We study a class of elliptic inequalities which arise in the study of blow-up rate estimates for parabolic problems, and obtain a sharp existence/nonexistence result. Namely, for any p ≥ 1, we show that the inequality ∆u + u p ≤ ε in R n with u(0) = 1 admits a radial, positive nonincreasing solution for all ε > 0, if and only if n ≥ 2. This solves a problem left open in [Souplet & Tayachi, Colloq. Math. 2001]. The result stands in contrast with the classical case ε = 0.