We study the uniqueness and expansion properties of the positive blow-up boundary solution of the logistic equation
△u + au = b(x)f(u) in a smooth bounded domain Ω. The absorbtion term f is a positive function satisfying the Keller–Osserman condition and such that the mapping f(u)/u is increasing on (0,+∞), b is nonnegative, while the values of the real parameter a are related to an appropriate semilinear eigenvalue problem. Our analysis is based on the Karamata regular variation theory.