In this talk, I present some results about the compactness, existence and multiplicity of the solutions to the prescribing fractional $Q$-curvature problem. At first, we consider the fractional order is $2\sigma$ on $n$-dimensional standard sphere when $n − 2\sigma = 2$, $\sigma = 1 + m/2$, $m\in \mathbb{N}_+$, we obtain some compactness and existness results. Secondly, by combining critical points at infinity approach with Morse theory we obtain existence results under suitable pinching conditions. Thirdly, we obtain some results on the density and multiplicity of positive solutions to the prescribed fractional $Q$-curvatures problems for $\sigma\in(0,1)$ and $\sigma\in(1,\frac{n}{2})$ is an integer. This is a joint work with Dr. Yan Li, Heming Wang and Ning Zhou.
