In this paper we study the Nirenberg problem on standard half spheres \((\mathbb {S}^n_+,g), \, n \ge 5\), which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent:

where \(K \in C^3(\mathbb {S}^n_+)\) is a positive function. This problem has a variational structure but the related Euler–Lagrange functional \(J_K\) lacks compactness. Indeed it admits critical points at infinity, which are limits of non compact orbits of the (negative) gradient flow. Through the construction of an appropriate pseudogradient in the neighborhood at infinity, we characterize these critical points at infinity, associate to them an index, perform a Morse type reduction of the functional \(J_K\) in their neighborhood and compute their contribution to the difference of topology between the level sets of \(J_K\), hence extending the full Morse theoretical approach to this non compact variational problem. Such an approach is used to prove, under various pinching conditions, some existence results for \((\mathcal {P})\) on half spheres of dimension \(n \ge 5\).

在本文中，我们研究了标准半球\((\mathbb {S}^n_+,g), \, n \ge 5\)上的 Nirenberg 问题，其中包括找到指定标量曲率和零边界均值的共形度量边界上的曲率。这个问题相当于解决以下涉及临界 Sobolev 指数的边值问题：

其中\(K \in C^3(\mathbb {S}^n_+)\)是一个正函数。该问题具有变分结构，但相关的欧拉-拉格朗日泛函\(J_K\)缺乏紧凑性。事实上，它承认无穷远的临界点，这是（负）梯度流的非紧凑轨道的极限。通过适当的构造pseudogradient在在无限远处附近，我们表征这些在无穷远处的临界点，准到他们的指标，执行莫尔斯式减速功能的\（J_K \）并计算它们对\(J_K\)的水平集之间拓扑差异的贡献，从而将完整的莫尔斯理论方法扩展到这个非紧致变分问题。这种方法用于证明在各种收缩条件下，\((\mathcal {P})\)在维度为\(n \ge 5\) 的半球上的一些存在结果。