and $6$. On ${𝕊}^5$ 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 ${𝕊}^6$ we prove the existence of at least one solution when an index formula associated to this conformal invariant is different from zero.

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