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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Blow-up examples for second order elliptic PDEs of critical Sobolev growth


Authors: Olivier Druet and Emmanuel Hebey
Journal: Trans. Amer. Math. Soc. 357 (2005), 1915-1929
MSC (2000): Primary 58E35
Published electronically: September 2, 2004
MathSciNet review: 2115082
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Abstract: Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$, and $\Delta_g = -div_g\nabla$ be the Laplace-Beltrami operator. Let also $2^\star$ be the critical Sobolev exponent for the embedding of the Sobolev space $H_1^2(M)$ into Lebesgue's spaces, and $h$ be a smooth function on $M$. Elliptic equations of critical Sobolev growth such as

\begin{displaymath}(E)\qquad\qquad\qquad\qquad\qquad\qquad\Delta_gu + hu = u^{2^\star-1} \qquad\qquad\qquad\qquad\qquad\qquad\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 equations has been available since the 1980's. The $C^0$-theory was recently developed by Druet-Hebey-Robert. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of $(E)$. It was used as a key point by Druet to prove compactness results for equations such as $(E)$. An important issue in the field of blow-up analysis, in particular with respect to previous work by Druet and Druet-Hebey-Robert, is to get explicit nontrivial examples of blowing-up sequences of solutions of $(E)$. We present such examples in this article.


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Additional Information

Olivier Druet
Affiliation: Département de Mathématiques - UMPA, 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, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France
Email: Emmanuel.Hebey@math.u-cergy.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03681-5
Received by editor(s): September 5, 2003
Published electronically: September 2, 2004
Article copyright: © Copyright 2004 American Mathematical Society