The Nirenberg problem and its generalizations: a unified approach

Abstract

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.