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

Abstract

This paper shows the existence and the uniqueness of the positive solution 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 ) ⩾ 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 ) at t = 0 in the special case when there exist H > 0 and p > 1 such that h ( u ) ∼ H u p 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 ) = H u p for all u ⩾ 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].