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A New Approach to the Existence of Quasiperiodic Solutions for Second-Order Asymmetric p-Laplacian Differential Equations.

1. Introduction

In recent years, all kinds of nonlinear dynamic behavior, such as the existence of positive solutions [1-16] and signchanging 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

[mathematical expression not reproducible], (1)

where [mathematical expression not reproducible], and [lambda] and [mu] are positive constants satisfying

[mathematical expression not reproducible] (2)

where [psi](t, x) [member of] [C.sup.0,1] ([S.sup.P] x R) is a continuous function, 2[[pi].sub.p]-periodic in the first argument and continuously differentiable in the second one, where [S.sup.p] = R/2[[pi].sub.p]Z. Since the pioneering works of Aubry [33] and Mather [34], the existence of AubryMather 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]).

If p = 2, then [[pi].sub.p] = [pi] and (1) reduces to the following piecewise linear equation:

[mathematical expression not reproducible](3)

and (2) becomes

[mathematical expression not reproducible] (4)

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 [omega] [member of] [Q.sup.+] in (4), provided that [mathematical expression not reproducible], and the perturbation term [phi](x) [member of] [C.sup.2](R) [intersection] [L.sup.[infinity]] (R) satisfies some growth conditions. Recently, this result was extended to a much weaker smoothness nonlinearity [psi](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 w [member of] [R.sup.+] in (4) and [psi](t,x) [member of] [C.sup.0,1](SP x 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 pLaplacian-like nonlinear differential equation:

[mathematical expression not reproducible] (5)

where p > 1, [lambda] and [mu] are positive constants satisfying (2) with [mathematical expression not reproducible] is a 2[[pi].sub.p]-periodic function. They considered the existence of AubryMather 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 [C.sup.2] smooth in t.

Now a natural question to ask is whether the smoothness of the function h(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 Poincare 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 Poincaree 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 [psi](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, [psi](t, x) [member of] [C.sup.0,1](SP x R) satisfies the following conditions:

([A.sub.1]) The limit is

[mathematical expression not reproducible]. (6)

([A.sub.2]) There exist constants d [greater than or equal to] 0, [beta] [greater than or equal to] 0, such that

[mathematical expression not reproducible] (7)

Then there exists [epsilon].sub.-0] > 0, such that, for any [mathematical expression not reproducible], (1)possesses an Aubry-Mather type solution [z.sub.[alpha]](t) = ([x.sub.[alpha]](t), [x'.sub.[alpha]] (t)) with rotation number [alpha]; that is,

(i) if [alpha] = k/m is rational and (k,m) = 1, the solutions [mathematical expression not reproducible], are mutually unlinked periodic solutions of period m;

(ii) if [alpha] is irrational, the solution [z.sub.[alpha]] (t) is either a usual quasiperiodic solution or a generalized one.

Remark 2. A solution is called generalized quasiperiodic one if the closed set

[mathematical expression not reproducible] (8)

is Denjoy's minimal set (see its definition in [47]).

Remark 3. Using the rule of L'Hospital to condition ([A.sub.1]), it can easily be seen that

[mathematical expression not reproducible]

Remark 4. We noticed that the perturbations g(x) and h(t) in [39] need to be bounded. But from ([A.sub.1]) and ([A.sub.2]) of this paper, it is easy to verify that the perturbation [psi](t, x) can be either a bounded function or an unbounded function. For example, we can set [psi](t, x) to be a bounded function arctan x ? (1 + [sin.sup.2](([pi]/[[pi].sub.p])t)) or an unbounded function [mathematical expression not reproducible] when d =1 and [beta] = [pi]/4 - 1/2 in Theorem 1. Moreover, positive constant a = n [member of] N satisfying (1.2) in [39] has been extended to the case to e [R.sup.+] in this paper. Thus, our situation is more general than the results obtained in [39] for p [greater than or equal to] 2.

Remark 5. If p = 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 ([[phi].sub.p](x'))' 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 Poincare 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 AubryMather theorem [37].

