The axisymmetric σ-Nirenberg problem
The axisymmetric σ-Nirenberg problem
复制标题
轴对称 Ï-Nirenberg 问题
DOI：
10.1016/j.jfa.2021.109198
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发表时间：
2021
影响因子：
1.7
通讯作者：
Wang, Bo
中科院分区：
文献类型：
--
作者：
Li, YanYan;Nguyen, Luc;Wang, Bo
研究了轴对称球面Sn上2≤ k< n/2，n≥ 5的共形度量的σ k曲率的确定问题.利用曲率函数K在南北两极附近的性质，证明了该方程的紧性、非紧性、存在性和不存在性。例如，考虑北极和南极是平坦阶β∈[2，n）的K的局部极大点的情况。除其他外，我们证明以下陈述。(1)当β> n− 2 k时，解集是紧的，有一个非零的总度计数，因此非空。(2)当β= n− 2 k时，存在一个与K相关的显式正常数C（K）。如果C（K）> 1，则解集是紧的，总度计数为非零，因此非空。当C（K）< 1时，解集是紧的，但总度数为0，解集有时为空，有时为非空. (3)当2 n− 2 k≤ β< n− 2 k时，解集是紧的，但总度计数为零，解集有时为空，有时为非空。(4)当β< n− 2 k 2时，存在K，对于K存在一个能量无界的解的爆破序列。在β的这个相同范围内，也存在一些K，其解集是空的。
We study the problem of prescribing σ k-curvature for a conformal metric on the standard sphere S n 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− 2 k, the solution set is compact, has a nonzero total degree counting and is therefore non-empty.(2) When β= n− 2 k, 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 2 n− 2 k≤ β< n− 2 k, 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− 2 k 2, 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.