Compactness and existence results of the prescribing fractional $Q$-curvatures problem on $\mathbb{S}^n$

Abstract

This paper is devoted to establishing the compactness and existence results of the solutions to the prescribing fractional Q-curvatures problem of order 2σ on n-dimensional standard sphere whenThe compactness results are novel and optimal. In addition, we prove a degreecounting 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.