Sharp condition for the Liouville property in a class of nonlinear elliptic inequalities
Volume 164 / 2021
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 , we show that the inequality \Delta u+ u^p \leq \varepsilon in \mathbb R ^n with u(0)=1 admits a radial, positive nonincreasing solution for all \varepsilon \gt 0 if and only if n\ge 2. This solves a problem left open in [Souplet & Tayachi, Colloq. Math. 88 (2001)]. The result stands in contrast with the classical case \varepsilon =0.