On the Existence of Solutions for Prescribing Fractional Q-curvature Problem on \({\mathbb S}^{n}\)

The aim of this paper is to investigate the existence of solutions to the prescribing fractional Q-curvature problem on \({\mathbb S}^{n}\) under some reasonable assumption of the Laplacian sign at the critical point of prescribing curvature function K. Due to the lack of compactness, we choose to return to the basic elements of variational theory and study the deformation along the flow lines. The novelty of the paper is that we obtain the existence without assuming any symmetry and periodicity on K. In addition, to overcome the loss of compactness for high-order operator problem, we need more delicate estimates with the second order cases.