## 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 $$\mathbb R^N$$. 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 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.

### MSC:

 35J60 Nonlinear elliptic equations 34C99 Qualitative theory for ordinary differential equations 35B40 Asymptotic behavior of solutions to PDEs