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International Journal of Differential Equations
Volume 2011, Article ID 383294, 11 pages
http://dx.doi.org/10.1155/2011/383294
Research Article

Positive Solutions of Sturm-Liouville Boundary Value Problems in Presence of Upper and Lower Solutions

1Department of Foundation Courses, Beijing Union University, Beijing 100101, China
2Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received 17 May 2011; Revised 3 September 2011; Accepted 4 September 2011

Academic Editor: A. M. El-Sayed

Copyright © 2011 Li Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a kind of Sturm-Liouville boundary value problems. Using variational techniques combined with the methods of upper-lower solutions, the existence of at least one positive solution is established. Moreover, the upper solution and the lower solution are presented.

1. Introduction

The Sturm-Liouville boundary value problems (for short, BVPs) have received a lot of attention. Many works have been carried out to discuss the existence of at least one solution or multiple solutions. The methods used therein mainly depend on the Leray-Schauder continuation theorem and the Mawhin continuation theorem. Since it is very difficult to give the corresponding Euler functional for Sturm-Liouville BVPs and verify the existence of critical points for the Euler functional, few people consider the existence of solutions for Sturm-Liouville BVPs by critical point theory and many works considered the existence of solutions for Dirichlet BVPs. For example, by a three-critical-point theorem due to Ricceri [1], Bonanno [2] considered Dirichlet problems. Moreover, Afrouzi and Heidarkhani [3] also considered the existence of three solutions for a kind of Dirichlet BVP. By using an appropriate variational framework, the authors [4] considered the existence of positive solutions for the Dirichlet BVP.

In this paper, using variational methods combined with the methods of upper-lower solutions, we consider the positive solutions of the following BVP: where , , , , .

The paper is organized as follows. In the forthcoming section, we give the Euler functional of BVP(1.1) and some basic lemmas. In Section 3, firstly, we give an upper solution of BVP(1.1), then, by the mountain pass lemma, the lower solution of BVP(1.1) is obtained. At last, we show the existence of at least one positive solution of BVP(1.1) based on the upper solution and the lower solution we obtain.

2. Preliminary

The Sobolev space is defined by and is endowed with the norm Then, is a separable and reflexive Banach space [5].

Lemma 2.1 (see [6]). There exists a positive constant such that for any . Here .
For , suppose that , .

Lemma 2.2 (see [7]). If , then, .

Lemma 2.3 (see [8]). For , let ; then, the following properties hold:(i);(ii);(iii);(iv)if uniformly converges to in , then, uniformly converges to ;(v), .In the following, we state the (C ) condition [9].
(C) Every sequence such that the following conditions hold:(i) is bounded;(ii), has a subsequence which converges strongly in .

With a similar proof of Lemma 2.5 [8], one has the following lemma.

Lemma 2.4. If is a critical point of the Euler functional then, is a solution of BVP (1.1). Here, .

Remark 2.5. While , the Euler functional does not include , while , does not include . Hence, in order to be convenient, we assume that .

With little modification to the proof of Theorem 1.4 in [7], we obtain the following.

Remark 2.6. is continuously differentiable on , and, by computation, one has

Definition 2.7. is an upper solution of BVP (1.1) if it satisfies If is not a solution of BVP(1.1), then, is a strict upper solution.

Definition 2.8. is a lower solution of BVP(1.1) if it satisfies If is not a solution of BVP(1.1), then, is a strict lower solution.

Definition 2.9. is said to be a positive solution of BVP(1.1) if , , .

3. Existence of Positive Solutions

Choose and , satisfying , then, where are constants. If we choose , , satisfies , . Moreover, is continuous.

Lemma 3.1. Assume(A1),is satisfied; then, is a strict upper solution of BVP (1.1). Here is some positive constant.

Proof. From (A1), there exists a constant such that Hence, holds for and some large positive constant . Then, that is, , . Obviously, , . Therefore, from Definition 2.7, one has that is a strict upper solution of BVP (1.1).

In the following, we assume the following conditions.(A2)There exist and satisfying for , , for , . (A3)There exists such that , , .

Consider the auxiliary BVP Obviously, the corresponding Euler functional of BVP(3.4) is Obviously, is continuously differentiable on , and, by computation, one has

Lemma 3.2. If is a solution of BVP (3.4), then, .

Proof. Let be a solution of the BVP (3.4). If there exists a subset , , for , then from the BVP (3.4), one has for which contradicts with the assumptions. Moreover, is an absolutely continuous function on , and so the fundamental theorem of calculus ensures the existence of a set such that and is differentiable on , , Therefore, for a.e. , . Since is absolutely continuous on , then, for .

Lemma 3.3. Assume that (A2), (A3) hold; then, BVP(3.4) has a solution , that is, BVP(3.4) has a positive solution .

Proof. Assume that satisfies (i) and (ii) of the (C) condition; then, Here, is some positive constant and , .
First, we show that is bounded. Indeed, from (3.8), we have Choose ; then, Hence, is bounded.
Moreover, For large , Hence, is bounded; then, is uniformly bounded in . By the compactness of the embedding , the sequence has a subsequence, again denoted by for convenience, such that Moreover, Since in , then, , , , , . Moreover,
From and is bounded in , , , , and one has . Hence,
If , from Lemma 2.1, there exists a positive constant such that
If , by Lemma 2.1, the Hölder inequality, and the boundedness of in , one has From (3.17) and (3.18), we have as . Then, , that is, is a Cauchy sequence in . By the completeness of , we have in . From the discussion above, satisfies the (C) condition.
For , , one has that is, is nonincreasing in . Assume that , Hence, Obviously, there exists such that, for , .
On the other hand, for ; then, by Lemma 2.2 Let be some large positive constant. Since , . Moreover, . From the mountain pass lemma [10], possesses a critical value , that is, there exists such that , . Then, from Lemma 2.4, one has that BVP(3.4) has a positive solution and , .

Lemma 3.4. Assume that (A2), (A3) hold; then, BVP(1.1) has a strict lower solution where is some positive constant and is the positive solution of BVP(3.4) one obtains that in Lemma 3.3.

Proof. Assume is small enough such that and , , . Then, Moreover, , . Hence, is a strictly lower solution of BVP(1.1) and , , .

Theorem 3.5. Assume that (A1)–(A3) hold; then, BVP(1.1) has a positive solution and .

Proof. Let . Make a truncation function of as and assume that . Consider the following BVP: The corresponding Euler functional of BVP(3.25) is It is obvious that is weakly lower semicontinuous. Since and are continuous on , is continuous, is coercive. Hence, can attain its infimum in . Without loss of generality, we may assume that attains its infimum in . In the following, we show that is a solution of BVP(1.1).
Assume that has a negative minimum, and let .
If , then, which reaches a contradiction. Similarly, .
If , there exist an open interval and with , , , . Hence, From the discussion above, one has . Similarly, . Then, . Since and are the strictly lower and upper solutions of BVP(1.1), respectively, , . Therefore, we obtain a positive solution of BVP (1.1).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 10671012) and Natural Science Foundation of Beijing Union University (zk201011x).

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