Prescribing Q-curvature problem on Sn

AbstractLet Pn be the n-th order Paneitz operator on Sn, n⩾3. We consider the following prescribing Q-curvature problem on Sn:Pnu+(n−1)!=Q(x)enuon Sn, where Q is a smooth positive function on Sn satisfying the following non-degeneracy condition:(ΔQ)2+|∇Q|2≠0. Let G∗:Sn→Rn+1 be defined byG∗(x)=(−ΔQ(x),∇Q(x)). We show that if Q>0 is non-degenerate and deg(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