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

$\varphi \left(\delta \right)$ which depends only on

$p$. (Here

$\delta \left(x\right)$ denotes the distance of

$x$ to the boundary.) In this paper the authors consider a secondary effect in the asymptotic behaviour of solutions, namely, the behaviour of

$u/\varphi \left(\delta \right)-1$ as

$\delta \to 0$. They derive estimates for this expression, which are valid for a large class of nonlinearities and extend a recent result of Lazer and McKenna.