Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach

We study the uniqueness and expansion properties of the positive solution of the logistic equation ∆u + au = b(x)f(u) in a smooth bounded domain Ω, subject to the singular boundary condition u = +∞ on ∂Ω. The absorption term f is a positive function satisfying the Keller–Osserman condition and such that the mapping f(u)/u is increasing on (0,+∞). We assume that b is non-negative, 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.