The Hardy–Schrödinger operator with interior singularity: the remaining cases

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem L γ u - λ u = u 2 ∗ ( s ) - 1 | x | s on a smooth bounded domain Ω in R n ( n ≥ 3 ) having the singularity 0 in its interior. Here γ < ( n - 2 ) 2 4 , 0 ≤ s < 2 , 2 ∗ ( s ) : = 2 ( n - s ) n - 2 and 0 ≤ λ < λ 1 ( L γ ) , the latter being the first eigenvalue of the Hardy–Schrödinger operator L γ : = - Δ - γ | x | 2 . There is a threshold λ ∗ ( γ , Ω ) ≥ 0 beyond which the minimal energy is achieved, but below which, it is not. It is well known that λ ∗ ( Ω ) = 0 in higher dimensions, for example if 0 ≤ γ ≤ ( n - 2 ) 2 4 - 1 . Our main objective in this paper is to show that this threshold is strictly positive in “lower dimensions” such as when ( n - 2 ) 2 4 - 1 < γ < ( n - 2 ) 2 4 , to identify the critical dimensions (i.e., when the situation changes), and to characterize it in terms of Ω and γ . If either s > 0 or if γ > 0 , i.e., in the truly singular case , we show that in low dimensions, a solution is guaranteed by the positivity of the “Hardy-singular internal mass” of Ω , a notion that we introduce herein. On the other hand, and just like the case when γ = s = 0 studied by Brezis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983 ) and completed by Druet (Ann Inst H Poincaré Anal Non Linéaire 19(2):125–142, 2002 ), n = 3 is the critical dimension, and the classical positive mass theorem is sufficient for the merely singular case , that is when s = 0 , γ ≤ 0 .