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 $P_\sigma(v)= Kv^{\frac{n+2\sigma}{n-2\sigma}}$ on the standard unit sphere $\mathbb{S}^n$ for all $\sigma\in (0,n/2)$, 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.