The axisymmetric σk-Nirenberg problem

Abstract

We study the problem of prescribing σk-curvature for a conformal metric on the standard sphere Sn with 2≤k<n/2 and n≥5 in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function K near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of K of flatness order β∈[2,n). We prove among other things the following statements. (1) When β>n−2k, the solution set is compact, has a nonzero total degree counting and is therefore non-empty. (2) When β=n−2k, there is an explicit positive constant C(K) associated with K. If C(K)>1, the solution set is compact with a nonzero total degree counting and is therefore non-empty. If C(K)<1, the solution set is compact but the total degree counting is 0, and the solution set is sometimes empty and sometimes non-empty. (3) When 2n−2k≤β<n−2k, the solution set is compact, but the total degree counting is zero, and the solution set is sometimes empty and sometimes non-empty. (4) When β<n−2k2, there exists K for which there exists a blow-up sequence of solutions with unbounded energy. In this same range of β, there exists also some K for which the solution set is empty.