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

and Applied Analysis 3 Theorem 1.2. Let an arbitrary integerN > 0 be fixed. Then as ξ → 0, λ π2 1 − 1 8 ξ2 1 192 ( 1 1 8π2 ) ξ4 N ∑ n 3 anξ 2n o ( ξ2N ) , 1.11 where {an} n 3, 4, . . . 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 L2-framework. Suppose that f u u p > 1 in 1.6 . Then we know from 9 that, for a given α > 0, there exists a unique solution pair uα, λ α ∈ C2 I × R of 1.6 – 1.8 satisfying ‖uα‖2 α. Furthermore, λ α is an increasing function of α > 0 and as α → ∞, λ α αp−1 C0α p−1 /2 O 1 . 1.12 We see from 1.9 and 1.12 the difference between the asymptotic formulas for λ ξ and λ α when f u u 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 λ α and λ ξ of 1.1 – 1.3 when ξ 1 and α 1. Now we consider 1.1 – 1.3 in L2-framework. Let α > 0 be a given constant. Assume that there exists a solution pair uα, λ α ∈ C2 I ×R satisfying ‖uα‖2 α. Then, it is natural to expect that for t ∈ I, as α → ∞, uα t α −→ √ 2 sinπt. 1.13 Therefore, we expect that ‖uα‖∞ ∼ √ 2 ‖uα‖2 for α 1. 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 Mα : { v ∈ H1 0 I : ‖v‖2 α } , 1.14 where ‖v‖2 is the usual L2-norm of v, α > 0 is a parameter, andH1 0 I is the usual real Sobolev space. Then consider the following minimizing problem, which depends on α > 0: Minimize K v : 1 2 ∥v′∥22 ∫ I 1 − cosv t dt under the constraint v ∈ Mα. 1.15 Let β α : min v∈Mα K v . 1.16 Then by Lagrange multiplier theorem, for a given α > 0, there exists a pair uα, λ α ∈ Mα × R which satisfies 1.1 – 1.3 with K uα β α . Here, λ α , which is called the variational 4 Abstract and Applied Analysis eigenvalue, is the Lagrange multiplier. By this variational framework, we parameterize the solution u, λ of 1.1 – 1.3 by α, that is, u, λ uα, λ α ∈ Mα × R . Then we know from the arguments in 10, 11 that λ α is continuous function for 0 < α 1 and α 1. Our next aim is to study precisely the asymptotic behavior of λ α as α → ∞. Theorem 1.3. As α → ∞ λ α π2 23/4π−1/2α−3/2 cos (√ 2α − 3 4 π ) − π−3α−2 sin (√ 2α − 3 4 π ) cos (√ 2α − 3 4 π ) 21/4π−1/2α−5/2 { − 8 sin (√ 2α − 3 4 π ) − 1 √ 2π2 cos ( 2 √ 2α − 1 4 π ) 1 π2 cos (√ 2α ) cos (√ 2α − 1 4 π ) − 1 4 π−5cos3 (√ 2α − 3 4 π )}


Introduction
We consider the following nonlinear eigenvalue problem: −u t sin u t λu t , t ∈ I : 0, 1 , where λ > 0 is a parameter. This problem comes from sine-Gordon equation and has been investigated from a view point of bifurcation theory in L ∞ -framework. Indeed, by using implicit function theorem, it has been shown in 1 that for ξ > 0, there exists a continuous function λ λ ξ such that u ξ , λ ξ ∈ C 2 I × R satisfies 1.1 -1.3 with u ξ ∞ ξ. Moreover, the solution set of of 1.1 -1.3 is given by Γ : { u ξ , λ ξ ∈ C 2 I × R ; ξ > 0}. 2 Abstract and Applied Analysis Furthermore, it is well known that u ξ t ∼ ξ sin πt for ξ 1 and 0 < ξ 1. Therefore, we have Equations 1.1 -1.3 are the special case of the following semilinear equation: −u t f u t λu t , t ∈ I, 1.6 u t > 0, t ∈ I, 1.7 The structures of the global behavior of the bifurcation curves of 1.6 -1.8 have been studied by many authors in L ∞ -framework. We refer to 2-6 and the references therein. In particular, if f u /u is strictly increasing as u → ∞, then we know from 3 that λ ξ is also strictly increasing for ξ > 0 and the asymptotic behavior of λ ξ as ξ → ∞ is mainly determined by f ξ /ξ. For example, if f u u p p > 1 in 1.6 , then as ξ → ∞ cf. 7 , where δ > 0 is a constant. However, since sin u /u is not strictly increasing but oscillating as a function of u ≥ 0, it is interesting to study whether the oscillation property of sin u has effect on the asymptotic shape of λ ξ for ξ > 0 or not. Motivated by this, we first establish the precise asymptotic formula for λ ξ as ξ → ∞.

