Prescribing Q-curvature problem on Sn

Abstract

Let Pn be the n-th order Paneitz operator on Sn, n ≥ 3. We consider the following prescribing Q-curvature problem on Sn:Pn u + (n - 1) ! = Q (x) en u on Sn, where Q is a smooth positive function on Sn satisfying the following non-degeneracy condition:(Δ Q)2 + | ∇ Q |2 ≠ 0 . Let G* : Sn → Rn + 1 be defined byG* (x) = (- Δ Q (x), ∇ Q (x)) . We show that if Q > 0 is non-degenerate and deg (frac(G*, | G* |), Sn) ≠ 0, then the above equation has a solution. When n is even, this has been established in our earlier work [J. Wei, X. Xu, On conformal deformation of metrics on Sn, J. Funct. Anal. 157 (1998) 292-325]. When n is odd, Pn becomes a pseudo-differential operator. Here we develop a unified approach to treat both even and odd cases. The key idea is to write it as an integral equation and use Liapunov-Schmidt reduction method. © 2009.