2. Preliminaries

2.1. The Action and Angle Variables. Let Sp(t) = sinpf be the solution of

([[phi].sub.p](u'))' + [[phi].sub.p] (u) = 0 (9)

satisfying the initial condition u(0) = 0, u'(0) = 1. Then it follows from [48] that [S.sub.p](t) = [sin.sub.p]t is a 2[[pi].sub.p]-periodic [C.sup.2] odd function with [sin.sub.p] ([[pi].sub.p] - t) = [sin.sub.p] t, for t [member of] [0,[[pi].sub.p]/2], and [mathematical expression not reproducible]. Moreover, for t [member of] [mathematical expression not reproducible] can be implicitly given by

[mathematical expression not reproducible] (10)

Introducing a new variable y = -[[phi].sub.p](x'), then (9) is equivalent to the planar system

X = -[[phi].sub.q] (y), y' = [[phi].sub.p (x), (11)

where q is the conjugate exponent of p : [p.sup.-1] + [q.sup.-1] = 1. Letting (x, y) = ([C.sub.p](t), [S.sub.p](t)) be the unique solution of (11) satisfying ([C.sub.p](0), [S.sub.p](0)) = (1,0), then the functions [C.sub.p](t) and Sp(t) are much similar to cosine and sine. It follows from [49] that [C.sub.p](t) e [C.sup.2] and Sp(t) [member of] [C.sup.1] are 2[[pi].sub.p]-periodic, and for [for all] n [member of] Z, [C.sub.p](t) = 0 iff t = [[pi].sub.p]/2 + n[[pi].sub.p], and [S.sub.p](t) = 0 iff t = n[[pi].sub.p]. Moreover, [C.sub.p](t) = - [[phi].sub.q]([S.sub.p](t)) and [S'.sub.p](t) = [[phi].sub.q] ([C.sub.p](t)), and (1/p)\[C.sub.p][(t).sup.1/P] + (1/q)[absolute value of [S.sub.p] (t)].sup.1/q] [equivalent to]1/p.

Now we consider (1). Set y = -[[phi].sub.p](x') in (1); then (1) can be rewritten as a planar system

[mathematical expression not reproducible] (12)

where q = p/(p - 1) is the conjugate exponent of p.

Lemma 6. For p [greater than or equal to] 2 and for any ([x.sub.0], [y.sub.0]) [member of] [R.sup.2], [t.sub.0] [member of] R, the solution

z (t) = (x (t, [t.sub.0], [X.sub.0], [y.sub.0]), y (t, [t.sub.0], [X.sub.0], [y.sub.0])) (13)

of (12) satisfying the initial condition z([t.sub.0]) = ([x.sub.0], [y.sub.0]) 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: X

[mathematical expression not reproducible] (14)

Then, by using (2) and direct computation, one obtains the following.

Lemma 7. (i) Both C(t) [member of] [C.sup.2] and S(t) [member of] [C.sup.1] are 2[[pi].sub.p]/w-periodic functions, and C(t) can be given by

[mathematical expression not reproducible] (15)

Now we introduce an action-angle variables transformation by the mapping [PSI]: [S.sup.p] x (0, +[infinity]) [right arrow] [R.sup.2] {(0,0)}, where (x, y) = [PSI]([theta], I) defined by the formula

[mathematical expression not reproducible] (16)

where [gamma] = [[lambda].sup.-1] wp 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 transformation W and using Lemma 7 (iii), (12) is changed into

[mathematical expression not reproducible] (17)

where [mathematical expression not reproducible].

We notice that the relation between (17) and (12) is that if [theta](t) = [theta](t; [[theta].sub.0], [I.sub.0]),I(t) = I(t;[[theta].sub.0], [I.sub.0]) are the solutions of (17) with the initial value condition [theta](0) = [[theta].sub.0] I(0) = [I.sub.0], then

[mathematical expression not reproducible] (18)

and

[mathematical expression not reproducible] (19)

are the solutions of (12) with initial data x(0) = x(0, [[theta].sub.0], [I.sub.0]),y(0) = y(0; [[theta].sub.0], [I.sub.0]). By Lemma 6, (17) has a unique solution for [I.sub.0] > 0 and [[theta].sub.0] [member of] R. Moreover, this solution has continuous derivatives with respect to initial data [[theta].sub.0] and [I.sub.0].

