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.

We study concentrated positive bound states of the following nonlinear Schrödinger equation:where p is subcritical. We prove that, at a local maximum point x 0 of the potential function V (x) and for arbitrary positive integer K(K > 1), there always exist solutions with K interacting bumps concentrating near x 0 . We also prove that at a nondegenerate local minimum point of V (x) such solutions do not exist.

The cross section of a diblock copolymer in the cylindrical phase is made up of a large number of microdomains of small discs with high concentration of the minority monomers. Often several ring like microdomains appear among the discs. We show that a ring like structure may exist as a stable solution of a free boundary problem derived from the Ohta-Kawasaki theory of diblock copolymers. The existence of such a stable, single ring structure explains why rings exist for a long period of time before they eventually disappear or become discs in a diblock copolymer. A variant of Lyapunov-Schmidt reduction process is carried out that rigorously reduces the free boundary problem to a finite dimensional problem. The finite dimensional problem is solved numerically. A stability criterion on the parameters determines whether the ring solution is stable.

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