Existence of Positive Solutions of a Singular Nonlinear Boundary Value Problem

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.

Existence and multiplicity of solutions of singular problem

where is allowed to have singularities at and , have been studied by several authors, see Asakawa [1], Agarwal and ORegan [2], ORegan [3], Habets and Zanolin [4], Xu and Ma [5], Yang [6], and the references therein. The main tools in [1–6] are the method of lower and upper solutions, Leray-Schauder continuation theorem, and the fixed point index theory in cones. Recently, Ma [7] studied the existence of nodal solutions of the singular boundary value problem

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.

Let be Banach space defined by

with the norm

Let

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

for some -Carathéodory functions defined on with

uniformly for a.e. , and

for some -Carathéodory functions defined on with

uniformly for a.e. .

(H2) for a.e. and .

(H3) There exists function such that

Remark 1.2.

If , , , and , then (1.8) implies that

and (1.10) implies that

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

Theorem A (Rabinowitz, [8]).

Let be a real reflexive Banach space. Let be completely continuous, such that , . Let , such that is an isolated solution of the following equation:

for and , where , are not bifurcation points of (1.14). Furthermore, assume that

where is an isolating neighborhood of the trivial solution. Let

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 ]).

Let , then the eigenvalue problem

has a sequence of eigenvalues as follows:

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

Remark 1.4.

Note that and for each . Therefore, there exist constants , such that

Our main result is the following.

Theorem 1.5.

Let (H1)–(H3) hold. Assume that either

or

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.

Lemma 2.1 (see [15, Proposition ]).

For any , the linear problem

has a unique solution and , such that

where

Furthermore, if , then

Let be the Banach space with the norm , and

Let be an operator defined by

where

Then, from Lemma 2.1, is well defined.

Lemma 2.2.

Let and be the first eigenfunction of (1.17). Then for all , one has

Proof.

For any , integrating by parts, we have

Since and , then

Therefore, we only need to prove that

Let us deal with the first equality, the second one can be treated by the same way. Note that , then

which implies that . Then is bounded on . Now, we claim that

Suppose on the contrary that , then for small enough, we have

Therefore,

which is a contradiction. Combining (1.19) with (2.13), we have

This completes the proof.

Remark 2.3.

Under the conditions of Lemma 2.2, for the later convenience, (2.8) is equivalent to

Lemma 2.4 (see [1, Lemma ]).

For every , the subset defined by

is precompact in .

Let be the closure of the set of positive solutions of the problem

We extend the function to an -Carathéodory function defined on by

Then for and a.e. . For , let be an arbitrary solution of the problem

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 .

Let be an -Carathéodory function. Let be the Nemytskii operator associated with the function as follows:

Lemma 2.5.

Let on . Let be such that in , . Then,

Moreover, , whenever .

Let be the Nemytskii operator associated with the function as follows:

Then (2.21), with , is equivalent to the operator equation

that is,

Lemma 2.6.

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

Proof.

From (1.10) in (H1), there exists , such that, for a.e. and ,

Since is an -Carathéodory function, then there exists , such that, for a.e. and , . Therefore, for a.e. and , we have

For convenience, let . We first show that is continuous. Suppose that in as . Clearly, as for a.e. and there exists such that for every . It is easy to see that

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.

In the following, we will apply the Leray-Schauder degree theory mainly to the mapping ,

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

Lemma 2.7.

Let be a compact interval with , then there exists a number with the property

Proof.

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

Set . Then and . Now, from condition (H1), we have the following:

and accordingly

Let and denote the nonnegative eigenfunctions corresponding to and , respectively, then we have from the first inequality in (2.33) that

From Lemma 2.2, we have that

Since in , from (1.12), we have that

By the fact that , we conclude that in . Thus,

Combining this and (2.35) and letting in (2.34), it follows that

and consequently

Similarly, we deduce from second inequality in (2.33) that

Thus, . This contradicts .

Corollary 2.8.

For and , .

Proof.

Lemma 2.7, applied to the interval , guarantees the existence of such that for

This together with Lemma 2.6 implies that for any ,

which ends the proof.

Lemma 2.9.

Suppose , then there exists such that with , ,

where is the nonnegative eigenfunction corresponding to .

Proof.

Suppose on the contrary that there exist and a sequence with and in such that for all . As

and in , it concludes from Lemma 2.2 that

Notice that has a unique decomposition

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

Choose such that

By (H1), there exists , such that

Therefore, for a.e. ,

Since , there exists , such that

and consequently

Applying (2.51), it follows that

Thus,

This contradicts (2.47).

