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Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$, and $\Delta_g = -div_g\nabla$ the Laplace-Beltrami operator. Also let $2^\star$ be the critical Sobolev exponent for the embedding of the Sobolev space $H_1^2(M)$ into Lebesgue spaces, and $h$ a smooth function on $M$. Elliptic equations of critical Sobolev growth like \[\Delta_gu + hu = u^{2^\star-1}\] have been the target of investigation for decades. A very nice $H_1^2$-theory for the asymptotic behaviour of solutions of such an equation is available since the 1980's. In this announcement we present the $C^0$-theory we have recently developed. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of the above equation.

Copyright 2003 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 09 (2003), pp. 19-25
• Publisher Identifier: S 1079-6762(03)00108-2
• 2000 Mathematics Subject Classification. Primary 35J60; Secondary 58J05
• Key words and phrases. Critical elliptic equations, blow-up behaviour, bubbles
• Received by editors November 4, 2002
• Received by editors in revised form December 16, 2002
• Posted on February 3, 2003
• Communicated by Tobias Colding
• Comments (When Available)

Olivier Druet

Département de Mathématiques, Ecole Normale Supérieure de Lyon, 46 allée d'Italie, 69364 Lyon cedex 07, France

E-mail address: Olivier.Druet@umpa.ens-lyon.fr

Emmanuel Hebey

Département de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France

E-mail address: Emmanuel.Hebey@math.u-cergy.fr

Frédéric Robert

Department of Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland

E-mail address: Frederic.Robert@math.ethz.ch