On second-order effects in the boundary behaviour of large solutions of semilinear elliptic problems

Abstract

Let $D$ be a bounded smooth domain in $\mathbb{R}^N$. It is well known that large solutions of an equation such as $\Delta u= u^p, \;p>1$ in $D$ blow up at the boundary at a rate $\phi(\delta)$ which depends only on $p$. (Here $\delta(x)$ denotes the distance of $x$ to the boundary.) In this paper we consider a secondary effect in the asymptotic behaviour of solutions, namely, the behaviour of $u/\phi(\delta)-1$ as $\delta \to 0$. We derive estimates for this expression, which are valid for a large class of nonlinearities and extend a recent result of Lazer and McKenna ([7]).