Hardy-Sobolev critical elliptic equations with boundary singularities

Unlike the non-singular case s = 0, or the case when 0 belongs to the interior of a domain Omega in R-n (n greater than or equal to 3), we show that the value and the attainability of the best Hardy-Sobolev constant on a smooth domain Omega, mu(s)(Omega) :=inf{integral(Omega) \delu\(2) dx; u is an element of H-0(1) (Omega) and integral(Omega) \u\(2*(s))/\x\(s) = 1} when 0<s<2, 2*(s)= 2(n-s)/n-2, and when 0 is on the boundary partial derivativeOmega are closely related to the properties of the curvature of partial derivativeOmega at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form: -Deltau = u(p-1)/\x\(s) + f(x, u) in Omega subset of 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. (C) 2004 Elsevier SAS. All rights reserved.