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−∆0w = f e − 1, where ∆0 denotes the Laplace operator associated with the standard metric g0 on S2. J. Moser [22] proved that this equation has a solution if the function f satisfies the condition f(x) = f(−x) for all x ∈ S. The general case was studied by S.-Y. A. Chang, M. Gursky, and P. Yang [10, 11, 14]. A. Bahri and J. M. Coron [4, 5] and R. Schoen and D. Zhang [24] constructed metrics with prescribed scalar curvature on S. J. F. Escobar and R. Schoen [18] studied the prescribed scalar curvature problem on manifolds which are not necessarily conformally equivalent to the standard sphere. Our aim is to generalize these results to higher dimensions. Let g be a conformal metric on S. We denote by R the scalar curvature of g and by Ric the Ricci tensor of g. Moreover, we denote by ∆ the Laplace operator with respect to the metric g. A natural conformal invariant in dimension four is Q = − 6 (∆R−R2 + 3 |Ric|2).

Prescribing a Higher Order Conformal Invariant on S<sup>n</sup>