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Electronic Research Announcements
Electronic Research Announcements
ISSN 1079-6762

A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth


Authors: Olivier Druet, Emmanuel Hebey and Frédéric Robert
Journal: Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 19-25
MSC (2000): Primary 35J60; Secondary 58J05
Published electronically: February 3, 2003
MathSciNet review: 1988868
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Abstract: 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

\begin{displaymath}\Delta_gu + hu = u^{2^\star-1}\end{displaymath}

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.


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Additional Information

Olivier Druet
Affiliation: Département de Mathématiques, Ecole Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon cedex 07, France
Email: Olivier.Druet@umpa.ens-lyon.fr

Emmanuel Hebey
Affiliation: Département de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France
Email: Emmanuel.Hebey@math.u-cergy.fr

Frédéric Robert
Affiliation: Department of Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland
Email: Frederic.Robert@math.ethz.ch

DOI: http://dx.doi.org/10.1090/S1079-6762-03-00108-2
PII: S 1079-6762(03)00108-2
Keywords: Critical elliptic equations, blow-up behaviour, bubbles
Received by editor(s): November 4, 2002
Received by editor(s) in revised form: December 16, 2002
Published electronically: February 3, 2003
Communicated by: Tobias Colding
Article copyright: © Copyright 2003 American Mathematical Society