Asymptotics for the blow-up boundary solution of the logistic equation with absorption

Abstract

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. To cite this article: F.-C. Cîrstea, V. Rădulescu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).