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We consider the problem of existence of conformal metrics with prescribed Q-curvature on standard sphere , ≥ 5. Under the assumption that the order of flatness at critical points of prescribed Q-curvature function ( ) is ∈ ]1, − 4], we give precise estimates on the losses of the compactness, and we prove new existence and multiplicity results through an Euler-Hopf type formula.

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Abstract:In this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S , ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16]. MSC:58E05, 35J65, 53C21, 35B40