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 0 1 (Ω) and ∫ Ω |u| 2 ∗ (s) |x| s =1 when 0 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: − Δ u= u p−1 |x| s +f(x,u) in Ω⊂ 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.