On the prescribed scalar curvature on 3-half spheres: Multiplicity results and Morse inequalities at infinity
type: Item
instance of: scholarly article
title: On the prescribed scalar curvature on 3-half spheres: Multiplicity results and Morse inequalities at infinity
zbMATH Open document ID: 1159.58007
author: Mohamed Ben Ayed
author: Mohameden Ould Ahmedou
published in: Discrete and Continuous Dynamical Systems
publication date: 2009-02-19T00:00:00Z
review text: This paper deals with a generalization of the scalar curvature problem on closed Riemann manifolds. The authors are interested in the existence of a smooth positive solution in the case of a nonlinear boundary condition. The main analytic difficulties of the problem considered in the present paper are due to the presence of critical exponents. In such a case, since the Sobolev embeddings are not compact, the associated energy functional fails to satisfy the Palais-Smale compactness condition. By means of a Morse-type lemma at infinity, the authors argue that noncompact orbits of the gradient can be treated as usual critical points. It is also established that the noncompactness of the flow lines of the gradient flow does occur only if the manifold is conformally equivalent to the standard half-sphere, at least in the three dimensional case. The proofs strongly rely on the critical point theory at infinity, introduced and developed by Abbas Bahri.
reviewed by: Vicenţiu D. Rădulescu
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 5509399
zbMATH Keywords: critical point at infinity
zbMATH Keywords: gradient flow
zbMATH Keywords: Morse inequalities
zbMATH Keywords: prescribed scalar curvature
zbMATH Keywords: topology at infinity
MaRDI profile type: Publication
full work available at URL: dcds.2009.23.655
OpenAlex ID: W2073858342
DOI: 10.3934/DCDS.2009.23.655
description: scientific article; zbMATH DE number 5509399