- Research Article
- Open Access

# Existence of Positive Solutions of a Singular Nonlinear Boundary Value Problem

- Ruyun Ma
^{1}Email author and - Jiemei Li
^{1, 2}

**Received:**21 May 2010**Accepted:**11 August 2010**Published:**18 August 2010

## Abstract

We are concerned with the existence of positive solutions of singular second-order boundary value problem , , , which is not necessarily linearizable. Here, nonlinearity is allowed to have singularities at . The proof of our main result is based upon topological degree theory and global bifurcation techniques.

## Keywords

- Banach Space
- Global Bifurcation
- Reflexive Banach Space
- Nodal Solution
- Fixed Point Index

## 1. Introduction

by applying Rabinowitz's global bifurcation theorem, where is allowed to have singularities at and is linearizable at as well as at . It is the purpose of this paper to study the existence of positive solutions of (1.1), which is not necessarily linearizable.

Definition 1.1.

A function is said to be an -Carathéodory function if it satisfies the following:

(i)for each , is measurable;

(ii)for a.e. , is continuous;

(iii)for any , there exists such that

In this paper, we will prove the existence of positive solutions of (1.1) by using the global bifurcation techniques under the following assumptions.

(H1) Let be an -Carathéodory function and there exist functions , , , and such that

uniformly for a.e. .

(H2) for a.e. and .

Remark 1.2.

The main tool we will use is the following global bifurcation theorem for problem which is not necessarily linearizable.

Theorem A (Rabinowitz, [8]).

then there exists a continuum (i.e., a closed connected set) of containing , and either

(i) is unbounded in , or

(ii) .

To state our main results, we need the following.

Lemma 1.3 (see [1, Proposition ]).

Moreover, for each , is simple and its eigenfunction has exactly zeros in .

Remark 1.4.

Our main result is the following.

Theorem 1.5.

then (1.1) has at least one positive solution.

Remark 1.6.

For other references related to this topic, see [9–14] and the references therein.

## 2. Preliminary Results

Lemma 2.1 (see [15, Proposition ]).

Then, from Lemma 2.1, is well defined.

Lemma 2.2.

Proof.

This completes the proof.

Remark 2.3.

Lemma 2.4 (see [1, Lemma ]).

is precompact in .

Since for a.e. , Lemma 2.2 yields for . Thus, is a nonnegative solution of (2.19), and the closure of the set of nontrivial solutions of (2.21) in is exactly .

Lemma 2.5.

Moreover, , whenever .

Lemma 2.6.

Let (H1) and (H2) hold. Then the operator is completely continuous.

Proof.

By the Lebesgue dominated convergence theorem, we have that in as . Thus, is continuous.

Let be a bounded set in . Lemma 2.4 together with (2.28) shows that is precompact in . Therefore, is completely continuous.

For , let , let denote the degree of on with respect to .

Lemma 2.7.

Proof.

Suppose to the contrary that there exist sequences and in in , such that for all , then, in .

Thus, . This contradicts .

Corollary 2.8.

For and , .

Proof.

which ends the proof.

Lemma 2.9.

where is the nonnegative eigenfunction corresponding to .

Proof.

where and . Since on and , we have from (2.46) that .

This contradicts (2.47).

Corollary 2.10.

For and , .

Proof.

Now, using Theorem A, we may prove the following.

Proposition 2.11.

Proof.

For fixed with , let us take that , and . It is easy to check that, for , all of the conditions of Theorem A are satisfied. So there exists a connected component of solutions of (2.30) containing , and either

(i) is unbounded, or

(ii) .

By Lemma 2.7, the case (ii) can not occur. Thus, is unbounded bifurcated from in . Furthermore, we have from Lemma 2.7 that for any closed interval , if , then in is impossible. So must be bifurcated from in .

## 3. Proof of the Main Results

Proof of Theorem 1.5.

We note that for all since is the only solution of (2.30) for and .

Case 1.

We divide the proof into two steps.

