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On second-order effects in the boundary behaviour of large solutions of semilinear elliptic problems. (English) Zbl 1042.35535
Let $D$ be a bounded smooth domain in ${ℝ}^{N}$. 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.

##### MSC:
 35J60 Nonlinear elliptic equations 34C99 Qualitative theory of solutions of ODE 35B40 Asymptotic behavior of solutions of PDE