Conformal metrics with prescribed fractional scalar curvature on conformal infinities with positive fractional Yamabe constants

Let $(X, g^+)$ be an asymptotically hyperbolic manifold and $(M, [\hat{h}])$ its conformal infinity. Our primary aim in this paper is to introduce the prescribed fractional scalar curvature problem on $M$ and provide solutions under various geometric conditions on $X$ and $M$. We also obtain the existence results for the fractional Yamabe problem in the endpoint case, e.g., $n = 3$, $\gamma = 1/2$ and $M$ is non-umbilic, etc. Every solution we find turns out to be smooth on $M$.