Hardy–Sobolev critical elliptic equations with boundary singularities

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, µ s (Ω) := inf Ω |∇u| 2 dx; u ∈ H 1 0 (Ω) and Ω |u| 2 * (s) |x| s = 1 when 0 < s < 2, 2 * (s) = 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: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. RésuméContrairement 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,quand 0 < s < 2, 2 * (s) = 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 : x, u) in Ω ⊂ 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.  2004 Elsevier SAS. All rights reserved.

Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities

“…Now we use Proposition 3.1 and the dilation invariance to obtain non-existence results in some domains with the boundary being smooth at 0. The following nonexistence results are similar to those of [27]. …”

“…When p = 2 and is a cone, some existence and non-existence results for solutions of (1.1) were obtained in [22] and [8,9] under the condition α = 0 or β = 0, respectively. More recently, in [27] (1.1) is treated with α = 0, p = 2, 0 ∈ ∂ , and being a more general domain. The results there show that the existence and non-existence of the solutions to (1.1) depend on the principal curvature of ∂ at 0, which is in striking contrast to the case that is a cone.…”

Nonlinear elliptic problems with a singular weight on the boundary

“…If we consider 0 ∈ ∂ , Theorem 1.3 is independent of the geometry of the domain in a neighborhood of the singularity, in contrast to the results in [17] and [18]. A sufficient geometrical condition for existence, as in the critical problem, is far from being clear at the moment.…”

“…with 0 ∈ ∂ , 0 < α < 2 and p * (α) = 2(N − α) N − 2 , has been studied by Ghoussoub-Kang in [17] and by Ghoussoub-Robert in [18], with variational methods. We recall that p * (α) is the critical exponent in the embedding of H 1 0 ( ) in the corresponding weighted space (see for instance [7]).…”

“…In [17] and [18] the authors give sufficient local conditions on the boundary at 0, precisely a condition of negativity of the curvatures and the mean curvature at 0, respectively, for the best constant in this embedding to be attained, which yields existence of a solution to (1.2).…”