AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 753857 10.1155/2012/753857 753857 Research Article Asymptotic Behavior of Bifurcation Curve for Sine-Gordon-Type Differential Equation Shibata Tetsutaro Zheng Sining Laboratory of Mathematics, Institute of Engineering Hiroshima University Higashi-Hiroshima 739-8527 Japan hiroshima-u.ac.jp 2012 30 12 2012 2012 23 10 2012 11 12 2012 2012 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.

We consider the nonlinear eigenvalue problems for the equation u(t)+sin u(t)=λu(t), u(t)>0, tI=:(0,1), u(0)=u(1)=0, where λ>0 is a parameter. It is known that for a given ξ>0, there exists a unique solution pair (uξ,λ(ξ))C2(I¯)×+ with uξ=ξ. We establish the precise asymptotic formulas for bifurcation curve λ(ξ) as ξ and ξ0 to see how the oscillation property of sin u has effect on the behavior of λ(ξ). We also establish the precise asymptotic formula for bifurcation curve λ(α)  (α=uλ2) to show the difference between λ(ξ) and λ(α).

1. Introduction

We consider the following nonlinear eigenvalue problem: (1.1)-u′′(t)+sinu(t)=λu(t),  tI=:(0,1),(1.2)u(t)>0,tI,(1.3)u(0)=u(1)=0, 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  that for ξ>0, there exists a continuous function λ=λ(ξ) such that (uξ,λ(ξ))C2(I-)×+ satisfies (1.1)–(1.3) with uξ=ξ. Moreover, the solution set of of (1.1)–(1.3) is given by Γ:={(uξ,λ(ξ))C2(I-)×+;  ξ>0}. Furthermore, it is well known that uξ(t)~ξsinπt for ξ1 and 0<ξ1. Therefore, we have (1.4)λ(ξ)π2        (ξ),(1.5)λ(ξ)π2+1        (ξ0). Equations (1.1)–(1.3) are the special case of the following semilinear equation: (1.6)-u′′(t)+f(u(t))=λu(t),      tI,(1.7)u(t)>0,      tI,(1.8)u(0)=u(1)=0. 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  and the references therein. In particular, if f(u)/u is strictly increasing as u, then we know from  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)=up (p>1) in (1.6), then as ξ (cf. ), (1.9)λ(ξ)=ξp-1+O(e-δξ), where δ>0 is a constant. However, since (sinu)/u is not strictly increasing but oscillating as a function of u0, it is interesting to study whether the oscillation property of sinu 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.

Theorem 1.1.

As ξ, (1.10)λ(ξ)=π2+22πξ-3/2cos(ξ-34π)+22πξ-5/2{-38sin(ξ-34π)-12π2cos(2ξ-14π)+22πξ-5/2+1π2cosξcos(ξ-14π)}+o(ξ-5/2).

The local behavior of λ(ξ) as ξ0 can be obtained formally by the method in . 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 ξ0.

Theorem 1.2.

Let an arbitrary integer N>0 be fixed. Then as ξ0, (1.11)λ=π2+1-18ξ2+1192(1+18π2)ξ4+n=3Nanξ2n+o(ξ2N), 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)=up(p>1) in (1.6). Then we know from  that, for a given α>0, there exists a unique solution pair (uα,λ(α))C2(I-)×+ of (1.6)–(1.8) satisfying uα2=α. Furthermore, λ(α) is an increasing function of α>0 and as α, (1.12)λ(α)=αp-1+C0α(p-1)/2+O(1).

We see from (1.9) and (1.12) the difference between the asymptotic formulas for λ(ξ) and λ(α) when f(u)=up 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 ξ1 and α1.

