Let Ω be a smooth bounded domain in R N . Let b ⩾ 0 , b ≢ 0 be a continuous function on Ω ¯ and consider a closed subset D 0 ≠ ∅ of [ b = 0 ] . We study the logistic problem Δ u + a u = b ( x ) f ( u ) in Ω ∖ D 0 , B u = 0 on ∂ Ω, and u = + ∞ on ∂ D 0 , where a is a real number, B denotes either the Dirichlet or the mixed boundary operator, and f ⩾ 0 is a smooth function such that f ( u ) / u is increasing on ( 0 , ∞ ) . In this Note we establish the existence of extremal singular solutions to the above problem, a uniqueness result, and we describe the blow-up at the boundary. To cite this article: F.-C. Cîrstea, V. Rădulescu, C. R. Acad. Sci. Paris, Ser. I 339 (2004).