# A New Approach to the Existence of Quasiperiodic Solutions for Second-Order Asymmetricp-Laplacian Differential Equations

Authors: Xiaoming Wang; Lixia Wang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

Forp≥2 and ϕp(s):=sp-2s, we propose a new estimate approach to study the existence of Aubry-Mather sets and quasiperiodic solutions for the second-order asymmetric p-Laplacian differential equations ϕpx′′+λϕp(x+)-μϕp(x-)=ψ(t,x), where λ and μ are two positive constants satisfying λ-1/p+μ-1/p=2/ω with ω∈R+, ψ(t,x)∈C0,1(Sp×R) is a continuous function, 2πp-periodic in the first argument and continuously differentiable in the second one, x±=max⁡{±x,0}, πp=2π(p-1)1/p/psin⁡π/p, and Sp=R/2πpZ. Using the Aubry-Mather theorem given by Pei, we obtain the existence of Aubry-Mather sets and quasiperiodic solutions under some reasonable conditions. Particularly, the advantage of our approach is that it not only gives a simpler estimation procedure, but also weakens the smoothness assumption on the function ψ(t,x) in the existing literature.


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## Body

## 1. Introduction

In recent years, all kinds of nonlinear dynamic behavior, such as the existence of positive solutions [1–16] and sign-changing solutions [17, 18], the existence and uniqueness of solutions [19–25], the existence and multiplicity results [26–30], and the existence of unbounded solutions[31, 32], have been widely investigated for some nonlinear ordinary differential equations and partial differential equations due to the application in many fields such as physics, mechanics, and the engineering technique fields. In the present paper, we deal with the existence of Aubry-Mather sets and quasiperiodic solutions for the second-order differential equations with a p-Laplacian and an asymmetric nonlinear term(1)ϕpx′′+λϕpx+-μϕpx-=ψt,x,where ϕp(s):=|s|p-2s,p≥2, x±=max{±x,0}, and λ and μ are positive constants satisfying(2)1λ1/p+1μ1/p=2ω,ω∈R+,where ψ(t,x)∈C0,1(Sp×R) is a continuous function, 2πp-periodic in the first argument and continuously differentiable in the second one, where Sp=R/2πpZ. Since the pioneering works of Aubry [33] and Mather [34], the existence of Aubry-Mather sets and quasiperiodic solutions for a variety of differential equations, such as Hamiltonian systems [35–41], and reversible systems [42–44] had been widely investigated due to the application in many fields such as one-dimensional crystal model of solid state physics, differential geometry, and dynamical systems (see [45, 46]).Ifp=2, then πp=π and (1) reduces to the following piecewise linear equation:(3)x′′+λx+-μx-=ψt,x and (2) becomes(4)1λ+1μ=2ω,ω∈R+. The first result is due to Capietto and Liu [38], who proved that the existence of Aubry-Mather sets and quasiperiodic solutions of (3) for some ω∈Q+ in (4), provided that ψ(t,x)=e(t)-ϕ(x), e(t)∈C2(R/2πZ), and the perturbation term ϕ(x)∈C2(R)∩L∞(R) satisfies some growth conditions. Recently, this result was extended to a much weaker smoothness nonlinearity ψ(t,x). In [41], by using the Aubry-Mather theorem generalized by Pei [37], the present author [41] studied the existence of Aubry-Mather sets and quasiperiodic solutions of (3), under the condition that ω∈R+ in (4) and ψ(t,x)∈C0,1(Sp×R) can be allowed to be either a bounded function or an unbounded function, which differs from above existing results.In [39], (3) has been generalized to the following p-Laplacian-like nonlinear differential equation:(5)p-1-1ϕpx′′+λϕpx+-μϕpx-+gx=ht, where p>1, λ and μ are positive constants satisfying (2) with ω=n∈N, g(x)∈C1(R)∩L∞(R), and h(t)∈C2(Sp) is a 2πp-periodic function. They considered the existence of Aubry-Mather sets and quasiperiodic solutions of (5) when g(x) satisfies some further approximate properties at infinity. We notice that in [39], to overcome the barriers of weak smoothness, they made use of exchange of the role of time and angle variables skills and showed the existence of Aubry-Mather sets and quasiperiodic solutions by employing a version of