- Research Article
- Open Access
Existence of Positive Solutions of a Singular Nonlinear Boundary Value Problem
© R. Ma and J. Li. 2010
- Received: 21 May 2010
- Accepted: 11 August 2010
- Published: 18 August 2010
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.
- Banach Space
- Global Bifurcation
- Reflexive Banach Space
- Nodal Solution
- Fixed Point Index
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.
In this paper, we will prove the existence of positive solutions of (1.1) by using the global bifurcation techniques under the following assumptions.
The main tool we will use is the following global bifurcation theorem for problem which is not necessarily linearizable.
Theorem A (Rabinowitz, ).
To state our main results, we need the following.
Lemma 1.3 (see [1, Proposition ]).
Our main result is the following.
then (1.1) has at least one positive solution.
Lemma 2.1 (see [15, Proposition ]).
This completes the proof.
Lemma 2.4 (see [1, Lemma ]).
which ends the proof.
This contradicts (2.47).
Now, using Theorem A, we may prove the following.
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
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 .
Proof of Theorem 1.5.
We divide the proof into two steps.
under the following assumptions:
The language of eigenvalue of singular linear eigenvalue problem did not occur until Asakawa  in 2001. The first part of Theorem 1.5 is new.
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.
- 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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