Critical points at infinity in Yamabe changing-sign equations^∗

Department of Mathematics, Rutgers University – Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.
karine.beauchard@ens-rennes.fr

Received: 6 June 2016
Revised: 7 June 2016

Abstract

In the well-known paper [A. Bahri and J.M. Coron, Commun. Pure Appl. Math. 41 (1988) 253–294], Bahri and Coron develop the theory of critical points at innity and find the solutions of Yamabe problem via Morse theory. This is a very delicate problem because of the lack of compactness caused by the invariance under the conformal group. To obtain the desired results, one needs a careful analysis on the change of the topology of the level sets. In this work, the author continues to use these ideas and give a preliminary study of the topological features for the Yamabe sign-changing variational problem on domains of R³ or on spheres S³. One of key points consists to understand the Morse relations at innity based on the expansion of the energy functional in a neighborhood of innity. In particular, one study weather the relation ∂^∞°∂^∞ = 0 holds where ∂^∞ is the intersection operator at innity. Although I could not understand completely the details, I believe such study is very delicate and the ideas and techniques developed could be also useful in the others context, in particular, some conformal invariant problems like Yang-Mills equations and harmonic maps. I recommend strongly the publication of the paper.