Abstract and Applied Analysis

Volume 2012 (2012), Article ID 753857, 16 pages

doi:10.1155/2012/753857

Research Article

Asymptotic Behavior of Bifurcation Curve for Sine-Gordon-Type Differential Equation

Tetsutaro Shibata

Laboratory of Mathematics, Institute of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

Received 23 October 2012; Accepted 11 December 2012

Academic Editor: Sining Zheng

Copyright © 2012 Tetsutaro Shibata. 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 the nonlinear eigenvalue problems for the equation , , , , where is a parameter. It is known that for a given , there exists a unique solution pair with . We establish the precise asymptotic formulas for bifurcation curve as and to see how the oscillation property of has effect on the behavior of . We also establish the precise asymptotic formula for bifurcation curve    to show the difference between and .

1. Introduction

We consider the following nonlinear eigenvalue problem:

where is a parameter. This problem comes from sine-Gordon equation and has been investigated from a view point of bifurcation theory in -framework. Indeed, by using implicit function theorem, it has been shown in [1] that for , there exists a continuous function such that satisfies (1.1)–(1.3) with . Moreover, the solution set of of (1.1)–(1.3) is given by . Furthermore, it is well known that for and . Therefore, we have

Equations (1.1)–(1.3) are the special case of the following semilinear equation:

The structures of the global behavior of the bifurcation curves of (1.6)–(1.8) have been studied by many authors in -framework. We refer to [2–6] and the references therein. In particular, if is strictly increasing as , then we know from [3] that is also strictly increasing for and the asymptotic behavior of as is mainly determined by . For example, if () in (1.6), then as (cf. [7]),

where is a constant. However, since is not strictly increasing but oscillating as a function of , it is interesting to study whether the oscillation property of has effect on the asymptotic shape of for or not.

Motivated by this, we first establish the precise asymptotic formula for as .

Theorem 1.1. As ,

The local behavior of as can be obtained formally by the method in [8]. However, it seems rather hard task to obtain the higher terms of the asymptotic expansion of , since it is necessary to solve the equations derived from the asymptotic expansion of step by step.

Here, we introduce a simpler way on how to obtain the asymptotic expansion formula for as .

Theorem 1.2. Let an arbitrary integer be fixed. Then as ,

where are the constants determined inductively.

Next, since (1.1)–(1.3) is regarded as an eigenvalue problem, we focus our attention on studying the structure of the solution set in -framework. Suppose that in (1.6). Then we know from [9] that, for a given , there exists a unique solution pair of (1.6)–(1.8) satisfying . Furthermore, is an increasing function of and as ,

We see from (1.9) and (1.12) the difference between the asymptotic formulas for and when in (1.6). We refer to [4, 7, 9] for the works in this direction.

Motivated by this, it seems interesting to compare the asymptotic behavior of and of (1.1)–(1.3) when and .

Now we consider (1.1)–(1.3) in -framework. Let be a given constant. Assume that there exists a solution pair satisfying . Then, it is natural to expect that for , as ,

Therefore, we expect that for . To obtain the existence, we apply the variational method to our situation, namely, we consider the constrained minimization problem associated with (1.1)–(1.3). Let

where is the usual -norm of , is a parameter, and is the usual real Sobolev space. Then consider the following minimizing problem, which depends on :

Let

Then by Lagrange multiplier theorem, for a given , there exists a pair which satisfies (1.1)–(1.3) with . Here, , which is called the variational eigenvalue, is the Lagrange multiplier. By this variational framework, we parameterize the solution of (1.1)–(1.3) by , that is, . Then we know from the arguments in [10, 11] that is continuous function for and . Our next aim is to study precisely the asymptotic behavior of as .

Theorem 1.3. As

By Theorems 1.1 and 1.3, we clearly understand the difference between and .

The remainder of this paper is organized as follows. In Section 2, we prove Theorem 1.1. We prove Theorem 1.2 in Section 3. Section 4 is devoted to the proof of Theorem 1.3.

2. Proof of Theorem 1.1

In what follows, denotes various positive constants independent of . We write for simplicity. We know from [1] that if satisfies (1.1)–(1.3), then

By (1.1), for ,

This implies that for ,

By this, (2.2) and putting , we obtain

By this and (2.3), for ,

Then by putting , we obtain

where

We put

Lemma 2.1. For

Proof. By putting in (2.11), integration by parts and l'Hopital's rule,

By [12, page 962],

where is Bessel function of the first kind. For , by [12, page 972], we have

By this, (2.14) and (2.15), we obtain (2.13). Thus, the proof is complete.

Remark 2.2. Taking (1.4) into account, (2.13) is written as

After we obtain (2.31) later, then (2.13) will be improved in the form (2.32).

Lemma 2.3. For ,

Proof. For and , by mean value theorem,

By this and Lebesgue's convergence theorem, we have

where

We know

Taking (2.22) into account and integration by parts in , we obtain that

where

Then by l'Hopital's rule,

We next calculate . We know from [12, pages 442 and 972] that for ,

where is Bessel function. Integration by parts in (2.25), applying the l'Hopital's rule, putting and taking (2.28) into account, we obtain

Clearly,

By (1.4), (2.20), (2.23), (2.27), (2.29), and (2.30), we obtain (2.18). Thus the proof is complete.

Proof of Theorem 1.1. By (2.8), Lemmas 2.1 and 2.3,

By this and Lemma 2.1,

By this, (2.31) and Lemmas 2.1 and 2.3,

By this, we obtain (1.10). Thus, the proof is complete.

3. Proof of Theorem 1.2

We write for simplicity. We prove (1.11) by showing the calculation to get . The argument to obtain is the same as that to obtain . The argument in this section is a variant used in [11, Section 2]. By (2.8) and (2.10), we have

Since , by Taylor expansion, for , we obtain

By this and (3.1),

By using this, direct calculation gives us Theorem 1.2. For completeness, we calculate (1.11) up to the third term.

Step 1. We have

By (3.3), (3.4), and Taylor expansion,

By this, (1.5) and direct calculation, we obtain

This implies

Step 2. Now we calculate the third term of . First, we note that

By this, (1.5), (3.3), (3.7), Taylor expansion, and the same calculation as that to obtain (3.5),

By this, we obtain (1.11) up to the third term. Thus, the proof is complete.

4. Proof of Theorem 1.3

In this section, we assume that . We write for simplicity. We consider the solution pair . We obtain from the same argument as that in [10, Theorem 1.2] that

uniformly on as . By this, we have

Furthermore, by [13, Lemma 2.4], we see that is continuous for . By multiplying by (1.1) and integration by parts, we obtain

By this and (1.16), for ,

This along with (4.1) implies that is continuous for .

Lemma 4.1. For ,

where

Proof. By (2.7), (2.10), and putting and ,

Now, the result follows easily from (4.7). Thus, the proof is complete.

Lemma 4.2. For ,

Proof. By (2.10) and (4.6),

By this, (2.8), Lemma 2.1, and Taylor expansion,

By this, (4.5), (2.12), (2.13), (2.18), Taylor expansion, and (4.2),

Thus, the proof is complete.

Proof of Theorem 1.3. By Lemma 4.2, we put

Then substitute (4.12) for (1.10) and use Taylor expansion to obtain

Thus, we obtain (1.17) and the proof is complete.

Acknowledgment

The author thanks the referee for the helpful suggestions that improved the paper.

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