Aubry-Mather theorem obtained by Pei [37]. Moreover, the results in [39] need the smoothness requirement of the perturbation function at least to C2 smooth in t.Now a natural question to ask is whether the smoothness of the functionh(t) in (5) is further reduced; we can also obtain the same results as [13]. In this paper, we will deal with this interesting problem and answer this question in the form of Theorem 1 with more general case (1) than that of (5). Because of the presence of weak smoothness nonlinearity, the methods of seeking the existence of Aubry-Mather sets and quasiperiodic solutions for problems as [38, 39] do not seem to be applicable to (1). This phenomenon provokes some mathematical difficulties, which make the study of (1) particularly interesting. Our approach here is mainly based on the direct proof of the Poincaré map of the transformed system satisfying monotone twist property and is developed from the present author (see the recent papers [41, 44]) but is more subtle than the ones in [38–40]. More efforts have to be made to estimate the monotone twist property for the Poincaré map of the transformed system, but the procedure is a little simpler than those in [38–40]. One important advantage of our approach is that it does not require any high smoothness assumptions on function ψ(t,x). Our results improve and generalize some results of the previous studies [39, 41] to some extent.The main result of this paper is the following theorem.Theorem 1.
Suppose that (2) holds. Moreover, ψ(t,x)∈C0,1(Sp×R) satisfies the following conditions:
( A 1 ) The limit is(6)limx→+∞ψxt,x=0,uniformly  int∈0,2πp.
( A 2 ) There exist constants d≥0,β>0, such that (7)sgnxp-1ψt,x-xψxt,x>β,for∀x≥dand∀t∈0,2πp.
Then there existsε0>0, such that, for any α∈(2ωπp,2ωπp+ε0), (1) possesses an Aubry-Mather type solution zα(t)=(xα(t),xα′(t)) with rotation number α; that is,
(i) ifα=k/m is rational and (k,m)=1, the solutions zαi(t)=zα(t+2πpi),0≤i≤m-1, are mutually unlinked periodic solutions of period m;
(ii) ifα is irrational, the solution zα(t) is either a usual quasiperiodic solution or a generalized one.Remark 2.
A solution is called generalized quasiperiodic one if the closed set(8)Mα≡zα2πpi,i∈Z¯ is Denjoy’s minimal set (see its definition in [47]).Remark 3.
Using the rule of L’Hospital to condition(A1), it can easily be seen that ψ0limx→+∞ψt,xx=0,uniformlyint∈0,2πp.Remark 4.
We noticed that the perturbationsg(x) and h(t) in [39] need to be bounded. But from (A1) and (A2) of this paper, it is easy to verify that the perturbation ψ(t,x) can be either a bounded function or an unbounded function. For example, we can set ψ(t,x) to be a bounded function arctan⁡x·(1+sin2(π/πpt)) or an unbounded function x1/3·(1+cos2(π/πpt)) for t∈[0,2πp] when d=1 and β=π/4-1/2 in Theorem 1. Moreover, positive constant ω=n∈N satisfying (1.2) in [39] has been extended to the case ω∈R+ in this paper. Thus, our situation is more general than the results obtained in [39] for p≥2.Remark 5.
Ifp=2, let us point out that the results in Theorem 1 have covered the conclusions obtained by Wang [41]. Besides, the estimation process in this paper is much more meticulous than that in [41] since the p-Laplacian ϕpx′′ of a function x(t), with p>2, is no longer linear. Therefore, the results obtained in this paper are natural generalizations and refinements of the results obtained in [41].The main idea of our proof is acquired from [39, 41]. The proof of Theorem 1 is based on an Aubry-Mather theorem due to Pei [37]. The rest of this manuscript is as follows. In Section 2, we introduce some action-angle variables transformation to transform system (1) into an equivalent integral Hamiltonian system and then present some growth properties on the corresponding action and angle variables functions. In Section 3, we provide some crucial estimates by some lemmas which say that the Poincaré mapping of the new system is monotone twist around the infinity. At last, Section 4 gives the proof of Theorem 1 by using Pei’s Aubry-Mather theorem [37].