For notional convenience, hereinafter, we write x, y, [theta], I instead of [mathematical expression not reproducible], respectively.

Firstly, by some simple calculations, we have the following.

Lemma 8. [mathematical expression not reproducible]

Now we are concerned with the growth estimates with regard to I(t; [[theta].sub.0], [I.sub.0]) and [theta](t; [[theta].sub.0], [I.sub.0]).

Lemma 9. The limit

[mathematical expression not reproducible] (20)

holds uniformly on t [member of] [0,2[[pi].sub.p]].

Proof. In view of ([[psi].sub.0]) and (16), there exist constants D >0, K >0, such that

[mathematical expression not reproducible] (21)

Then, by the Gronwall inequality, one has

[mathematical expression not reproducible] (22)

for all t [member of] (0, 2[[pi].sub.p].

So, by (22), I(t; [[theta].sub.0], [I.sub.0]) [right arrow] +[infinity] as [I.sub.0] [right arrow] +[infinity] uniformly for t [member of] [0,2[[pi].sub.p]].

According to (22), it is easy to see the following.

Corollary 10. [for all] [[theta].sub.0] [member of] R and [for all] t [member of] [0,2[[pi].sub.p]], there exist constants [[rho].sub.2] > [[rho].sub.1] > 0 and [bar.I] > 0, such that

[mathematical expression not reproducible] (23)

when [I.sub.0] [greater than or equal to] I [bar.I].

Lemma 11. [for all] [[theta] [member of] R and [for all t [member of] [0,2[[pi].sub.p]], there exists constant [bar.I] > 0, such that

[mathematical expression not reproducible] (24)

if [I.sub.0] [greater than or equal to] [bar.I].

Proof. Since ([[psi].sub.0]) holds, then, for every [epsilon] > 0, there exists M = M([epsilon]) >0, such that

[mathematical expression not reproducible] (25)

if [absolute value of x] [greater than or equal to] M and [for all] t [member of] [0,2[[pi].sub.p]. Hence,

[mathematical expression not reproducible] (26)

Thus, by using action-angle variables transformation (16) and Lemma 9, there exists [[bar.I].sub.1] > 0 such that d[theta]/dt [greater than or equal to] w/2 if I[greater than or equal to] [[bar.I].sub.1].

For if [absolute value of x] [less than or equal to] M, we assume that [absolute value of [psi](t,x)] [less than or equal to] [[psi[.sub.[infinity], where [mathematical expression not reproducible],and then

[mathematical expression not reproducible] (27)

So, by (16), Lemma 8 (ii), and Lemma 9, there exists a constant [[bar.I].sub.2] > 0, such that d[theta]/[d.sub.t] [greater than or equal to] w/2 if [I.sub.0] [greater than or equal to] [[bar.I].sub.2]

If we choose[mathematical expression not reproducible].

Exploiting the same arguments, one can show that the inequality on the right side of (i) holds.

3. Twist Property and Proof of Theorem 1

Let the Poincare mapping P of equation (17) be

[mathematical expression not reproducible]. (28)

In order to apply the Aubry-Mather theorem developed by Pei [37], we only need to show that the Poincare mapping P is a monotone twist map around the infinity; that is, it is enough to show [mathematical expression not reproducible]. In the following we are going to give its detailed proofs by some lemmas.

Similarly, for notional convenience, hereinafter, we also write x,y,[theta],I instead [mathematical expression not reproducible], respectively

Lemma 12. The following convergences hold uniformly on t e [0,2[[pi].sub.p]]:

[mathematical expression not reproducible]

Proof. If ([A.sub.1] and ([[psi].sub.0]) hold, then to each [epsilon] > 0 there corresponds a positive number M = M([epsilon]) > 0, such that

[mathematical expression not reproducible] (29)

and

[mathematical expression not reproducible] (30)

when [mathematical expression not reproducible] is a constant given in (16).

