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 (t) of the singular boundary value problemwhere 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 (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].

Blow-up rate of the unique solution for a class of one-dimensional equations with a weakly superlinear nonlinearity

“…In [2], Cano-Casanova and López-Gómez studied the existence, uniqueness and the blow-up rate of large solutions to (1.1), when b satisfies Under the conditions (B 1 ) and (i), (1.1) possesses a unique positive solution l(t). Further, suppose that the following conditions are satisfied:…”

Boundary blow-up solutions of p-Laplacian elliptic equations with lower order terms

“…Their blow-up solutions are relatively well understood. References for the case m = 1 include [2,3,[6][7][8][9][10][11]18,19,21,27,31,32,38,39,45], and those for the case m < 1 include [12,13,28,29,41]. We remark here that the original study of this problem (corresponding to p = 2 and m = 1) is developed in papers [8,9], motivated by a problem raised by H. Brezis.…”

Uniqueness results and asymptotic behavior of solutions with boundary blow-up for logistic-type porous media equations

“…For studies of other boundary blow-up problems, we also refer the reader to [1,2,5,7,17,18,[20][21][22]25,29,33] and the references therein. [13,15].…”