## 2. Preliminaries

### 2.1. The Action and Angle Variables

LetSp(t)=sinpt be the solution of(9)ϕpu′′+ϕpu=0satisfying the initial condition u(0)=0,u′(0)=1. Then it follows from [48] that Sp(t)=sinpt is a 2πp-periodic C2 odd function with sinp(πp-t)=sinpt, for t∈[0,πp/2], and sinp(2πp-t)=-sinpt for t∈[πp,2πp]. Moreover, for t∈(0,πp/2),Sp(t)>0,Sp′(t)>0 and Sp:[0,πp/2]→[0,(p-1)1/p] can be implicitly given by (10)∫0sinptds1-sp/p-11/p=t.Introducing a new variabley=-ϕpx′, then (9) is equivalent to the planar system(11)x′=-ϕqy,y′=ϕpx,where q is the conjugate exponent of p:p-1+q-1=1. Letting (x,y)=(Cp(t),Sp(t)) be the unique solution of (11) satisfying (Cp(0),Sp(0))=(1,0), then the functions Cp(t) and Sp(t) are much similar to cosine and sine. It follows from [49] that Cp(t)∈C2 and Sp(t)∈C1 are 2πp-periodic, and for ∀n∈Z, Cp(t)=0 iff t=πp/2+nπp, and Sp(t)=0 iff t=nπp. Moreover, Cp′(t)=-ϕq(Sp(t)) and Sp′(t)=ϕp(Cp(t)), and 1/pCpt1/p+1/qSpt1/q≡1/p.Now we consider (1). Set y=-ϕp(x′) in (1); then (1) can be rewritten as a planar system(12)x′=-ϕqy,y′=λϕpx+-μϕpx--ψt,x,where q=p/p-1 is the conjugate exponent of p.Lemma 6.
Forp≥2 and for any (x0,y0)∈R2, t0∈R, the solution (13)zt=xt,t0,x0,y0,yt,t0,x0,y0of (12) satisfying the initial condition z(t0)=(x0,y0) is unique and exists on the whole t-axis.Proof.
The proof of uniqueness can be established similarly to the proof of Proposition 2 in [50]; the global existence result can be acquired similarly to Lemma 3.1 in [51].Let(C(t),S(t)) be the solution of the following homogeneous system:(14)x′=-ϕqy,y′=λϕpx+-μϕpx-,x0=1,y0=0.Then, by using (2) and direct computation, one obtains the following.Lemma 7.
(i) BothC(t)∈C2 and S(t)∈C1 are 2πp/ω-periodic functions, and C(t) can be given by (15)Ct=Cpλ1/pt,0≤t≤πp2λ1/p,Cpμ1/pt-πp2λ1/p+πp2μ1/p,πp2λ1/p≤t≤πpω.
(ii)C′(t)=-ϕq(S(t)) and S′(t)=λϕp(C(t)+)-μϕp(C(t)-).
(iii)1/qStq+1/p(λCt+p+μCt-p)=λ/p.Now we introduce an action-angle variables transformation by the mappingΨ:Sp×(0,+∞)→R2∖{(0,0)}, where (x,y)=Ψ(θ,I) defined by the formula(16)x=γI1/pCθω,y=γI1/qSθω,where γ=λ-1ωp is a constant. This transformation is said to be a generalized symplectic transformation because its Jacobian is equal to 1.


