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Comptes Rendus Mathématique
Volume 337, n° 1
pages 37-42 (juillet 2003)
Doi : 10.1016/S1631-073X(03)00269-3
Received : 13 May 2003 ;  accepted : 13 May 2003
Higher order energy expansions for some singularly perturbed Neumann problems

Juncheng  Wei a ,  Matthias  Winter b
aDepartment of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong 
bFachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany 

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We consider the following singularly perturbed semilinear elliptic problem:   where   is a bounded smooth domain in  ,   is a small constant and   is a subcritical exponent. Let   be its energy functional, where  . Ni and Takagi proved that for a single boundary spike solution  , the following asymptotic expansion holds   where   is a generic constant,   is the unique local maximum point of   and   is the boundary mean curvature function. In this Note, we obtain the following higher order expansion of  :   where  ,   are generic constants and   is the Ricci scalar curvature at  . In particular  . Applications of this expansion will be given. To cite this article: J. Wei, M. Winter, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

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Nous étudions le problème suivant de perturbations singulières :   où   est un domaine ouvert dans  ,   est une constante petite et   est un exposant souscritique. L'énergie s'écrit alors  , où  . Ni et Takagi montrent que pour une solution   avec une pic sur la frontière du domaine, on a le développement asymptotique suivant :   où   est une constante générique,   est le point unique de maximum local de   et   est la fonction de la courbure moyenne sur la frontière. On établit que :  où  ,   sont les constantes génériques et   est la courbure scalaire de Ricci en  . En particulier  . Nous présentons des applications de ce développement asymptotique. Pour citer cet article : J. Wei, M. Winter, C. R. Acad. Sci. Paris, Ser. I 337 (2003).




© 2003  Académie des sciences@@#104156@@

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