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 Pσ(v)=Kvn+2σn-2σ on the standard unit sphere Sⁿ for all σ∈ (0 , n/ 2) , where P_σ is the intertwining operator of order 2 σ. 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 σ= 1 , it is the prescribing scalar curvature problem or the Nirenberg problem, and when σ= 2 , it is the prescribing Q-curvature problem.