### 2.2. Some Properties on Action and Angle Variables Functions

Under the transformationΨ and using Lemma 7 (iii), (12) is changed into(17)θ˙=Φ1t,θ,I,I˙=Φ2t,θ,I,where Φ1(t,θ,I)=ω+x(θ,I)ψ(t,x(θ,I))/pI,Φ2(t,θ,I)=ϕq(y(θ,I))ψ(t,x(θ,I))/ω.We notice that the relation between (17) and (12) is that if θ(t)=θ(t;θ0,I0),I(t)=I(t;θ0,I0) are the solutions of (17) with the initial value condition θ(0)=θ0,I(0)=I0, then (18)xt=xt;θ0,I0=xθt;θ0,I0,It;θ0,I0=γIt;θ0,I01/pCθt;θ0,I0ωand(19)yt=yt;θ0,I0=yθt;θ0,I0,It;θ0,I0=γIt;θ0,I01/qSθt;θ0,I0ωare the solutions of (12) with initial data x(0)=x(0;θ0,I0),y(0)=y(0;θ0,I0). By Lemma 6, (17) has a unique solution for I0>0 and θ0∈R. Moreover, this solution has continuous derivatives with respect to initial data θ0 and I0.For notional convenience, hereinafter, we writex,y,θ,I instead of x(θ(t;θ0,I0),I(t;θ0,I0)), y(θ(t;θ0,I0),I(t;θ0,I0)),θ(t;θ0,I0),I(t;θ0,I0), respectively.Firstly, by some simple calculations, we have the following.Lemma 8.
(i)∂x/∂I=x/pI,∂y/∂I=y/qI, ∂x/∂θ=-1/ωϕq(y),∂y/∂θ=λϕp(x+)-μϕp(x-)/ω.
(ii)|C(t)|≤max{1,(λ/μ)1/p}:=C∞,St≤(qλ/p)1/q:=S∞.Now we are concerned with the growth estimates with regard toI(t;θ0,I0) and θ(t;θ0,I0).Lemma 9.
The limit(20)limI0→+∞It;θ0,I0=+∞holds uniformly on t∈[0,2πp].Proof.
In view ofψ0 and (16), there exist constants D>0,K>0, such that (21)I′t=ϕqyθ,Iψt,xθ,Iω≤DIt+K,∀I≠0.
Then, by the Gronwall inequality, one has(22)e-2πDI0-KD1-e-2πD≤It≤e2πDI0+KDe2πD-1for all t∈[0,2πp].
So, by (22), I(t;θ0,I0)→+∞ as I0→+∞ uniformly for t∈[0,2πp].According to (22), it is easy to see the following.Corollary 10.
∀   θ 0 ∈ R and ∀t∈[0,2πp], there exist constants ρ2>ρ1>0 and I¯>0, such that (23)ρ1I0≤It;θ0,I0≤ρ2I0when I0≥I¯.Lemma 11.
∀   θ 0 ∈ R and ∀t∈[0,2πp], there exists constant I¯>0, such that (24)ω2≤θ′t;θ0,I0≤2ωif I0≥I¯.Proof.
Sinceψ0 holds, then, for every ε>0, there exists M=M(ε)>0, such that (25)ψt,x≤εxif |x|≥M and ∀t∈[0,2πp]. Hence,(26)dθdt=ω+xψt,xpI≥ω-εx2pI.Thus, by using action-angle variables transformation (16) and Lemma 9, there exists I1¯>0 such that dθ/dt≥ω/2 if I0≥I1¯.
For ifx≤M, we assume that ψt,x≤ψ∞, where ψ∞=max⁡ψt,x:t∈0,2πp,x≤M, and then (27)dθdt=ω+xψt,xpI≥ω-ψ∞xpI.
So, by (16), Lemma 8 (ii), and Lemma 9, there exists a constant I2¯>0, such that dθ/dt≥ω/2 if I0≥I2¯.
If we chooseI¯=max⁡{I1¯,I2¯}, then I0≥I¯ implies dθ/dt≥ω/2.
Exploiting the same arguments, one can show that the inequality on the right side of (i) holds.


## 2.1. The Action and Angle Variables

LetSp(t)=sinpt be the solution of(9)ϕpu′′+ϕpu=0satisfying the initial condition u(0)=0,u′(0)=1. Then it follows from [48] that Sp(t)=sinpt is a 2πp-periodic C2 odd function with sinp(πp-t)=sinpt, for t∈[0,πp/2], and sinp(2πp-t)=-sinpt for t∈[πp,2πp]. Moreover, for t∈(0,πp/2),Sp(t)>0,Sp′(t)>0 and Sp:[0,πp/2]→[0,(p-1)1/p] can be implicitly given by (10)∫0sinptds1-sp/p-11/p=t.Introducing a new variabley=-ϕpx′, then (9) is equivalent to the planar system(11)x′=-ϕqy,y′=ϕpx,where q is the conjugate exponent of p:p-1+q-1=1. Letting (x,y)=(Cp(t),Sp(t)) be the unique solution of (11) satisfying (Cp(0),Sp(0))=(1,0), then the functions Cp(t) and Sp(t) are much similar to cosine and sine. It follows from [49] that Cp(t)∈C2 and Sp(t)∈C1 are 2πp-periodic, and for ∀n∈Z, Cp(t)=0 iff t=πp/2+nπp, and Sp(t)=0 iff t=nπp. Moreover, Cp′(t)=-ϕq(Sp(t)) and Sp′(t)=ϕp(Cp(t)), and 1/pCpt1/p+1/qSpt1/q≡1/p.Now we consider (1). Set y=-ϕp(x′) in (1); then (1) can be rewritten as a planar system(12)x′=-ϕqy,y′=λϕpx+-μϕpx--ψt,x,where q=p/p-1 is the conjugate exponent of p.Lemma 6.
Forp≥2 and for any (x0,y0)∈R2, t0∈R, the solution (13)zt=xt,t0,x0,y0,yt,t0,x0,y0of (12) satisfying the initial condition z(t0)=(x0,y0) is unique and exists on the whole t-axis.Proof.
The proof of uniqueness can be established similarly to the proof of Proposition 2 in [50]; the global existence result can be acquired similarly to Lemma 3.1 in [51].Let(C(t),S(t)) be the solution of the following homogeneous system:(14)x′=-ϕqy,y′=λϕpx+-μϕpx-,x0=1,y0=0.Then, by using (2) and direct computation, one obtains the following.Lemma 7.
(i) BothC(t)∈C2 and S(t)∈C1 are 2πp/ω-periodic functions, and C(t) can be given by (15)Ct=Cpλ1/pt,0≤t≤πp2λ1/p,Cpμ1/pt-πp2λ1/p+πp2μ1/p,πp2λ1/p≤t≤πpω.
(ii)C′(t)=-ϕq(S(t)) and S′(t)=λϕp(C(t)+)-μϕp(C(t)-).
(iii)1/qStq+1/p(λCt+p+μCt-p)=λ/p.Now we introduce an action-angle variables transformation by the mappingΨ:Sp×(0,+∞)→R2∖{(0,0)}, where (x,y)=Ψ(θ,I) defined by the formula(16)x=γI1/pCθω,y=γI1/qSθω,where γ=λ-1ωp is a constant. This transformation is said to be a generalized symplectic transformation because its Jacobian is equal to 1.


