Non-degeneracy of the Bubble Solutions for the Fractional Prescribed Curvature Problem and Applications

We consider the following fractional prescribed scalar curvature problem {(-delta)(s)u = K (y)u(2 & lowast;s-1), in R-N, (P) u > 0, u is an element of ?H-s(R-N), where N > 2s+2, 1/2 < s < 1, 2(s)(& lowast;) = 2N/N -2s is the critical Sobolev exponent, ?H-s(R-N) is the completion of C-0(infinity) (R-N) under the semi-norm [u](H)s(R-N)(=) (integral (N)(R) |(-delta) (s/2) u|(2)dx)(1/2), and K(y) is a positive radial function. We first prove the non-degeneracy result for the positive bubble solutions of the above equation via the local Pohozaev identities. Then we apply the non-degeneracy result to construct new bubble solutions by finite dimensional reduction method. It should be mentioned that due to the non-localness of the fractional operator, we should establish the local Pohozaev identities for corresponding harmonic extension instead of u. This difference not only makes the sharp estimates for several integrals in the local Pohozaev identities, but also forces us to use the Pohozaev identities in quite a different way.