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On the Keller-Osserman conjecture in one dimensional case
- Lin-Lin Wang^{1} and
- Yong-Hong Fan^{1}Email author
- Received: 23 March 2016
- Accepted: 22 August 2016
- Published: 1 September 2016
Abstract
A sufficient and necessary condition for existence of solution for the boundary blow-up problem in one dimensional case is obtained. This problem can be seen as the Keller-Osserman conjecture, which comes from the study on elliptic equations.
Keywords
- Keller-Osserman conjecture
- boundary blow-up problem
- symmetrical method
1 Introduction
Theorem A
The condition (1.2) plays an important role in the study of the boundary blow-up problem, it was first proposed by Keller [2] and Osserman [3], thus lately, this condition was called the Keller-Osserman condition. Also see [4] and [5]. For later investigations on boundary blow-up problems we refer to [6–10] etc. An interesting problem is the following.
Keller-Osserman conjecture
When \(N=1\), the conclusion of Theorem A is also true.
Remark 1
- (i)
for \(f(u)=-u(\ln u+1)\), there are no solutions,
- (ii)
for \(f(u)=-u [ 2 ( \ln u ) ^{3}+3(\ln u)^{2} ] \), there exist solutions for some \(\lambda>0\).
It is easy to verify that \(f(u)=-u(\ln u+1)\) does not satisfy the Keller-Osserman condition (1.2), but \(f(u)=-u [ 2 ( \ln u ) ^{3}+3(\ln u)^{2} ] \) satisfies this Keller-Osserman condition.
Recently, Wang [12] used the same method as Anuradha’s and generalized the results of Anuradha’s; he obtained a more suitable condition for the existence of solutions for (1.3). The gap between the class of functions that satisfy the necessary condition but not the sufficient condition becomes smaller than that of Anuradha’s, but we still have a distance to Keller-Osserman’s conjecture. Later Zhang [13] considered this problem again. In this paper, we want to study problem (1.3) again and solve this conjecture.
First we introduce the definition of regularly varying which can be found in [14].
Definition 1
Remark
- (a)
Any function \(R\in \mathbb{R} _{q}\) can be written in terms of a slowly varying function. Indeed, set \(R(u)=u^{q}L(u)\), then L varies slowly.
- (b)
For any \(m>0\), \(u^{m}L(u)\rightarrow+\infty\), \(u^{-m}L(u)\rightarrow0\) as \(u\rightarrow+\infty\).
Here come our main results.
Theorem 1
If \(f(0)=0\), then equation (1.4) has a positive solution if and only if the Keller-Osserman condition (1.2) holds true.
Theorem 2
If \(f(0)>0\), then equation (1.4) has a positive solution if and only if the Keller-Osserman condition (1.2) and \(\max _{a\geq0}g(a)\geq\frac{\sqrt{2}}{2}\) hold true, where the function g is defined in (2.3).
Theorem 3
Theorem 4
Assume that \(f(0)=0\), \(\frac{f(u)}{u}\) is increasing on (\(0,+\infty\)) and \(f(t)\in \mathbb{R} _{q}\), \(q>1\). Then equation (1.4) has only one positive solution.
From Theorem 1, we can easily obtain the following.
Corollary 1
If \(f(0)=0\), then for any \(\lambda>0\), equation (1.3) has a positive solution if and only if the Keller-Osserman condition (1.2) holds true.
Corollary 1 generalizes Theorem 3.1, Theorem 3.2 and Theorem 3.6 in [11], also generalizing Theorem 2.1 and Theorem 2.2 in [12].
2 The proof of main results
In order to prove our main results, we need some lemmas, presented below.
The proof of this lemma is trivial, we omit it here.
Lemma 2
If \(u(t)\) is a solution of (1.4), then there exists only one point \(t_{0}\in(0,1)\) such that \(u^{\prime}(t_{0})=0\), in fact, \(t_{0}=\frac{1}{2}\).
The proof of this lemma can be obtained by the generalized Rolle theorem. \(t_{0}=\frac{1}{2}\) can be obtained easily by Lemma 1.
Lemma 3
Proof
The proof is trivial, we omit it. □
Lemma 4
Proof
Lemma 5
If \(f(0)=0\), then equation (2.1) has a solution if and only if the Keller-Osserman condition holds true.
Proof
From (2.8) and Lemma 4, we can easily obtain the following.
Lemma 6
If \(f(0)>0\), then equation (2.1) has a solution if and only if the Keller-Osserman condition and \(\max_{a\geq0}g(a)\geq \frac{\sqrt{2}}{2}\) are all satisfied.
3 Asymptotical stability
In this section, we want to study the behavior of large solutions near the boundary and the uniqueness of the solution. We start with the following comparison theorem.
Lemma 7
(Comparison theorem)
Proof
Remark
In Lemma 7, if we replace \(\frac{1}{2}\) and 1 by any other real numbers a and b (\(a< b\)), then the conclusion is also true.
Corollary 2
(Comparison theorem)
Proof
In Lemma 7, let \(v(t)\equiv0\), then we can immediately get the conclusion. □
Proof of Theorem 3
Proof of Theorem 4
Remark
- (H)There exists \(q>1\) such that for any \(\varepsilon>0\) sufficiently small, the function f satisfies$$ f \bigl[ (1+\varepsilon)s \bigr] \geq(1+\varepsilon)^{q}f(s)\quad \mbox{and}\quad f \bigl[ (1-\varepsilon)s \bigr] \leq(1- \varepsilon)^{q}f(s),\quad s\in (0,+\infty). $$
Declarations
Acknowledgements
The authors thank the reviewers for their insightful and detailed comments. This work is supported by NSF of China (11201213, 11371183), NSF of Shandong Province (ZR2015AM026, ZR2013AM004) and the Project of Shandong Provincial Higher Educational Science and Technology (J15LI07).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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