We study the semilinear elliptic inequality –Δu ≥ φ(δK (x))f(u) in ℝN / K, where φ, f are positive and non-increasing continuous functions. Here K ⊂ ℝN (N ≥ 3) is a compact set with finitely many components, each of which is either the closure of a C2 domain or an isolated point, and δK (x) = dist(x, ∂K). We obtain optimal conditions in terms of φ and f for the existence of C2-positive solutions. Under these conditions we prove the existence of a minimal solution and we investigate its behaviour around ∂K as well as the removability of the (possible) isolated singularities.