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-)×+ satisfying uα2=α. Then, it is natural to expect that for tI-, as α, (1.13)uα(t)α2sinπt. Therefore, we expect that uα~2  uα2 for α1. 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 (1.14)Mα:={vH01(I):v2=α}, where v2 is the usual L2-norm of v, α>0 is a parameter, and H01(I) is the usual real Sobolev space. Then consider the following minimizing problem, which depends on α>0: (1.15)Minimize  K(v):=12v22+I(1-cosv(t))dtunder  the  constraint  vMα. Let (1.16)β(α)minvMαK(v). Then by Lagrange multiplier theorem, for a given α>0, there exists a pair (uα,λ(α))Mα×+ which satisfies (1.1)–(1.3) with K(uα)=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)=(uα,λ(α))Mα×+. Then we know from the arguments in [10, 11] that λ(α) is continuous function for 0<α1 and α1. Our next aim is to study precisely the asymptotic behavior of λ(α) as α.

Theorem 1.3.

As α(1.17)λ(α)=π2+23/4π-1/2α-3/2cos(2α-34π)-π-3α-2sin(2α-34π)cos(2α-34π)+21/4π-1/2α-5/2{-38sin(2α-34π)-12π2cos(22α-14π)+21/4π-1/2α-5/2+1π2cos(2α)cos(2α-14π)-14π-5cos3(2α-34π)}+o(α-5/2).

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.

2. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>

In what follows, C denotes various positive constants independent of ξ1. We write λ=λ(ξ) for simplicity. We know from  that if (uξ,λ(ξ))C2(I-)×+ satisfies (1.1)–(1.3), then (2.1)uξ(t)=uξ(1-t),0t1,(2.2)uξ(12)=max0t1uξ(t)=ξ,(2.3)uξ(t)>0,0t<12. By (1.1), for tI-, (2.4)  [uξ′′(t)+λuξ(t)-sin uξ(t)]uξ(t)=0. This implies that for tI-, (2.5)ddt[12uξ(t)2+12λuξ(t)2+cosuξ(t)]=0. By this, (2.2) and putting t=1/2, we obtain (2.6)12uξ(t)2+12λuξ(t)2+cos uξ(t)  constant  =12λξ2+cos ξ. By this and (2.3), for 0t1/2, (2.7)uξ(t)=λ(ξ2-uξ(t)2)+2(cosξ-cosuξ(t)). Then by putting s=uξ(t)/ξ, we obtain (2.8)12=01/2dt=01/2uξ(t)λ(ξ2-uξ(t)2)+2(cosξ-cosuξ(t))dt=1λ0111-s2+2(cosξ-cosξs)/(λξ2)ds=1λ{0111-s2ds+(0111-s2+Bds-0111-s2ds)}=1λ(π2+V), where (2.9)V:=-01B1-s2+B1-s2(1-s2+B+1-s2)ds,(2.10)B:=2λξ2(cosξ-cosξs). We put (2.11)V1=-1λξ201cosξ-cosξs(1-s2)3/2ds,(2.12)V2=V-V1.

Lemma 2.1.

For ξ1(2.13)V1=π21λξ3/2[(1+15128ξ2(1+o(1)))cos(ξ-34π)V1=π21λξ3/2-38ξ(1+o(1))  sin(ξ-34π)].

Proof.

By putting s=sin θ in (2.11), integration by parts and l'Hopital's rule, (2.14)-V1=1λξ20π/21cos2θ(cosξ-cos(ξsin θ))dθ=1λξ20π/2(tan θ)(cos ξ-cos(ξsinθ))dθ=1λξ2limtπ/2[tant(cos ξ-cos(ξsint))]0t-1λξ0π/2tanθcosθsin(ξsinθ)dθ=-1λξ0π/2sinθ sin(ξsinθ)dθ. By [12, page 962], (2.15)0π/2sinθsin(ξsinθ)dθ=π2J1(ξ), where J1(ξ) is Bessel function of the first kind. For ξ1, by [12, page 972], we have (2.16)J1(ξ)=2πξ[(1+15128ξ2(1+o(1)))cos(ξ-34π)-38ξ(1+o(1))sin(ξ-34π)]. 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 (2.17)V1=2-1/2π-3/2ξ-3/2(1+o(1))[(1+15128ξ2(1+o(1)))cos(ξ-34π)V1=2-12π-32ξ-32(1+o(1))-38ξ(1+o(1))sin(ξ-34π)]. After we obtain (2.31) later, then (2.13) will be improved in the form (2.32).

