Hardy–Sobolev critical elliptic equations with boundary singularities
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Bibliographic record
Abstract
Unlike the non-singular case s=0 , or the case when 0 belongs to the interior of a domain Ω in ℝ^n ( n⩾3 ), we show that the value and the attainability of the best Hardy–Sobolev constant on a smooth domain Ω , \mu _{s}(\Omega ): = \inf \left\{\int \limits_{\Omega }\left|\nabla u\right|^{2}dx;u∊H_{0}^{1}(\Omega )\text{ and }\int \limits_{\Omega }\frac{\left|u\right|^{2*(s)}}{\left|x\right|^{s}} = 1\right\} when 0<s<2 , 2^∗(s)=\frac{2(n - s)}{n - 2} , and when 0 is on the boundary ∂Ω are closely related to the properties of the curvature of ∂Ω at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form: - \Delta u = \frac{u^{p - 1}}{\left|x\right|^{s}} + f(x,u)\text{ in }\Omega \subset ℝ^{n}, where f is a lower order perturbative term at infinity and f(x,0)=0 . We show that the positivity of the sectional curvature at 0 is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at 0. Résumé Contrairement au cas non-singulier s=0 , ou au cas d’une singularité à l’intérieur d’ un domaine Ω de ℝ^n ( n⩾3 ), on montre que la valeur de la meilleure constante dans l’inégalité de Hardy–Sobolev sur un domaine régulier, \mu _{s}(\Omega ): = \inf \left\{\int \limits_{\Omega }\left|\nabla u\right|^{2}dx;u∊H_{0}^{1}(\Omega )\text{ et }\int \limits_{\Omega }\frac{\left|u\right|^{2*(s)}}{\left|x\right|^{s}} = 1\right\} quand 0<s<2 , 2^∗(s)=\frac{2(n - s)}{n - 2} , et quand 0 appartient à la frontière, est étroitement liée aux propriétés de la courbure de ∂Ω en 0. Ces mêmes conditions sur la courbure sont aussi pertinentes pour l’existence de solutions d’équations à potentiel singulier de la forme : - \Delta u = \frac{u^{p - 1}}{\left|x\right|^{s}} + f(x,u)\text{ in }\Omega \subset ℝ^{n}, où f est une perturbation d’ordre inférieur à l’infini et f(x,0)=0 . On montre que la positivité de la courbure sectionelle est suffisante pour l’existence de solutions des problèmes avec conditions de Dirichlet au bord, tandis que pour les problèmes de Neumann, c’est la positivité de la coubure moyenne qui compte.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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