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
DOI:
https://doi.org/10.1090/S1079-6762-03-00108-2
Published electronically:
February 3, 2003
MathSciNet review:
1988868
Full-text PDF Free Access
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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 \[ \Delta _gu + hu = u^{2^\star -1}\] 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.
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Dru1 Druet, O., The best constants problem in Sobolev inequalities, Math. Ann., 314, 327-346, 1999.
Dru2 ---, Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Amer. Math. Soc., 130, 2351-2361, 2002.
Dru3 ---, From one bubble to several bubbles. The low-dimensional case, Preprint, 2002.
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SchZha Schoen, R., and Zhang, D.,Prescribed scalar curvature on the $n$-sphere, Calc. Var. Partial Differential Equations, 4, 1-25, 1996.
Str Struwe, M., A global compactness result for elliptic boundary problems involving limiting nonlinearities, Math. Z., 187, 511-517, 1984.
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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
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