The axisymmetric $sigma_k$-Nirenberg problem

We study the problem of prescribing $sigma_k$-curvature for a conformal metric on the standard sphere $mathbb{S}^n$ with $2 leq k < n/2$ and $n geq 5$ 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 $beta in [2,n)$. We prove among other things the following statements. (1) When $beta>n-2k$, the solution set is compact, has a nonzero total degree counting and is therefore non-empty. (2) When $ beta = 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 $frac{2}{n-2k}le beta < n-2k$, 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 $beta < frac{n-2k}{2}$, there exists $K$ for which there exists a blow-up sequence of solutions with unbounded energy. In this same range of $beta$, there exists also some $K$ for which the solution set is empty.