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

Autor: Mohamed Ben Ayed, Mohameden Ould Ahmedou
Rok vydání: 2020
Předmět:
DOI: 10.48550/arxiv.2012.12973
Popis: 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)-Δgu+n(n-2)4u=Kun+2n-2,u>0inS+n,∂u∂ν=0on∂S+n.where$$K \in C^3(\mathbb {S}^n_+)$$K∈C3(S+n)is a positive function. This problem has a variational structure but the related Euler–Lagrange functional$$J_K$$JKlacks compactness. Indeed it admitscritical points at infinity, which arelimitsof non compact orbits of the (negative) gradient flow. Through the construction of an appropriatepseudogradientin theneighborhood at infinity, we characterize thesecritical points at infinity, associate to them an index, perform aMorse type reductionof the functional$$J_K$$JKin their neighborhood and compute their contribution to the difference of topology between the level sets of$$J_K$$JK, hence extending the full Morse theoretical approach to thisnon 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.
Databáze: OpenAIRE
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