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

Abstract

In this paper we study the Nirenberg problem on standard half spheres $$(\mathbb {S}^n_+,g), \, n \ge 5$$ ( S + n , g ) , n ≥ 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: $$\begin{aligned} (\mathcal {P}) \quad {\left\{ \begin{array}{ll} -\Delta _{g} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 &{}\quad \text{ in } \mathbb {S}^n_+, \\ \frac{\partial u}{\partial \nu }\, =\, 0 &{}\quad \text{ on } \partial \mathbb {S}^n_+. \end{array}\right. } \end{aligned}$$ ( P ) - Δ g u + n ( n - 2 ) 4 u = K u n + 2 n - 2 , u > 0 in S + n , ∂ u ∂ ν = 0 on ∂ S + n . where $$K \in C^3(\mathbb {S}^n_+)$$ K ∈ C 3 ( S + n ) is a positive function. This problem has a variational structure but the related Euler–Lagrange functional $$J_K$$ 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 $$J_K$$ J K in their neighborhood and compute their contribution to the difference of topology between the level sets of $$J_K$$ 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 $$(\mathcal {P})$$ ( P ) on half spheres of dimension $$n \ge 5$$ n ≥ 5 .

Non simple blow ups for the Nirenberg problem on half spheres

“…Remark 1.1 We point out that the assumption that z is not a local maximum is necessary indeed one can prove that at local maxima of K 1 all blow ups are isolated simple. See [1]. Furthermore under the assumption that ∂K/∂ν ≤ 0 at critical point of K 1 , other than local maxima, the existence of non simple blow up can be ruled out.…”

“…Furthermore under the assumption that ∂K/∂ν ≤ 0 at critical point of K 1 , other than local maxima, the existence of non simple blow up can be ruled out. See [1] Actually it turns out that the existence of solutions blowing at the same boundary point with the energy of m boundary bubbles, m ≥ 2 is related to the existence of critical points of the following Kirchhoff-Routh type function:…”

“…In a subsequent paper, we will compute the difference of topology induced by these non simple blow up points and shows that the computation is much more involved compared the simple blow up case. See [2].…”

“…In fact we need only to improve the estimates of (24) and the first part of (20). Observe that, using [3], it holds 2) .…”

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

“…Lemma 3.8 There exists a bounded pseudogradient W 2 3 such that the following holds : There is a constant…”