1.10
The local behavior of λ ξ as ξ → 0 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 λ ξ , since it is necessary to solve the equations derived from the asymptotic expansion of λ ξ step by step.
Here, we introduce a simpler way on how to obtain the asymptotic expansion formula for λ ξ as ξ → 0. where {a n } n 3, 4, . . . 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 L 2 -framework. Suppose that f u u p p > 1 in 1.6 . Then we know from 9 that, for a given α > 0, there exists a unique solution pair u α , λ α ∈ C 2 I × R of 1.6 -1.8 satisfying u α 2 α. Furthermore, λ α is an increasing function of α > 0 and as α → ∞, We see from 1.9 and 1.12 the difference between the asymptotic formulas for λ ξ and λ α when f u u p 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 λ α and λ ξ of 1.1 -1.3 when ξ 1 and α 1. Now we consider 1.1 -1.3 in L 2 -framework. Let α > 0 be a given constant. Assume that there exists a solution pair u α , λ α ∈ C 2 I × R satisfying u α 2 α. Then, it is natural to expect that for t ∈ I, as α → ∞, Therefore, we expect that u α ∞ ∼ √ 2 u α 2 for α 1. 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 v 2 is the usual L 2 -norm of v, α > 0 is a parameter, and H 1 0 I is the usual real Sobolev space. Then consider the following minimizing problem, which depends on α > 0: Then by Lagrange multiplier theorem, for a given α > 0, there exists a pair u α , λ α ∈ M α × R which satisfies 1.1 -1.3 with K u α β α . Here, λ α , which is called the variational eigenvalue, is the Lagrange multiplier. By this variational framework, we parameterize the solution u, λ of 1.1 -1.3 by α, that is, u, λ u α , λ α ∈ M α × R . Then we know from the arguments in 10, 11 that λ α is continuous function for 0 < α 1 and α 1. Our next aim is to study precisely the asymptotic behavior of λ α as α → ∞.

1.17
By Theorems 1.1 and 1.3, we clearly understand the difference between λ ξ and λ α . 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.

Proof of Theorem 1.1
In what follows, C denotes various positive constants independent of ξ 1. We write λ λ ξ for simplicity. We know from 1 that if u ξ , λ ξ ∈ C 2 I × R satisfies 1.1 -1.3 , then This implies that for t ∈ I, d dt Abstract and Applied Analysis 5 By this, 2.2 and putting t 1/2, we obtain By this and 2.3 , for 0 ≤ t ≤ 1/2, Then by putting s u ξ t /ξ, we obtain B : 2 λξ 2 cos ξ − cos ξs .

2.10
We put 2.14 By 12, page 962 , where J 1 ξ is Bessel function of the first kind. For ξ 1, by 12, page 972 , we have π .

2.16
By this, 2.14 and 2.15 , we obtain 2.13 . Thus, the proof is complete.

2.17
After we obtain 2.31 later, then 2.13 will be improved in the form 2.32 .

2.18
Abstract and Applied Analysis 7 Proof. For ξ 1 and 0 ≤ s ≤ 1, by mean value theorem,

2.19
By this and Lebesgue's convergence theorem, we have

2.27
We next calculate V 5 . We know from 12, pages 442 and 972 that for z 1,

2.28
Abstract and Applied Analysis 9 where J 0 z is Bessel function. Integration by parts in 2.25 , applying the l'Hopital's rule, putting θ π/2 − η and taking 2.28 into account, we obtain

2.33
By this, we obtain 1.10 . Thus, the proof is complete.

Proof of Theorem 1.2
We write λ λ ξ for simplicity. We prove 1.11 by showing the calculation to get a 2 . The argument to obtain a n n ≥ 3 is the same as that to obtain a 2 . The argument in this section is a variant used in 11, Section 2 . By 2.8 and 2.10 , we have Since 0 < ξ 1, by Taylor expansion, for 0 ≤ s ≤ 1, 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 3.4 Abstract and Applied Analysis 11 By 3.3 , 3.4 , and Taylor expansion, 3.5 By this, 1.5 and direct calculation, we obtain This implies Step 2. Now we calculate the third term of λ ξ . First, we note that By this, we obtain 1.11 up to the third term. Thus, the proof is complete.

Proof of Theorem 1.3
In this section, we assume that α 1. We write λ λ α for simplicity. We consider the solution pair λ α , u α ∈ R ×M α . We obtain from the same argument as that in 10, Theorem 1.2 that This along with 4.1 implies that λ α is continuous for α 1.