On the Keller-Osserman conjecture in one dimensional case
type: Item
instance of: scholarly article
title: On the Keller-Osserman conjecture in one dimensional case
zbMATH Open document ID: 1362.34043
author: Linlin Wang
author: Yong-Hong Fan
published in: Boundary Value Problems
publication date: 2016-10-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B40
zbMATH DE Number: 6643017
zbMATH Keywords: Keller-Osserman conjecture
zbMATH Keywords: blow-up boundary value problem
zbMATH Keywords: existence
zbMATH Keywords: qualitative properties
Wikidata QID: Q59466544
reviewed by: S. G. Pyatkov
MaRDI profile type: Publication
full work available at URL: s13661-016-0667-7
OpenAlex ID: W2514525646
cites work: On solutions of δu=f(u)
cites work: On the inequality \(\Delta u \geqq f (u)\)
cites work: Uniqueness of the blow-up boundary solution of logistic equations with absorbtion.
cites work: Boundary blow-up solutions of \(p\)-Laplacian elliptic equations with lower order terms
cites work: Existence and global behavior of positive solution for semilinear problems with boundary blow-up
cites work: Asymptotic estimates of boundary blow-up solutions to the infinity Laplace equations
cites work: Boundary blow-up solutions of \(p\)-Laplacian elliptic equations with a weakly superlinear nonlinearity
cites work: A boundary blow-up problem with a nonlocal reaction
cites work: Boundary blow-up solutions to degenerate elliptic equations with non-monotone inhomogeneous terms
cites work: Explosive nonnegative solutions to two point boundary value problems
cites work: Existence and multiplicity of boundary blow-up nonnegative solutions to two-point boundary value problems
cites work: Existence of explosive solutions to two-point boundary value problems.
cites work: Regularly varying functions
DOI: 10.1186/S13661-016-0667-7
description: scientific article; zbMATH DE number 6643017