Sharp condition for the Liouville property in a class of nonlinear elliptic inequalities

Abstract

. 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. 88 (2001)]. The result stands in contrast with the classical case ε = 0 .