Boundary blow-up in nonlinear elliptic equations of Bieberbach--Rademacher type

Abstract

We establish the uniqueness of the positive solution for equations of the form $-\Delta u=au-b(x)f(u)$ in $\Omega$, $u|\_{\partial\Omega}=\infty$. The special feature is to consider nonlinearities $f$ whose variation at infinity is \emph{not regular} (e.g., $\exp(u)-1$, $\sinh(u)$, $\cosh(u)-1$, $\exp(u)\log(u+1)$, $u^\beta \exp(u^\gamma)$, $\beta\in {\mathbb R}$, $\gamma>0$ or $\exp(\exp(u))-e$) and functions $b\geq 0$ in $\Omega$ vanishing on $\partial\Omega$. The main innovation consists of using Karamata's theory not only in the statement/proof of the main result but also to link the non-regular variation of $f$ at infinity with the blow-up rate of the solution near $\partial\Omega$.