In this paper we prescribe a fourth order conformal invariant on the standard n-sphere, with n ≥ 5, and study the related fourth order elliptic equation. We prove new existence results based on a new type of Euler-Hopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal metrics having the same Q-curvature.

Abstract. In this paper, we consider the problem of existence of conformal metrics with prescribed mean curvature on the unit ball of R n , n ≥ 3. Under the assumption that the order of flatness at critical points of prescribed mean curvature function H(x) is β ∈]1, n − 2], we give precise estimates on the losses of the compactness and we prove new existence result through an Euler-Hopf type formula.

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In the present work, we consider the prescribed Q-curvature problem on the unit sphere S n , n ≥ 5. Under the hypothesis that the prescribed function satisfies a flatness condition of order β = n, we give a complete description of the lack of compactness of the problem and we provide an existence result in terms of an Euler-Hopf index.