Denote [mathematical expression not reproducible].

(i) By action-angle variables transformation (16) anc Lemma 8 (ii) and p [greater than or equal to] 2, we have

[mathematical expression not reproducible] (31)

Then, given [bar.I] > 0 choosing [I.sub.0] so that [I.sub.0] [greater than or equal to] [bar.I], by using Corollary 10, provided

[mathematical expression not reproducible] (32)

we have

[mathematical expression not reproducible] (33)

Since [epsilon] >0 is arbitrary, the proof of (i) is complete.

For (ii), observe that p [greater than or equal to] 2, and, combining (16) and Lemma 8 (ii), one has

[mathematical expression not reproducible] (34)

Then, given [bar.I] > 0, choosing [I.sub.0] so that [I.sub.0] [greater than or equal to] [bar.I], by using Corollary 10, provided

[mathematical expression not reproducible] (35)

we have

[mathematical expression not reproducible] (36)

where [mathematical expression not reproducible]. Since [epsilon] > 0 is arbitrary, (ii) is proved.

(iii) Set [mathematical expression not reproducible], Lemma 7

(ii), and p [greater than or equal to] 2, we can get

[mathematical expression not reproducible] (37)

Then, given [bar.I] > 0, choosing [I.sub.0] so that [I.sub.0] [greater than or equal to] [bar.I], by using Corollary 10, provided

[mathematical expression not reproducible] (38)

we have

[mathematical expression not reproducible] (39)

Since [epsilon] >0 is arbitrary, the proof of (iii) is finished.

[mathematical expression not reproducible] (40)

As a result of Lemma 9, Corollary 10, and Lemma 12, we have the following.

Lemma 13. [for all] t, s [member of] [0,2 [[pi].suv.p]], the following conclusions hold:

(i) [b.sub.1] (t) = o(1/[I.sub.0]), as [I.sub.0] [right arrow] + [infinity].

(ii) [b.sub.2] (t) = o(1), as [I.sub.0] [right arrow] + [infinity]

(iii) [b.sub.1](t) x [b.sub.3](s) = o(1), as [I.sub.0] [right arrow] +[infinity].

Let us consider the variational equation of (17) with respect to the initial value I0. One can verify that

[mathematical expression not reproducible] (41)

Lemma 14. For all t [member of] (0,2[pi].sub.p], [I.sub.0] [right arrow] +[infinity], one has

([mathematical expression not reproducible]

Proof. From variational equations (41) and Lemma 13, one has

[mathematical expression not reproducible] (42)

and here we have used [mathematical expression not reproducible].

Hence, for all [mathematical expression not reproducible]. Thus, (i) and (ii) are proved.

To prove (iii), we consider the variational equation of (17) about [[theta].sub.0]; one can get

[mathematical expression not reproducible] (43)

By using a similar argument in (ii), we can also show that [mathematical expression not reproducible]. This completes the proof of Lemma 14.

Next, we will develop an estimate of upper bound and lower bound for [b.sub.1](t).

Lemma 15. Let d [greater than or equal to] 0 satisfy ([A.sub.2]).

(i) If [mathematical expression not reproducible], then there exists a constant [E.sub.d] > 0, such that \[mathematical expression not reproducible.

(ii) If [mathematical expression not reproducible] for all t e [0,2 [[pi].sub.p]], then there exists a constant [F.sub.d] > 0, such that [absolute value of ]. Moreover, if [mathematical expression not reproducible].

Proof. [mathematical expression not reproducible]. Then

[mathematical expression not reproducible] (44)

Writing [mathematical expression not reproducible].