## 2.2. Some Properties on Action and Angle Variables Functions

Under the transformationΨ and using Lemma 7 (iii), (12) is changed into(17)θ˙=Φ1t,θ,I,I˙=Φ2t,θ,I,where Φ1(t,θ,I)=ω+x(θ,I)ψ(t,x(θ,I))/pI,Φ2(t,θ,I)=ϕq(y(θ,I))ψ(t,x(θ,I))/ω.We notice that the relation between (17) and (12) is that if θ(t)=θ(t;θ0,I0),I(t)=I(t;θ0,I0) are the solutions of (17) with the initial value condition θ(0)=θ0,I(0)=I0, then (18)xt=xt;θ0,I0=xθt;θ0,I0,It;θ0,I0=γIt;θ0,I01/pCθt;θ0,I0ωand(19)yt=yt;θ0,I0=yθt;θ0,I0,It;θ0,I0=γIt;θ0,I01/qSθt;θ0,I0ωare the solutions of (12) with initial data x(0)=x(0;θ0,I0),y(0)=y(0;θ0,I0). By Lemma 6, (17) has a unique solution for I0>0 and θ0∈R. Moreover, this solution has continuous derivatives with respect to initial data θ0 and I0.For notional convenience, hereinafter, we writex,y,θ,I instead of x(θ(t;θ0,I0),I(t;θ0,I0)), y(θ(t;θ0,I0),I(t;θ0,I0)),θ(t;θ0,I0),I(t;θ0,I0), respectively.Firstly, by some simple calculations, we have the following.Lemma 8.
(i)∂x/∂I=x/pI,∂y/∂I=y/qI, ∂x/∂θ=-1/ωϕq(y),∂y/∂θ=λϕp(x+)-μϕp(x-)/ω.
(ii)|C(t)|≤max{1,(λ/μ)1/p}:=C∞,St≤(qλ/p)1/q:=S∞.Now we are concerned with the growth estimates with regard toI(t;θ0,I0) and θ(t;θ0,I0).Lemma 9.
The limit(20)limI0→+∞It;θ0,I0=+∞holds uniformly on t∈[0,2πp].Proof.
In view ofψ0 and (16), there exist constants D>0,K>0, such that (21)I′t=ϕqyθ,Iψt,xθ,Iω≤DIt+K,∀I≠0.
Then, by the Gronwall inequality, one has(22)e-2πDI0-KD1-e-2πD≤It≤e2πDI0+KDe2πD-1for all t∈[0,2πp].
So, by (22), I(t;θ0,I0)→+∞ as I0→+∞ uniformly for t∈[0,2πp].According to (22), it is easy to see the following.Corollary 10.
∀   θ 0 ∈ R and ∀t∈[0,2πp], there exist constants ρ2>ρ1>0 and I¯>0, such that (23)ρ1I0≤It;θ0,I0≤ρ2I0when I0≥I¯.Lemma 11.
∀   θ 0 ∈ R and ∀t∈[0,2πp], there exists constant I¯>0, such that (24)ω2≤θ′t;θ0,I0≤2ωif I0≥I¯.Proof.
Sinceψ0 holds, then, for every ε>0, there exists M=M(ε)>0, such that (25)ψt,x≤εxif |x|≥M and ∀t∈[0,2πp]. Hence,(26)dθdt=ω+xψt,xpI≥ω-εx2pI.Thus, by using action-angle variables transformation (16) and Lemma 9, there exists I1¯>0 such that dθ/dt≥ω/2 if I0≥I1¯.
For ifx≤M, we assume that ψt,x≤ψ∞, where ψ∞=max⁡ψt,x:t∈0,2πp,x≤M, and then (27)dθdt=ω+xψt,xpI≥ω-ψ∞xpI.
So, by (16), Lemma 8 (ii), and Lemma 9, there exists a constant I2¯>0, such that dθ/dt≥ω/2 if I0≥I2¯.
If we chooseI¯=max⁡{I1¯,I2¯}, then I0≥I¯ implies dθ/dt≥ω/2.
Exploiting the same arguments, one can show that the inequality on the right side of (i) holds.


