Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line

This paper shows the existence and the uniqueness of the positive solution ℓ(t)ℓ(t) of the singular boundary value problem{u″(t)=f(t)h(u(t)),t>0,u(0)=∞,u(∞)=0, where f   is a continuous non-decreasing function such that f(0)⩾0f(0)⩾0, and h   is a non-negative function satisfying the Keller–Osserman condition. Moreover, it also ascertains the exact blow-up rate of ℓ(t)ℓ(t) at t=0t=0 in the special case when there exist H>0H>0 and p>1p>1 such that h(u)∼Huph(u)∼Hup for sufficiently large u  . Naturally, the blow-up rate of the problem in such a case equals its blow-up rate for the very special, but important, case when h(u)=Huph(u)=Hup for all u⩾0u⩾0. So, our results are substantial improvements of some previous findings of [J. López-Gómez, Uniqueness of large solutions for a class of radially symmetric elliptic equations, in: S. Cano-Casanova, J. López-Gómez, C. Mora-Corral (Eds.), Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, World Scientific, 2005, pp. 75–110] and [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385–439].