Abstract :
Let Pn be the n-th order Paneitz operator on Sn, n 3. We consider the following prescribing Qcurvature
problem on Sn:
Pnu+ (n−1)! = Q(x)enu on Sn,
where Q is a smooth positive function on Sn satisfying the following non-degeneracy condition:
( Q)2 + |∇Q|2 = 0.
Let G∗ :Sn→Rn+1 be defined by
G∗(x) = − Q(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.
© 2009 Published by Elsevier Inc.
