Geodesic
Energy and Morse index of solutions of Yamabe type problems on thin annuli
Journal of the European Mathematical Society, Tome 7 (2005) no. 3, pp. 283-304.

Voir la notice de l'article provenant de la source EMS Press

In this paper we consider the following Yamabe type family of problem (Pε​):−Δuε​=uεn−2n+2​​,uε​>0 in Aε​, uε​=0 on ∂Aε​, where Aε​ is an annulus-shaped domain of Rn, n≥3, which becomes thinner when ε→0. We show that for every solution uε​, the energy ∫Aε​​∣∇uε​∣2, as well as the Morse index tends to infinity as ε→0. Such a result is proved through a fine blow-up analysis of some appropriate scalings of solutions whose limiting profiles are regular as well as singular solutions of some elliptic problem on Rn, a half space or an infinite strip. Our argument involves also a Liouville-type theorem for regular solutions on the infinite strip.
DOI : 10.4171/jems/29
Classification : 35-XX, 58-XX, 00-XX
Keywords: Elliptic PDE, critical Sobolev exponent, blow up analysis, Liouville type theorem 
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     title = {Energy and {Morse} index of solutions of {Yamabe} type problems on thin annuli},
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Mohammed Ben Ayed; Khalil El Mehdi; Mohameden Ould Ahmedou; Filomena Pacella. Energy and Morse index of solutions of Yamabe type problems on thin annuli. Journal of the European Mathematical Society, Tome 7 (2005) no. 3, pp. 283-304. doi : 10.4171/jems/29. http://geodesic.mathdoc.fr/articles/10.4171/jems/29/

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