Uniqueness of the blow-up boundary solution of logistic equations with absorbtion

Cı̂rstea, Florica-Corina; Rădulescu, Vicenţiu

Comptes Rendus. Mathématique, Tome 335 (2002) no. 5, pp. 447-452.

Soit $\Omega$ un domaine borné et régulier de ${ℝ}^N$. On suppose que f∈C¹[0,∞) est une fonction non-negative telle que f(u)/u soit strictement croissante sur (0,+∞). Soit a un réel et b⩾0, $b/\equiv 0$ une fonction continue sur $\overline{\Omega }$. On étudie l'équation logistique Δu+au=b(x)f(u) sur $\Omega$. Le but de cette Note est de montrer l'unicité de la solution explosant au bord de $\Omega$ dans un contexte général, qui apparaı̂t en théorie des probabilités.

Let $\Omega$ be a smooth bounded domain in ${ℝ}^N$. Assume f∈C¹[0,∞) is a non-negative function such that f(u)/u is increasing on (0,∞). Let a be a real number and let b⩾0, $b/\equiv 0$ be a continuous function such that b≡0 on $\partial \Omega$. We study the logistic equation Δu+au=b(x)f(u) in $\Omega$. The special feature of this work is the uniqueness of positive solutions blowing-up on $\partial \Omega$, in a general setting that arises in probability theory.

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