Asymptotic Behavior of Bifurcation Curve for Sine-Gordon-Type Differential Equation
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 .
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  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  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. ), 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 . 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  that, for a given , there exists a unique solution pair of (1.6)–(1.8) satisfying . Furthermore, is an increasing function of and as ,
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
2. Proof of Theorem 1.1
In what follows, denotes various positive constants independent of . We write for simplicity. We know from  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.
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.
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 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
Lemma 4.2. For ,
The author thanks the referee for the helpful suggestions that improved the paper.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, San Diego, Calif, USA, 5th edition, 1994.View at: MathSciNet