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

Volume 164 / 2021

Philippe Souplet Colloquium Mathematicum 164 (2021), 43-52 MSC: Primary 35J60, 35B08, 35B53; Secondary 35K55, 35B44. DOI: 10.4064/cm8147-1-2020 Published online: 7 August 2020


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 $p\ge 1$, 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 $\varepsilon =0$.


  • Philippe SoupletUniversité Sorbonne Paris Nord, CNRS UMR 7539
    Laboratoire Analyse, Géométrie et Applications
    93430 Villetaneuse, France

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