The Nirenberg problem and its generalizations: A unified approach

Making use of integral representations, we develop a unified approach to establish blow up profiles, compactness and existence of positive solutions of the conformally invariant equations $P_\sigma(v)= Kv^{\frac{n+2\sigma}{n-2\sigma}}$onthestandardunitsphere$$\mathbb{S}$^n$forall$\sigma\in (0,n/2)$,where$P_\sigma$ is the intertwining operator of order $2\sigma$. Finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures. When $\sigma =1$, it is the prescribing scalar curvature problem or the Nirenberg problem, and when $\sigma =2$, it is the prescribing $Q$-curvature problem.