A sharp inequality for Sobolev functions

Pedro M. Girão

doi:10.1016/S1631-073X(02)02215-X

A sharp inequality for Sobolev functions
[Une inégalité dans un espace de Sobolev]

Pedro M. Girão ¹

¹ Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

Comptes Rendus. Mathématique, Volume 334 (2002) no. 2, pp. 105-108.

Résumé
Abstract

Nous considérons N⩾5, a>0, $Ω$ un ouvert borné régulier de $ℝ^{N}$ , $2^{*} = \frac{2 N}{N - 2}$ , $2^{#} = \frac{2 (N - 1)}{N - 2}$ et ||u||²=|∇u|₂²+a|u|₂². Nous prouvons qu'il existe α₀>0 tel que, pour toute fonction $u \in H^{1} (Ω) ⧹ {0}$ ,

\frac{S}{2^{2 / N}} ⩽ \frac{{∥ u ∥}^{2}}{{| u |}_{2^{*}}^{2}} (1 + α_{0} \frac{{| u |}_{2^{#}}^{2^{#}}}{{∥ u ∥ \cdot | u |}_{2^{*}}^{2^{*} / 2}}) .

Cette inégalité implique l'inégalité de Cherrier.

Let N⩾5, a>0, $Ω$ be a smooth bounded domain in $ℝ^{N}$ , $2^{*} = \frac{2 N}{N - 2}$ , $2^{#} = \frac{2 (N - 1)}{N - 2}$ and ‖u‖²=|∇u|₂²+a|u|₂². We prove there exists an α₀>0 such that, for all $u \in H^{1} (Ω) ⧹ {0}$ ,

\frac{S}{2^{2 / N}} ⩽ \frac{{∥ u ∥}^{2}}{{| u |}_{2^{*}}^{2}} (1 + α_{0} \frac{{| u |}_{2^{#}}^{2^{#}}}{{∥ u ∥ \cdot | u |}_{2^{*}}^{2^{*} / 2}}) .

This inequality implies Cherrier's inequality.

Accepté le : 2001-11-15
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02215-X

Affiliations des auteurs :

Pedro M. Girão ¹

¹ Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

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     title = {A sharp inequality for {Sobolev} functions},
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Pedro M. Girão. A sharp inequality for Sobolev functions. Comptes Rendus. Mathématique, Volume 334 (2002) no. 2, pp. 105-108. doi : 10.1016/S1631-073X(02)02215-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02215-X/

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Claudianor O. Alves; Giovany M. Figueiredo; Riccardo Molle Multiple positive bound state solutions for a critical Choquard equation, Discrete and Continuous Dynamical Systems, Volume 41 (2021) no. 10, pp. 4887-4919 | DOI:10.3934/dcds.2021061 | Zbl:1468.81041
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David G. Costa; Pedro M. Girão Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation., Journal of Differential Equations, Volume 188 (2003) no. 1, pp. 164-202 | DOI:10.1016/s0022-0396(02)00070-0 | Zbl:1163.35372

Cité par 4 documents. Sources : zbMATH

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