In this talk， I present some results about the compactness and existence results of the solutions to the prescribing fractional Q-curvature problem. At first, we consider the fractional order is 2σ on n-dimensional standard sphere when n 2σ = 2, σ = 1 + m/2, m ∈ N+. The compactness results are novel and optimal. In addition, we proved a degree-counting formula of all solutions to achieve the existence. From our results, we can know where blow up occur. Furthermore, the sequence of solutions that blow up precisely at any finite distinct location can be constructed. It is worth noting that our results include the case of multiple harmonic. Secondly, by combining critical points at infinity approach with Morse theory we obtain new existence results under suitable pinching conditions. This is a joint work with Dr. Yan Li and Ning Zhou.
