**Abstract** : Let $\Omega$ be a smooth bounded domain in R^n (n ≥ 3) such that 0 ∈ ∂Ω. We consider issues of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem, −∆u − γ u |x|^{-2} − h(x)u = |u|^{2*(s)−2} u |x|^{-s} in Ω, u = 0 on ∂Ω \ {0}, (E) where 0 < s < 2, 2*(s) := 2(n−s) n−2 , γ ∈ R and h ∈ C^0 (Ω). We use sharp blow-up analysis on-possibly high energy-solutions of corresponding subcritical problems to establish, for example, that if γ < n^2/4 − 1 and the principal curvatures of ∂Ω at 0 are non-positive but not all of them vanishing, then Equation (E) has an infinite number of high energy (possibly sign-changing) solutions. This complements results of the first and third authors, who showed in [20] that if γ ≤ n^2/4 − 1/4 and the mean curvature of ∂Ω at 0 is negative, then (E) has a positive least energy solution. On the other hand, the sharp blow-up analysis also allows us to show that if the mean curvature at 0 is nonzero and the mass, when defined, is also nonzero, then there is a surprising stability of regimes where there are no variational positive solutions under C^1-perturbations of the potential h. In particular, and in sharp contrast with the non-singular case (i.e., when γ = s = 0), we prove non-existence of such solutions for (E) in any dimension, whenever Ω is star-shaped and h is close to 0, which include situations not covered by the classical Pohozaev obstruction.