On the Keller-Osserman conjecture in one dimensional case
The following blow-up boundary value problem is studied: \[ -u''(t)=f(u(t))\;(t\in (0,1)), \;\lim_{t\to 0} u(t)=\lim_{t\to 1} u(t)=+\infty, \eqno{(1)} \] where \(f(u)\geq 0\) is monotone and increasing for \(u\geq 0\). Let \(g(a)=\int_{a}^{\infty}\Bigl(\int_{0}^{t}f(s)\,ds\Bigr)^{\frac{-1}{2}}\,dt\). It is demonstrated that the problem (1) has a positive solution if and only if \(g(1)<\infty\) in the case of \(f(0)=0\) and if and only if \(g(1)<\infty\) and \(\max_{a>0}g(a)\geq 1/\sqrt{2}\) in the case of \(f(0)>0\). Moreover, some qualitative properties of solutions to the problem (1) are established.
scientific article; zbMATH DE number 6643017