## 3. Twist Property and Proof of Theorem1

Let the Poincaré mappingP of equation (17) be (28)P:θ0,I0⟼θ2πp,θ0,I0,I2πp,θ0,I0.In order to apply the Aubry-Mather theorem developed by Pei [37], we only need to show that the Poincaré mapping P is a monotone twist map around the infinity; that is, it is enough to show ∂θ(2π;θ0,I0)/∂I0<0 if I0≫1. In the following we are going to give its detailed proofs by some lemmas.Similarly, for notional convenience, hereinafter, we also writex,y,θ,I instead of x(θ(t;θ0,I0),I(t;θ0,I0)), y(θ(t;θ0,I0),I(t;θ0,I0)),θ(t;θ0,I0),I(t;θ0,I0), respectively.Lemma 12.
The following convergences hold uniformly ont∈[0,2πp]:(i)
xψ(t,x)/I→0;x2ψx(t,x)/I→0, as I0→+∞.(ii)
ϕq(y)ψ(t,x)/I→0;ϕq(y)xψx(t,x)/I→0, as I0→+∞.(iii)
xψ2t,xϕq′(y)ϕp(x)/I2→0;xψ(t,x)ψx(t,x)ϕq2(y)/I2→0; x2ψxt,xψt,xϕq′(y)ϕp(x)/I2→0; x2ψx2(t,x)ϕq2(y)/I2→0, as I0→+∞.Proof.
If(A1) and ψ0 hold, then to each ε>0 there corresponds a positive number M=M(ε)>0, such that (29)ψxt,x≤ε2γ2/pand (30)ψt,x≤ε2γ2/pxwhen x≥M and t∈[0,2πp], where γ=λ-1ωp is a constant given in (16).
DenoteV1(ε)=max⁡{ψt,x:t∈[0,2πp],x≤M}, V2(ε)=max⁡ψxt,x:t∈0,2πp,x≤M.
(i) By action-angle variables transformation (16) and Lemma 8 (ii) and p≥2, we have (31)xψt,xI≤MV1εI+εx22γ2/pI≤MV1εI+ε2I1-2/p;x2ψxt,xI≤M2V2εI+εx22γ2/pI≤M2V2εI+ε2I1-2/p.
Then, givenI¯>0, choosing I0 so that I0≥I¯, by using Corollary 10, provided (32)It≥max⁡2MV1εε;2M2V2εε;1,we have (33)xψt,xI≤ε2+ε2≤ε;x2ψxt,xI≤ε2+ε2≤ε.
Sinceε>0 is arbitrary, the proof of (i) is complete.
For (ii), observe thatp≥2, and, combining (16) and Lemma 8 (ii), one has(34)ϕqyψt,xI≤εyq-1x2γ2/pI+V1εyq-1I≤ε2C∞S∞q-1I2/p-1+V1εγ1/pS∞q-1I-1/q;ϕqyxψxt,xI≤εyq-1x2γ2/pI+Myq-1V2εI≤ε2C∞S∞q-1I2/p-1+Mγ1/pV2εS∞q-1I-1/q.
Then, givenI¯>0, choosing I0 so that I0≥I¯, by using Corollary 10, provided (35)I1/qt≥max⁡2V1εS∞q-1γ1/pε;2MV2εS∞q-1γ1/pε;1, we have (36)ϕqyψt,xI<σε2+ε2;ϕqyxψxt,xI<σε2+ε2, where σ=C∞S∞q-1. Since ε>0 is arbitrary, (ii) is proved.
(iii) Setη1=qC∞p+2S∞q-2, η2=qMp+2S∞q-2γ1-2/q, ν1=C∞2S∞2(q-1), ν2=MV2(ε)γ2/pS∞2(q-1). In view of (16), Lemma 7 (ii), and p≥2, we can get(37)xψ2t,xϕq′yϕpxI2=q-1xψ2t,xϕpxyq-2I2≤qxp+2yq-2ε24γ4/pI2+qMp+2yq-2I2≤ε24qI4/p-2C∞p+2S∞q-2+qMp+2S∞q-2γ1-2/qI1+2/q≤ε2η12I4/p-2+η22I1+2/q;xψt,xψxt,xϕq2yI2≤y2q-1x2ε24γ4/pI2+MV1εV2εy2q-1I2≤ε24I4/p-2C∞2S∞2q-1+MV1εV2εγ2/pS∞2q-1I2/q≤ε2ν1I4/p-22+V1εν22I2/q;x2ψxt,xψt,xϕq′yϕpxI2=q-1x2ψt,xψxt,xϕpxyq-2I2≤qxp+2yq-2ε24γ4/pI2+qMp+1V1εV2εyq-2I2≤ε24qI4/p-2C∞p+2S∞q-2+qMp+2V1εV2εS∞q-2γ1-2/qI1+2/q≤ε2η12I4/p-2+V1εV2εη22I1+2/q;x2ψx2t,xϕq2yI2≤x2y2q-1ε24γ4/pI2+M2V2εy2q-1I2≤ε24I4/p-2C∞2S∞2q-1+M2V2εγ2/pS∞2q-1I2/q≤ε2ν1I4/p-22+Mν22I2/q.