Lemma 2.3.

For ξ1, (2.18)V2=-2-1/2π-7/2(1+o(1))ξ-5/2{12cos(2ξ-14π)-cosξcos(ξ-14π)}+o(ξ-5/2).

Proof.

For ξ1 and 0s1, by mean value theorem, (2.19)|B|Cξ-1(1-s)Cξ-1(1-s2). By this and Lebesgue's convergence theorem, we have (2.20)V2=-2λξ201cosξ-cosξs1-s2×(11-s2+B(1-s2+B+1-s2)-11-s2(21-s2))ds=-(1+o(1))2λξ201cosξ-cosξs1-s2×2(1-s2)-(1-s2+B+1-s21-s2+B)1-s2+B(1-s2+B+1-s2)1-s221-s2ds=-(1+o(1))2λξ201cosξ-cosξs1-s2·1-s2-B-1-s21-s2+B4(1-s2)2ds=-(1+o(1))12λξ201cosξ-cosξs1-s2·(1-s2-B)2-(1-s2)(1-s2+B)(1-s2)2[(1-s2-B)+1-s21-s2+B]ds=34(1+o(1))1λξ201cosξ-cosξs1-s2·(1-s2)B(1-s2)3ds=32(1+o(1))1λ2ξ401(cosξ-cosξs)2(1-s2)5/2ds=32(1+o(1))1λ2ξ40π/2(cosξ-cos (ξ sin θ))2cos4θdθ=32(1+o(1))1λ2ξ4V3, where (2.21)V3:=0π/2(cos ξ-cos(ξ sin θ))2cos4θdθ. We know (2.22)1cos4θdθ=13sin θ(1cos3θ+2cos θ). Taking (2.22) into account and integration by parts in V3, we obtain that (2.23)V3=limθπ/2[13sinθ(1cos3θ+2cos θ)(cos ξ-cos(ξsinθ))2]0θ-23ξ0π/2sinθ(1cos2θ+2)(cos ξ-cos(ξsinθ))sin(ξsinθ)dθ:=13V4-23ξ(V5+V6), where (2.24)V4:=limθπ/2sinθ(1cos3θ+2cosθ)(cosξ-cos(ξsinθ))2(2.25)V5:=0π/2sinθcos2θ(cosξ-cos(ξsinθ))sin(ξsinθ)dθ,(2.26)V6:=20π/2sinθ(cosξ-cos(ξsinθ))sin(ξsinθ)dθ. Then by l'Hopital's rule, (2.27)V4=limθπ/2sinθ(1cos3θ+2cosθ)(cosξ-cos(ξsinθ))2=limθπ/2(1+2cos2θ)(cosθ-cos(ξsinθ))2cos3θ=limθπ/2(cosξ-cos(ξsinθ))2cos3θ=limθπ/2-2ξ(cosξ-cos(ξsinθ))sin(ξsinθ)3cosθsinθ=limθπ/2-2ξsinξ3(cosξ-cos(ξsinθ))cosθsinθ=limθπ/2-2ξ2sinξ3sin(ξsinθ)cosθcos(2θ)=0. We next calculate V5. We know from [12, pages 442 and 972] that for z1, (2.28)0π/2cos(zcosθ)dθ=π2J0(z)=π2(1+o(1))z-1/2cos(z-14π), where J0(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 (2.29)V5=limθπ/2[1cosθ(cosξ-cos(ξsinθ))sin(ξsinθ)]0θ-ξ0π/2(sin2(ξsinθ)+cosξcos(ξsinθ)-cos2(ξsinθ))dθ=ξ0π/2cos(2ξsinθ)dθ-ξcosξ0π/2cos(ξsinθ)dθ=ξ0π/2cos(2ξcosη)dη-ξcosξ0π/2cos(ξcosη)dη=π2ξ1/2(1+o(1))(12cos(2ξ-14π)-cosξcos(ξ-14π)). Clearly, (2.30)V6=O(1). 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 <xref ref-type="statement" rid="thm1.1">1.1</xref>.

