The axisymmetric sigma(k)-Nirenberg problem

摘要

We study the problem of prescribing sk-curvature for a conformal metric on the standard sphere Snwith 2 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 2n- 2k= 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 < n-2k2, there exists Kfor 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. (C) 2021 Elsevier Inc. All rights reserved.