Asymptotics for the blow-up boundary solution of the logistic equation with absorption
Florica C. Cîrstea, Vicenţiu D. Rădulescu · Comptes Rendus Mathématique · 2003
Let Ω be a smooth bounded domain in R N . Assume that f ⩾0 is a C 1 -function on [0,∞) such that f ( u )/ u is increasing on (0,+∞). Let a be a real number and let b ⩾0, b ≢0 be a continuous function such that b ≡0 on ∂ Ω . The purpose of this Note is to establish the asymptotic behaviour of the unique positive solution of the logistic problem Δ u + au = b ( x ) f ( u ) in Ω , subject to the singular boundary condition u ( x )→+∞ as dist ( x , ∂ Ω ) → 0 . Our analysis is based on the Karamata regular variation theory.