论文标题
Nirenberg高维半球上的问题：夹紧条件的效果
The Nirenberg problem on high dimensional half spheres: The effect of pinching conditions
论文作者
论文摘要
在本文中，我们研究了标准一半的Nirenberg问题$（\ Mathbb {s}^n _+，g），\，\，n \ geq 5 $，其中包括找到规定的标量曲率曲率和零边界均值曲率的共形度量。此问题等于解决涉及关键Sobolev指数的以下边界值问题：\ begin {equination*}（\ Mathcal {p}）\ quad \ quad \ begin {case} - \ d_ {g} u \，+\，\ frac {n（n-2）} {4} u \，= k \，u^，u^{\ frac {n+2} {n-2} {n-2} {n-2}}}}，\，\，\，u> 0 \ frac {\ partial u} {\partialν} \，= \，0＆\ mbox {on} \ partial \ mathbb {s}^n_+。 \ end {cases} \ end {equation*} 其中$ k \ in c^3（\ mathbb {s}^n _+）$是一个正函数。 这个问题具有变异结构，但是相关的Euler-Lagrange功能$ J_K $缺乏紧凑性。确实，它承认了\ emph {critical {critical}，它是（负）梯度流的非紧凑轨道的\ emph {limits}。通过在\ emph {emph {emph {neighborhighous}中构建适当的\ emph {pseudogradient}，我们表征了这些\ emph {无限}的临界点，与他们相关联，执行\ emph {morse type降低}的函数$ j_k $ j_k $ j_k $ j never untery $ j n eftery $ j never in n e TOPTOLICY $ j_k $ j_k n eftery的范围;莫尔斯（Morse）的理论方法{非紧凑的变分问题}。这种方法用于在各种捏合条件下证明$（\ MATHCAL {p}）$的某些存在结果$ n \ geq 5 $。
In this paper we study the Nirenberg problem on standard half spheres $(\mathbb{S}^n_+,g), \, n \geq 5$, 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: \begin{equation*} (\mathcal{P}) \quad \begin{cases} -\D_{g} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 & \mbox{in } \mathbb{S}^n_+, \frac{\partial u}{\partial ν}\, =\, 0 & \mbox{on } \partial \mathbb{S}^n_+. \end{cases} \end{equation*} 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 \emph{critical points at infinity}, which are \emph{limits} of non compact orbits of the (negative) gradient flow. Through the construction of an appropriate \emph{pseudogradient} in the \emph{neighborhood at infinity}, we characterize these \emph{critical points at infinity}, associate to them an index, perform a \emph{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 \emph{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 $n \geq 5$.