Step 1.

We show that is bounded.

Let denote the nonnegative eigenfunction corresponding to .

Step 2.

We show that joins to .

So joins to .

Case 2.

.

Again joins to and the result follows.

Remark 3.1.

under the following assumptions:

satisfying has at least one zero in and has no zeros in .

It is worth remarking that (A1)-(A2) imply Condition (1.21) in Theorem 1.5. However, Condition (1.21) is easier to be verified than (A1)-(A2) since and are easily estimated by Rayleigh's Quotient.

*The language of eigenvalue of singular linear eigenvalue problem* did not occur until Asakawa [1] in 2001. The first part of Theorem 1.5 is new.

## Declarations

### Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC 11061030, the Fundamental Research Funds for the Gansu Universities.

## Authors’ Affiliations

## References

- Asakawa H:
**Nonresonant singular two-point boundary value problems.***Nonlinear Analysis: Theory, Methods & Applications*2001,**44**(6):791-809. 10.1016/S0362-546X(99)00308-9MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, O'Regan D:
*Singular Differential and Integral Equations with Applications*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:xii+402.View ArticleMATHGoogle Scholar - O'Regan D:
*Theory of Singular Boundary Value Problems*. World Scientific, River Edge, NJ, USA; 1994:xii+154.View ArticleMATHGoogle Scholar - Habets P, Zanolin F:
**Upper and lower solutions for a generalized Emden-Fowler equation.***Journal of Mathematical Analysis and Applications*1994,**181**(3):684-700. 10.1006/jmaa.1994.1052MathSciNetView ArticleMATHGoogle Scholar - Xu X, Ma J:
**A note on singular nonlinear boundary value problems.***Journal of Mathematical Analysis and Applications*2004,**293**(1):108-124. 10.1016/j.jmaa.2003.12.017MathSciNetView ArticleMATHGoogle Scholar - Yang X:
**Positive solutions for nonlinear singular boundary value problems.***Applied Mathematics and Computation*2002,**130**(2-3):225-234. 10.1016/S0096-3003(01)00046-7MathSciNetView ArticleMATHGoogle Scholar - Ma R:
**Nodal solutions for singular nonlinear eigenvalue problems.***Nonlinear Analysis: Theory, Methods & Applications*2007,**66**(6):1417-1427. 10.1016/j.na.2006.01.028MathSciNetView ArticleMATHGoogle Scholar - Rabinowitz PH:
**Some aspects of nonlinear eigenvalue problems.***The Rocky Mountain Journal of Mathematics*1973,**3:**161-202. 10.1216/RMJ-1973-3-2-161MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, O'Regan D:
*An Introduction to Ordinary Differential Equations, Universitext*. Springer, New York, NY, USA; 2008:xii+321.View ArticleMATHGoogle Scholar - Agarwal RP, O'Regan D:
*Ordinary and Partial Differential Equations, Universitext*. Springer, New York, NY, USA; 2009:xiv+410.MATHGoogle Scholar - Ghergu M, Radulescu VD:
*Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications*.*Volume 37*. Oxford University Press, Oxford, UK; 2008:xvi+298.MATHGoogle Scholar - Kristály A, Radulescu V, Varga C:
*Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and Its Applications, no. 136*. Cambridge University Press, Cambridge, UK; 2010.View ArticleMATHGoogle Scholar - Lomtatidze AG:
**Positive solutions of boundary value problems for second-order ordinary differential equations with singularities.***Differentsial'nye Uravneniya*1987,**23**(10):1685-1692.MathSciNetGoogle Scholar - Kiguradze IT, Shekhter BL:
**Singular boundary-value problems for ordinary second-order differential equations.***Journal of Soviet Mathematics*1988,**43**(2):2340-2417. translation from: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh., vol. 30, pp. 105–201, 1987 10.1007/BF01100361View ArticleMATHGoogle Scholar - Coster CD, Habets P:
*Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering*. 2006.,**205:**MATHGoogle Scholar

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