(ii) If [mathematical expression not reproducible], with condition ([A.sub.2]), it is easy to know that [mathematical expression not reproducible]. Hence, [b.sub.1] (t) < 0 and

[mathematical expression not reproducible] (45)

Therefore, setting [F.sub.d] = d[beta]/[p.sup.2], we obtain [absolute value of [b.sub.1](t)] [greater than or equal to] [F.sub.d]/ [I.sup.2](t). The proof is complete. ?

Let [mathematical expression not reproducible]. To estimate that the integral of [b.sup.+.sub.1](t) on [0,2[pi]] is smaller than the integral of [b.sup.-.sub.1](t) on [0,2[pi]], we need the following lemma.

Lemma 16. Let d [greater than or equal to] 0 be as in Theorem 1. Define [mathematical expression not reproducible]. Then there exist [[bar.I].sub.0] > 0,T > 0, such that

[mathematical expression not reproducible], (46)

for all [I.sub.0] [greater than or equal to] [bar.I].sub.0].

Proof. According to Lemma 11, we see that [DELTA]t [right arrow] 0 if and only if [DELTA][theta] [right arrow] 0.

By the action-angle variables transformation (16), it is not difficult to verify that there exists [tau] >0 such that [absolute value of tan [DELTA][theta]] [less than or equal to] [tau]d/[I.sup.1/q] (t) when [DELTA][theta] [right arrow] 0. Therefore, by using Corollary 10, we know that there exist [[bar.I].sub.0] > 0,T >0, such that

[mathematical expression not reproducible] (47)

for all [I.sub.0] [greater than or equal to] [[bar.I].sub.0]. Thus, we prove Lemma 16.

The next lemma gives the estimates of [mathematical expression not reproducible] for [I.sub.0] [greater than or equal to] 1.

Lemma 17. For [I.sub.0] [much greater than] 1, one gets [mathematical expression not reproducible].

Proof. The following results immediately from Corollary 10, Lemma 15, and Lemma 16:

[mathematical expression not reproducible] (48)

So, if [mathematical expression not reproducible]

4. Proof of Theorem 1

Now we start to give the proof of Theorem 1.

Proof of Theorem 1. Based on Lemma 17 and the AubryMather theorem [37], we can see that the Poincare map P of system (17) is a monotone twist map when [I.sub.0] [much greater than] 1. At last, using similar arguments as in [37], we may broaden the Poincare map P to a new map [??] which is a whole monotone twist homeomorphism on the cylinder [S.sup.1] x R and agree with P on [S.sup.1] x [[I.sub.0], +[infinity]) with a fixed constant [I.sub.0] [much greater than] 1. Hence, the existence of Aubry-Mather sets [M.sub.[sigma]] of [??] is ensured by the Aubry-Mather theorem due to Pei [37]. Moreover, for some small [epsilon]0 > 0, all those Aubry-Mather sets with rotation number [alpha] [member of] (2w[[pi].sub.p], 2w[[pi].sub.p] + [epsilon]0) lie in the domain [S.sup.1] x [[I.sub.0], +[infinity]). Therefore, they happen to be the Aubry-Mather sets of the Poincare 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 [u.sub.[alpha]](t) = ([x.sub.[[alpha]](t), [x'.sub.[alpha]](t)) with rotation number a. This completes the proof of Theorem 1.

https://doi.org/10.1155/2018/8254758

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11461056), the Youth Natural Science Foundation of Jiangxi (Grant no. 20132BAB211008), and the Natural Science Foundation of Jiangxi Provincial Department of Education (Grant no. GJJ170926).

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Xiaoming Wang (iD) (1) and Lixia Wang (2)

(1) School of Mathematics & Computer Science, Shangrao Normal University, Shangrao 334001, China

(2) School of Sciences, Tianjin Chengjian University, Tianjin 300384, China

Correspondence should be addressed to Xiaoming Wang; wxmsuda03@163.com

Received 24 April 2018; Accepted 17 May 2018; Published 19 June 2018

Academic Editor: Liguang Wang
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Title Annotation:Research Article
Author:Wang, Xiaoming; Wang, Lixia
Publication:Journal of Function Spaces
Date:Jan 1, 2018
Words:5360
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