SHARP CONDITION FOR THE LIOUVILLE PROPERTY IN A CLASS OF NONLINEAR ELLIPTIC INEQUALITIES

Philippe Souplet (1)

1 Université Sorbonne Paris Nord

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.

Keywords: AMS Classification: Primary: 35J60; 35B08; 35B53; Secondary; 35K55; 35B44 Keywords: elliptic inequalities; Liouville property; parabolic inequalities; blowup rate

Domains: Analysis of PDEs [math.AP]

Philippe Souplet. SHARP CONDITION FOR THE LIOUVILLE PROPERTY IN A CLASS OF NONLINEAR ELLIPTIC INEQUALITIES. 2020. ⟨hal-02446681⟩