By (2.8), Lemmas 2.1 and 2.3, (2.31)λ=π2+4πV+4V2=π2+4πV1+O(ξ-5/2)=π2+O(ξ-3/2). By this and Lemma 2.1, (2.32)V1=π21ξ3/2(π2+O(ξ-3/2))-1×(cos(ξ-34π)-38(1+o(1))ξ-1sin(ξ-34π)+O(ξ-2))=2-1/2π-3/2ξ-3/2(cos(ξ-34π)-38ξ-1sin(ξ-34π))+o(ξ-5/2).

By this, (2.31) and Lemmas 2.1 and 2.3, (2.33)λ=π2+4π(V1+V2)+O(V2)=π2+4π{2-1/2π-3/2ξ-3/2(cos(ξ-34π)-38ξ-1sin(ξ-34π))=π2+4π-2-1/2π-7/2ξ-5/2(12cos(2ξ-14π)-cosξcos(ξ-14π))}+o(ξ-5/2).

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

3. Proof of Theorem <xref ref-type="statement" rid="thm1.2">1.2</xref>

We write λ=λ(ξ) for simplicity. We prove (1.11) by showing the calculation to get a2. The argument to obtain an  (n3) is the same as that to obtain a2. The argument in this section is a variant used in [11, Section 2]. By (2.8) and (2.10), we have (3.1)12=1λ0111-s2+Bds. Since 0<ξ1, by Taylor expansion, for 0s1, we obtain (3.2)cosξ-cosξs=k=1(-1)k(2k)!ξ2k(1-s2k). By this and (3.1), (3.3)λ=20111-s2(1+2λξ211-s2k=1(-1)k(2k)!ξ2k(1-s2k))-1/2ds. 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)1+2λξ211-s2k=1(-1)k(2k)!ξ2k(1-s2k)=1-1λ+112λ1-s41-s2ξ2+o(ξ2). By (3.3), (3.4), and Taylor expansion, (3.5)λ=20111-s2(1-1λ+112λ(1+s2)ξ2+o(ξ2))-1/2ds=2λλ-10111-s2(1-124(λ-1)(1+s2)ξ2+o(ξ2))ds. By this, (1.5) and direct calculation, we obtain (3.6)λ-1=π-116πξ2+o(ξ2). This implies (3.7)λ=π2+1-18ξ2+o(ξ2).

Step 2.

Now we calculate the third term of λ(ξ). First, we note that (3.8)011+s2+s41-s2ds=1516π,01(1+s2)21-s2ds=1916π. By this, (1.5), (3.3), (3.7), Taylor expansion, and the same calculation as that to obtain (3.5), (3.9)λ-1=20111-s2{381-12(112(λ-1)(1+s2)ξ2=20111-s2-1360(λ-1)(1+s2+s4)ξ4)+381144(λ-1)2(1+s2)2ξ4+o(ξ4)}ds=π-116πξ2(1+18π2ξ2+o(ξ2))+1360π1516ξ4+1192π31916ξ4+o(ξ4)=π-116πξ2+1384π(1-58π2)ξ4+o(ξ4). By this, we obtain (1.11) up to the third term. Thus, the proof is complete.

4. Proof of Theorem <xref ref-type="statement" rid="thm1.3">1.3</xref>

In this section, we assume that α1. We write λ=λ(α) for simplicity. We consider the solution pair (λ(α),uα)+×Mα. We obtain from the same argument as that in [10, Theorem 1.2] that (4.1)uα(t)α2sinπt uniformly on [0,1] as α. By this, we have (4.2)uα=2α(1+o(1)). Furthermore, by [13, Lemma 2.4], we see that β(α) is continuous for α>0. By multiplying uα by (1.1) and integration by parts, we obtain (4.3)λ(α)α2=uα22+01uα(t)sinuα(t)dt. By this and (1.16), for α1, (4.4)λ(α)α2=2β(α)+01uα(t)sinuα(t)dt-201(1-cosuα(t))dt. This along with (4.1) implies that λ(α) is continuous for α1.

