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 $b\ge 0$ 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 $\partial \Omega$.