Then, givenI¯>0, choosing I0 so that I0≥I¯, by using Corollary 10, provided (38)I1/qt≥max⁡η2ε1/q+2;V1εν2ε1/2;V1εV2εη2ε1/q+2;Mν2ε1/2;1, we have (39)xψ2t,xϕq′yϕpxI2≤η1ε22+ε2;xψt,xψxt,xϕq2yI2≤ν1ε22+ε2;x2ψxt,xψt,xϕq′yϕpxI2≤η1ε22+ε2;x2ψx2t,xϕq2yI2≤ν1ε22+ε2.
Sinceε>0 is arbitrary, the proof of (iii) is finished.∀   t ∈ [ 0,2 π p ], set (40)b1t=∂Φ1∂I=-xp-1ψt,x-xψxt,xp2I2,b2t=∂Φ1∂θ=-ϕqyωpIψt,x+xψxt,x,b3t=∂Φ2∂θ=λϕpx+-μϕpx-ϕq′yψt,xω2-ϕq2yψxt,xω2.As a result of Lemma9, Corollary 10, and Lemma 12, we have the following.Lemma 13.
∀   t , s ∈ [ 0,2 π p ], the following conclusions hold:(i)
b1(t)=o(1/I0), as I0→+∞.(ii)
b2(t)=o(1), as I0→+∞.(iii)
b1(t)·b3(s)=o(1), as I0→+∞.
Let us consider the variational equation of (17) with respect to the initial value I0. One can verify that (41)θ˙I0=b1t∂I∂I0+b2t∂θ∂I0,I˙I0=-b2t∂I∂I0+b3t∂θ∂I0.Lemma 14.
For allt∈(0,2πp], I0→+∞, one has(i)
θI0(t;θ0,I0)→0;(ii)
II0(t;θ0,I0)=1+o(1);(iii)
θθ0(t;θ0,I0)=1+o(1).Proof.
From variational equations (41) and Lemma 13, one has(42)θI0t=e∫0tb2sds·∫0te-∫0sb2tdtb1s·II0sds=1+o1∫0tb1s·II0sds,II0t=e-∫0tb2sds·1+∫0te∫0sb2tdtb3s·θI0sds=1+o1+1+o1∫0tb3s·∫0sb1t·II0tdtds=1+o1+o1∫0t∫0sII0tdtds,
and here we have usedθI0(0)=0 and II0(0)=1.
Hence, for allt∈(0,2πp], I0→+∞, we have II0(t)=1+o(1) and θI0(t)=(1+o(1))∫0t‍b1(s)ds→0. Thus, (i) and (ii) are proved.
To prove (iii), we consider the variational equation of (17) about θ0; one can get (43)θ˙θ0=b1t∂I∂θ0+b2t∂θ∂θ0,I˙θ0=-b2t∂I∂θ0+b3t∂θ∂θ0.
By using a similar argument in (ii), we can also show thatθθ0(t;θ0,I0)=1+o(1), ∀t∈(0,2πp], as I0→+∞. This completes the proof of Lemma 14.Next, we will develop an estimate of upper bound and lower bound forb1(t).Lemma 15.
Letd≥0 satisfy (A2).
(i) Ifxt;θ0,I0≤d for all t∈[0,2πp], then there exists a constant Ed>0, such that b1t≤Ed/I2(t).
(ii) Ifxt;θ0,I0≥d for all t∈[0,2πp], then there exists a constant Fd>0, such that b1t≥Fd/I2(t). Moreover, if xt;θ0,I0≥d, then b1(t)<0.Proof.
(i) Ifxt;θ0,I0≤d, take Md=max|x|≤d,t∈[0,2πp]p-1ψx,t-xψxx,t. Then (44)b1t=-xp-1ψt,x-xψxt,xp2I2≤xMdp2I2t≤dMdp2I2t.Writing Ed=dMd/p2, we have |b1(t)|≤Ed/I2(t).
