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 $P_{\sigma }(v)=K{v}^{\frac{n+2\sigma }{n-2\sigma }}$ 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.

60 pages. arXiv admin note: text overlap with arXiv:1111.1332

References in corpus (7)

• On higher order extensions for the fractional Laplacian
• Sign of Green's function of Paneitz operators and the Q curvature
• Asymptotic analysis for fourth order Paneitz equations with critical growth
• Infinitely many non-radial sign-changing solutions for a Fractional Laplacian equation with critical nonlinearity
• On the recursive structure of Branson's Q-curvature
• Prescribing integral curvature equation
• Indefinite fractional elliptic problem and Liouville theorems

Cited by in corpus (1)

• A nonlocal $\mathbf{Q}$-curvature flow on a class of closed manifolds of dimension $\mathbf{n}\ge 5$