Abstract and Applied Analysis

Volume 2013 (2013), Article ID 492026, 6 pages

http://dx.doi.org/10.1155/2013/492026

Research Article

Positive Solutions of the One-Dimensional -Laplacian with Nonlinearity Defined on a Finite Interval

Ruyun Ma, Chunjie Xie, and Abubaker Ahmed

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 24 November 2012; Revised 7 February 2013; Accepted 21 February 2013

Academic Editor: Kunquan Lan

Copyright © 2013 Ruyun Ma 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 use the quadrature method to show the existence and multiplicity of positive solutions of the boundary value problems involving one-dimensional -Laplacian , , , where , is a parameter, for some constant , in , and .

1. Introduction and the Main Results

Let be continuous and on any subset of , and let be a continuous function. Wang [1] proved the existence of positive solutions of nonlinear boundary value problems

under the following assumptions:

Since then, the existence and multiplicity of positive solutions of (1) and its generalized forms have been extensively studied via the fixed point theorem in cones. For example, Ge [2] showed a series of results on the existence and multiplicity of solutions of nonlinear ordinary differential equation of second order/higher order subjected with diverse boundary conditions via topological degree and fixed point theorem in cones; Wang [3] use fixed point theorem in cones to study the existence of positive solutions for the one dimensional p-Laplacian. For other recent results along this line, see [4–11] and the bibliographies in [2]. For the special case , beautiful results have been obtained via the quadrature method; see Fink et al. [12], Brown and Budin [13], Addou and Wang [14], Cheng and Shao [15], Karátson and Simon [16], and the references therein.

The nonlinearity that appeared in the above previously papers is assumed to be well defined in or . Of course, natural question is what would happen if is only well defined in a finite interval , where is a positive constant; that is, what would happen if (2) is replaced with the following limit

It is worth remarking that the fixed point theorem in cones method in [1–3] cannot be used to deal with the existence of positive solutions of the problem

under the restriction (3) any more since the appearance of singularity of at . The purpose of this paper is to use the quadrature method to show the existence and multiplicity of positive solutions of (4), in which is a parameter, and satisfies the following assumptions:

 (H1) ; (H2) in ; (H3) .

Let , .

The main result of the paper is the following.

Theorem 1. Let , and let (H1), (H2), and (H3) hold. Then,

(i)if , then there exist , such that (4) has at least two positive solutions for , has at least one positive solution for , and has no positive solution for ;(ii)if , then (4) has at least one positive solution for ;(iii)if , then (4), has at least one positive solution for , where

The proof of our main result is motivated by Laetsch [17] in which the existence and multiplicity of positive solutions of (4) with were studied via the quadrature method. Since then, there are plenty of research papers on the study of exact multiplicity of positive solutions of the -Laplacian problem with general and some more special nonlinearities; see [18, 19] and the references therein. To find the exact number of positive solutions, the nonlinearity needs to satisfy some restrictive conditions, such as the monotonic condition or convex condition …. Our conditions (H1)–(H3) are not strong enough to guarantee the problem exact number of positive solutions.

The rest of the paper is arranged as follows. In Section 2, we state and prove some preliminary results. Finally in Section 3, we give the proof of Theorem 1.

2. Preliminaries

To prove our main results, we will use the uniqueness results due to Reichel and Walter [20] on the initial value problem

where , and .

Lemma 2. Let (H1) and (H2) hold. Then,

(a)(6) with has a unique local solution, and, the extension remains unique as long as ;(b)(6) with and has a unique local solution.(c)(6) with and has a unique local solution .

Proof . (a) It is an immediate consequence of Reichel and Walter [20, Theorem 2].

(b) (H1) implies that f is local Lipschitz continuous. Combining this with the fact that and using [20, (iii) and (v) in the case of Theorem 4], it follows that (6) with and has a unique solution in some neighborhood of .

(c) Define

and consider the auxiliary problem

It follows from [20, (i) in the case of Theorem 4] that (8) has a unique local solution in some neighborhood of , and consequently, (6) with has a unique local solution in some neighborhood of .

Lemma 3. Let (H1) and (H2) hold. Let be a solution of

with . Let be such that . Then, on , on , and on , and

Proof . Since for , it follows from (9) that

This, together with the fact that , implies that is nondecreasing in , and is nonincreasing in .

We claim that

In fact, suppose on the contrary that there exists such that ; then,

is well defined. Moreover,

By Lemma 2 (c),

for some . However, this contradicts the definition of , see (13). Therefore, in . Similarly, we may show that in .

Notice that (H2), (12), and (9) yield that

Now, since is independent of , both and satisfy the initial value problem

From (16) and Lemma 2(a), it follows that both and can be uniquely extended to . Thus, we have from Lemma 2(b) that (17) has a unique solution on , and, accordingly, (10) is true.

Lemma 4. Let (H1) and (H2) hold. Assume that is a positive solution of the problem (9) with and . Let be such that . Then,

(a);(b) is the unique point on which attains its maximum;(c).

Proof . (a) Suppose on the contrary that , and say that ; then, , and

However, this contradicts (12). Therefore, .

Also (b) and (c) can be easily deduced from (16).

3. Proof of the Main Result

To prove Theorem 1, we need the following quadrature method.

Lemma 5. For any , there exists a unique such that

has a positive solution with . Moreover, is a continuous function on .

Proof . By Lemma 4, is a positive solution of (19), if and only if is a positive solution of

Suppose that is a solution of (20), with . Then,

and so

Putting , we obtain

Hence, (if exists) is uniquely determined by .

If , we define by (24) and by (23); it is straightforward to verify that is twice differentiable, satisfies (21), in , and . The continuity of is implied by (24), and this completes the proof.

Lemma 6. Let (H3) hold, and let . Then

Proof . By (H3), there are positive numbers and such that

Thus, if , (24) implies that

as , where

If , then

Lemma 7. Let (H1) and (H2) hold, and let . Then,

Proof . There are positive numbers and such that

Thus, if , (24) implies that

as , where

Lemma 8. Let (H1), (H2), and (H3) hold, and assume that . Then,

(a)if , then ;(b)if , then ;(c)if , then , where

Proof . (a) If , then for any positive constant , there exists such that

Thus, if , (24) implies that

which implies that .

(b) If , then for any , there exists such that

Thus, if , (24) implies that

which implies that .

(c) If , then for any , there exists such that

Thus, if , (24) and the second part of (39) imply that

which implies that

Similarly,

which implies that if , (24) and the first part of (39) imply that

Combining (41) and (43), it follows that

Proof of Theorem 1. (i) It is from Lemma 8(a) that

From Lemma 6 or Lemma 7, we have that

Combining this with (45) and using the facts that and is continuous, it concludes that there exists such that (4) has at least two positive solutions for , has at least one positive solution for , and has no positive solution for .

(ii) It is an immediate consequence of Lemma 6, and Lemma 7, and Lemma 8(b).

(iii) It is an immediate consequence of Lemma 6, and Lemma 7, and Lemma 8(c).

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This work is supported by the NSFC (no.11061030), SRFDP (no.20126203110004), and Gansu Provincial National Science Foundation of China (no. 1208RJZA258).

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