Corollary 2.10.

For and , .

Proof.

Let , where is the number asserted in Lemma 2.9. As is bounded in , there exists such that , . By Lemma 2.9, one has

This together with Lemma 2.6 implies that

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

Proposition 2.11.

is a bifurcation interval from the trivial solution for (2.30). There exists an unbounded component of positive solutions of (2.30) which meets . Moreover,

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 .

Proof of Theorem 1.5.

It is clear that any solution of (2.30) of the form yields solutions of (1.1). We will show that crosses the hyperplane in . To do this, it is enough to show that joins to . Let satisfy

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

Case 1.

consider the following:

In this case, we show that the interval

We divide the proof into two steps.

Step 1.

We show that is bounded.

Since , . From (H3), we have

Let denote the nonnegative eigenfunction corresponding to .

From (3.4), we have

By Lemma 2.2, we have

Thus,

Step 2.

We show that joins to .

From (3.1) and (3.7), we have that . Notice that (2.30) is equivalent to the integral equation

which implies that

We divide the both sides of (3.9) by and set . Since is bounded in , there exist a subsequence of and with and on , such that

relabeling if necessary. Thus, (3.9) yields that

Let and denote the nonnegative eigenfunctions corresponding to and , respectively, then it follows from the second inequality in (3.11) that

and consequently

Similarly, we deduce from the first inequality in (3.11) that

Thus,

So joins to .

Case 2.

.

In this case, if is such that

then

and moreover,

Assume that is bounded, applying a similar argument to that used in Step 2 of Case 1, after taking a subsequence and relabeling if necessary, it follows that

Again joins to and the result follows.

Remark 3.1.

Lomtatidze [13, Theorem ] proved the existence of solutions of singular two-point boundary value problems as follows:

under the following assumptions:

(A1)

where satisfies the following condition:

 (A2) For , let be the solution of singular IVPs

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.

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.

1. Asakawa, H: Nonresonant singular two-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications. 44(6), 791–809 (2001). PubMed Abstract | Publisher Full Text

2. Agarwal, RP, O'Regan, D: Singular Differential and Integral Equations with Applications,p. xii+402. Kluwer Academic Publishers, Dordrecht, The Netherlands (2003)

3. O'Regan, D: Theory of Singular Boundary Value Problems,p. xii+154. World Scientific, River Edge, NJ, USA (1994)

4. Habets, P, Zanolin, F: Upper and lower solutions for a generalized Emden-Fowler equation. Journal of Mathematical Analysis and Applications. 181(3), 684–700 (1994). Publisher Full Text

5. Xu, X, Ma, J: A note on singular nonlinear boundary value problems. Journal of Mathematical Analysis and Applications. 293(1), 108–124 (2004). Publisher Full Text

6. Yang, X: Positive solutions for nonlinear singular boundary value problems. Applied Mathematics and Computation. 130(2-3), 225–234 (2002). Publisher Full Text

7. Ma, R: Nodal solutions for singular nonlinear eigenvalue problems. Nonlinear Analysis: Theory, Methods & Applications. 66(6), 1417–1427 (2007). PubMed Abstract | Publisher Full Text

8. Rabinowitz, PH: Some aspects of nonlinear eigenvalue problems. The Rocky Mountain Journal of Mathematics. 3, 161–202 (1973). Publisher Full Text

9. Agarwal, RP, O'Regan, D: An Introduction to Ordinary Differential Equations, Universitext,p. xii+321. Springer, New York, NY, USA (2008)

10. Agarwal, RP, O'Regan, D: Ordinary and Partial Differential Equations, Universitext,p. xiv+410. Springer, New York, NY, USA (2009)

11. Ghergu, M, Rădulescu, VD: Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications,p. xvi+298. Oxford University Press, Oxford, UK (2008)

12. Kristály, A, Rădulescu, 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)

13. Lomtatidze, AG: Positive solutions of boundary value problems for second-order ordinary differential equations with singularities. Differentsial'nye Uravneniya. 23(10), 1685–1692 (1987)

14. Kiguradze, IT, Shekhter, BL: Singular boundary-value problems for ordinary second-order differential equations. Journal of Soviet Mathematics. 43(2), 2340–2417 translation from: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh., vol. 30, pp. 105–201, 1987 (1988)

    translation from: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh., vol. 30, pp. 105–201, 1987

    Publisher Full Text

15. Coster, CD, Habets, P: Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering. 205, (2006)