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 $\mathrm{\Delta }u+au=b(x)f(u)$ in a smooth bounded domain $\mathrm{\Omega }$, subject to the singular boundary condition $u=+\mathrm{\infty }$ on $\mathrm{\partial }\mathrm{\Omega }$. The absorption term $f$ is a positive function satisfying the Keller–Osserman condition and such that the mapping $f(u)\mathrm{/}u$ is increasing on $(0,+\mathrm{\infty })$. 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.