Critical points at infinity in Yamabe changing-sign equations

Bahri, Abbas

ESAIM: Control, Optimisation and Calculus of Variations, Tome 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
Mots clés : Critical points, Yamabe equation, sign-changing solutions

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

A. Bahri and J.M. Coron, Vers une Theorie des Points Critiques a l’Infini. Seminaire Bony-Sjostrand-Meyer, Expose (VIII) (1984). | Numdam | Zbl

A. Bahri and J.M. Coron, The scalar curvature problem on the standard three-dimensional sphere. J. Funct. Anal. 95 (1991) 106–172. | DOI | Zbl

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

A. Bahri and Y. Xu, Recent Progress in Conformal geometry. Imperial College Press, Advanced Texts in Mathematics. London (2007).

Y. Chen, On a nonlinear elliptic equation involving the critical Sobolev exponent. Nonlin. Anal. 33 (1998) 41–49. | DOI | Zbl

M.Del Pino, M. Musso, F. Pacard and A.Pistoia, Torus action on $S^n$ and sign changing solutions for conformally invariant equations. Ann. Sci. Norm. Super. Pisa XII, issue 1 (2010). | Numdam | Zbl

E. Sandier and S. Serfaty, Vortex Patterns in Ginzburg-Landau Minimizers. XVIth International Congress Om Mathematical Physics. Held 3-8 August 2009 in Prague, Czech Republic. Edited by Pavel Exner. Doppler Institute, Prague, Czech Republic. World Scientific Publishing (2010) 246–264.