The Nirenberg problem on high dimensional half spheres: The effect of pinching conditions

TL;DR

This paper investigates the Nirenberg problem on high-dimensional half spheres, addressing the existence of conformal metrics with prescribed scalar curvature under pinching conditions, by analyzing critical points at infinity and extending Morse theory to a non-compact setting.

Contribution

It introduces a Morse theoretical framework for the Nirenberg problem on half spheres, characterizing critical points at infinity and proving existence results under pinching conditions.

Findings

Characterization of critical points at infinity.

Extension of Morse theory to non-compact variational problems.

Existence results for prescribed scalar curvature on half spheres.

Abstract

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 \nu }\, =\, 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 $n \geq 5$.