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.


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