Critical points at infinity in Yamabe changing-sign equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 4, pp. 939-952.

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 R 3 or on spheres S 3 . 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.

Received:
DOI: 10.1051/cocv/2016048
Classification: 35B38, 37B30
Mots-clés : Critical points, Yamabe equation, sign-changing solutions
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Bahri, Abbas. Critical points at infinity in Yamabe changing-sign equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 4, pp. 939-952. doi : 10.1051/cocv/2016048. http://archive.numdam.org/articles/10.1051/cocv/2016048/

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A. Bahri and J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Commun. Pure Appl. Math. 41 (1988) 253–294. | DOI | Zbl

A. Bahri and J.M. Coron, Sur une equation elliptique non linaire avec l’exposant critique de Sobolev. C.R. Acad. Sci. Paris 301 (1985) 345–348. | Zbl

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