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Elliptic equations in dimension three: a conjecture of H. Brezis

✍ Scribed by Olivier Druet

Book ID
104447275
Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
52 KB
Volume
334
Category
Article
ISSN
1631-073X

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✦ Synopsis

We study the existence of minimizing solutions for an elliptic equation involving critical Sobolev exponent on domains of the three-dimensional Euclidean space. We solve in particular by the affirmative a conjecture of HaΓ―m Brezis. The similar situation in higher dimensions was completely understood thanks to previous works by H. Brezis and L. Nirenberg. To cite this article: O. Druet, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 643-647. ο›™ 2002 AcadΓ©mie des sciences/Γ‰ditions scientifiques et mΓ©dicales Elsevier SAS

Equations elliptiques en dimension 3 : une conjecture de H. Brezis RΓ©sumΓ©

On Γ©tudie l'existence de solutions minimisantes Γ  une EDP elliptique Γ  croissance de Sobolev critique sur des domaines de l'espace euclidien de dimension trois. On rΓ©sout en particulier une conjecture de H. Brezis sur le sujet. Les questions analogues en dimensions plus grandes Γ©taient parfaitement comprises depuis des travaux de H. Brezis et L. Nirenberg.

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We establish existence of an infinite family of exponentially-decaying non-radial \(C^{2}\) solutions to the equation \(\Delta u+f(u)=0\) on \(\mathbb{R}^{2}\) for a large class of nonlinearities \(f\). These solutions have the form \(u(r, \theta)=e^{\text {imit }} u(r)\), where \(r\) and \(\theta\)