Critical points at infinity in Yamabe changing-sign equations

Bahri, Abbas

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 $\partial {}^{\infty }\partial {}^{\infty }=0$ holds where $\partial {}^{\infty }$ 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.

Classification: 35B38, 37B30
Keywords: Critical points, Yamabe equation, sign-changing solutions

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