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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 $\Bbb R^N$. 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 the authors consider a secondary effect in the asymptotic behaviour of solutions, namely, the behaviour of $u/\phi (\delta ) - 1$ as $\delta \rightarrow 0$. They derive estimates for this expression, which are valid for a large class of nonlinearities and extend a recent result of Lazer and McKenna.

35J60Nonlinear elliptic equations
34C99Qualitative theory of solutions of ODE
35B40Asymptotic behavior of solutions of PDE