Lemma 4.1.

For α1, (4.5)uα2=(1-2λ(π4+U))-1α2, where (4.6)U=-011-s2B1-s2+B(1-s2+B+1-s2)ds.

Proof.

By (2.7), (2.10), and putting θ=uα and s=θ/uα, (4.7)uα2-α2=201/2(uα2-uα(t)2)uα(t)λ(uα2-uα(t)2)+2(cosuα-cosuα(t))dt=20uαuα2-θ2λ(uα2-θ2)+2(cosuα-cosθ)dθ=2uα2λ011-s21-s2+Bds=2uα2λ[011-s2ds+01(1-s21-s2+B-1-s2)ds]=2uα2λ(π4+U). Now, the result follows easily from (4.7). Thus, the proof is complete.

Lemma 4.2.

For α1, (4.8)uα=2α-2-3/4π-5/2α-1/2cos(2α-34π)+o(α-1/2).

Proof.

By (2.10) and (4.6), (4.9)|U|Cλuα2|01cosuα-cos(uαs)1-s2ds|C(uα-2). By this, (2.8), Lemma 2.1, and Taylor expansion, (4.10)1-2λ(π4+U)=1-2(π+2V)-1(π4+U)=12-2π(U-V2(1+o(1)))=12+1πV(1+o(1)). By this, (4.5), (2.12), (2.13), (2.18), Taylor expansion, and (4.2), (4.11)uα=(12+1πV(1+o(1)))-1/2α=2(1-1πV(1+o(1)))α=2α-2-3/4π-5/2α-1/2cos(2α-34π)+o(α-1/2). Thus, the proof is complete.

Proof of Theorem <xref ref-type="statement" rid="thm1.3">1.3</xref>.

By Lemma 4.2, we put (4.12)uα=2α+Aα-1/2+o(α-1/2),A=-2-3/4π-5/2cos(2α-34π). Then substitute (4.12) for (1.10) and use Taylor expansion to obtain (4.13)λ=π2+22π(2α+Aα-1/2+o(α-1/2))-3/2cos(2α+Aα-1/2+o(α-1/2)-34π)+22π(2α)-5/2{-38sin(2α-34π)-12π2cos(22α-14π)+22π(2α)-5/2+1π2cos(2α)cos(2α-14π)}+o(α-5/2)=π2+23/4π-1/2α-3/2(1+12Aα-3/2+o(α-3/2))-3/2×{cos(2α-34π)cos(Aα-1/2(1+o(1)))-sin(2α-34π)sin(Aα-1/2(1+o(1)))}+22π(2α)-5/2{-38sin(2α-34π)-12π2cos(22α-14π)+22π(2α)-5/2+1π2cos(2α)cos(2α-14π)}+o(α-5/2)=π2+23/4π-1/2α-3/2(1-322Aα-3/2+o(α-3/2))×{(1-12A2α-1(1+o(1)))cos(2α-34π)-Aα-1/2(1+o(1))sin(2α-34π)}+21/4π-1/2α-5/2{-38sin(2α-34π)-12π2cos(22α-14π)+21/4π-1/2α-5/2+1π2cos(2α)cos(2α-14π)}+o(α-5/2)=π2+23/4π-1/2α-3/2cos(2α-34π)-π-3α-2sin(2α-34π)cos(2α-34π)+21/4π-1/2α-5/2{-38sin(2α-34π)-12π2cos(22α-14π)+21/4π-1/2α-5/2+1π2cos(2α)cos(2α-14π)-14π-5cos3(2α-34π)}+o(α-5/2). 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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