(ii) Ifxt;θ0,I0≥d, with condition (A2), it is easy to know that (p-1)xψ(t,x)-x2ψx(t,x)>βx when x≥d and (p-1)xψ(t,x)-x2ψx(t,x)>-βx when x≤-d. Hence, b1(t)<0 and (45)b1t=-xp-1ψt,x-xψxt,xp2I2≥βxp2I2t≥dβp2I2t.
Therefore, settingFd=dβ/p2, we obtain b1t≥Fd/I2(t). The proof is complete.Letb1(t)=b1+(t)-b1-(t) with b1±(t)=max⁡{±b1(t),0}. To estimate that the integral of b1+(t) on [0,2π] is smaller than the integral of b1-(t) on [0,2π], we need the following lemma.Lemma 16.
Letd≥0 be as in Theorem 1. Define Δt={t∈0,2πp∣xt;θ0,I0≤d}. Then there exist I0¯>0,T>0, such that (46)Δt≤TI01/q, for all I0≥I0¯.Proof.
According to Lemma11, we see that Δt→0 if and only if Δθ→0.
By the action-angle variables transformation (16), it is not difficult to verify that there exists τ>0 such that tanΔθ≤τd/I1/q(t) when Δθ→0. Therefore, by using Corollary 10, we know that there exist I0¯>0,T>0, such that (47)Δt≤TI01/q,for all I0≥I0¯. Thus, we prove Lemma 16.The next lemma gives the estimates of∂θ(2πp;θ0,I0)/∂I0 for I0≫1.Lemma 17.
ForI0≫1, one gets θI0(2πp)<0.Proof.
The following results immediately from Corollary10, Lemma 15, and Lemma 16:(48)θI02πp=1+o1∫02πpb1sds=1+o1∫b1t≤0b1sds+∫b1t≥0b1sds≤-1+o1Fdρ22I022πp-Δt-Edρ12I02Δt≤1+o1ρ12Fd+ρ22EdTρ12ρ22I02+1/q-2πpFdρ22I02.
So, ifI01/q>(ρ12Fd+ρ22Ed)T/2πpLdρ12, we have θI0(2πp)<0.


## 4. Proof of Theorem1

Now we start to give the proof of Theorem1.Proof of Theorem1.
Based on Lemma17 and the Aubry-Mather theorem [37], we can see that the Poincaré map P of system (17) is a monotone twist map when I0≫1. At last, using similar arguments as in [37], we may broaden the Poincaré map P to a new map P^ which is a whole monotone twist homeomorphism on the cylinder S1×R and agree with P on S1×[I0,+∞) with a fixed constant I0≫1. Hence, the existence of Aubry-Mather sets Mσ of P^ is ensured by the Aubry-Mather theorem due to Pei [37]. Moreover, for some small ε0>0, all those Aubry-Mather sets with rotation number α∈(2ωπp,2ωπp+ε0) lie in the domain S1×[I0,+∞). Therefore, they happen to be the Aubry-Mather sets of the Poincaré map of P. From the above discussions, we have showed the existence of Aubry-Mather sets; this implies that (1) has an Aubry-Mather type solution uα(t)=(xα(t),xα′(t)) with rotation number α. This completes the proof of Theorem 1.



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*Source: 8254758-2018-06-19.xml*