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 or on spheres . 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 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.
DOI: 10.1051/cocv/2016048
Mots-clés : Critical points, Yamabe equation, sign-changing solutions
@article{COCV_2016__22_4_939_0, author = {Bahri, Abbas}, title = {Critical points at infinity in {Yamabe} changing-sign equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {939--952}, publisher = {EDP-Sciences}, volume = {22}, number = {4}, year = {2016}, doi = {10.1051/cocv/2016048}, zbl = {1355.35019}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016048/} }
TY - JOUR AU - Bahri, Abbas TI - Critical points at infinity in Yamabe changing-sign equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 939 EP - 952 VL - 22 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016048/ DO - 10.1051/cocv/2016048 LA - en ID - COCV_2016__22_4_939_0 ER -
%0 Journal Article %A Bahri, Abbas %T Critical points at infinity in Yamabe changing-sign equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 939-952 %V 22 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016048/ %R 10.1051/cocv/2016048 %G en %F COCV_2016__22_4_939_0
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/
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
and ,A. Bahri and J.M. Coron, Vers une Theorie des Points Critiques a l’Infini. Seminaire Bony-Sjostrand-Meyer, Expose (VIII) (1984). | Numdam | Zbl
The scalar curvature problem on the standard three-dimensional sphere. J. Funct. Anal. 95 (1991) 106–172. | DOI | Zbl
and ,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
and ,Sur une equation elliptique non linaire avec l’exposant critique de Sobolev. C.R. Acad. Sci. Paris 301 (1985) 345–348. | Zbl
and ,A. Bahri and Y. Xu, Recent Progress in Conformal geometry. Imperial College Press, Advanced Texts in Mathematics. London (2007).
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 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.
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