On Conformal Deformations of Metrics onSn

OnSn, there is a naturally metric definednth order conformal invariant operatorPn. Associated with this operator is a so-calledQ-curvature quantity. When two metrics are pointwise conformally related, their associated operators, together with theirQ-curvatures, satisfy the natural differential equations. This paper is devoted to the question of which function can be aQ-curvature candidate. This is the so-calledprescribing Q-curvature problem. Our main result is that ifQis positive, nondegenerate and the naturally defined mapping associated withQhas nonzero degree, then our problem has a solution. This is the natural generalization of prescribing Gaussian curvature onS2intoSn.