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Abstract
In this paper, we study the following second-order periodic system:
x″+V′(x)+p(t)|x|α=0,
where V(x) has a singularity. Under some assumptions on the V(x) and p(t) by Ortega’s small twist theorem, we obtain the existence of quasi-periodic solutions
and boundedness of all the solutions.
Keywords:
boundedness of solutions; singularity; small twist theorem
1 Introduction and main result
In 1991, Levi [1] considered the following equation:
x″+V′(x,t)=0,(1.1)
where V(x,t) satisfies some growth conditions and V(x,t)=V(x,t+1). The author reduced the system to a normal form and then applied Moser twist theorem
to prove the existence of quasi-periodic solution and the boundedness of all solutions.
This result relies on the fact that the nonlinearity V(x,t) can guarantee the twist condition of KAM theorem. Later, several authors improved
the Levi’s result; we refer to [2-4] and the references therein.
Recently, Capietto, Dambrosio and Liu [5] studied the following equation:
x″+V′(x)=F(x,t),(1.2)
with F(x,t)=p(t) is a π-periodic function and V(x)=12x+2+1(1−x−2)ν−1, where x+=max{x,0}, x−=max{−x,0} and ν is a positive integer. Under the Lazer-Leach assumption that
1+12∫0πp(t0+θ)sinθdθ>0,∀t0∈R,(1.3)
they prove the boundedness of solutions and the existence of quasi-periodic solution
by KAM theorem. It is the first time that the equation of the boundedness of all solution
is treated in case of a singular potential.
We observe that F(x,t)=p(t) in (1.2) is smooth and bounded, so a natural question is to find sufficient conditions
on F(x,t) such that all solutions of (1.2) are bounded when F(x,t) is unbounded. The purpose of this paper is to deal with this problem.
Motivated by the papers [1,5,6], we consider the following equation:
x″+V′(x)+p(t)|x|α=0,(1.4)
where p(t) is a π-periodic function,
V(x)=12x+2+11−x−2−1,0<α<1,x>−1.(1.5)
We suppose Lazer-Leach assumption hold:
∫0πp(t0+θ)(sinθ)1+αdθ>0,∀t0∈R.(1.6)
Our main result is the following theorem.
Theorem 1Under the assumptions (1.5) and (1.6), all the solutions of (1.4) are bounded.
The main idea of our proof is acquired from [6]. The proof of Theorem 1 is based on a small twist theorem due to Ortega [7]. It mainly consists of two steps. The first one is to transform (1.4) into a perturbation
of integrable Hamilton system. The second one is to show that Poincaré map of the
equivalent system satisfies Ortega’s twist theorem, then some desired result can be
obtained.
Moreover, we have the following theorem on solutions of Aubry-Mather type.
Theorem 2Assume thatp(t)∈C(R)satisfies (1.6); then, there is anϵ0>0such that, for anyω∈(1π,1π+ϵ0), tequation (1.4) has a solution(xω(t),xω′(t))of the Mather type with rotation numberω. More precisely:
Case 1: ω=pqis rational. The solutions(xω(t+2iπ),xω′(t+2iπ)), 1≤i≤q−1are independent periodic solutions of periodicqπ; moreover, in this case,
limq→∞mint∈R(|xω(t)|+|xω′(t)|)=+∞.
Case 2: ωis irrational. The solution(xω(t),xω′(t))is either a usual quasi-periodic solution or a generalized one.
We will apply Aubry-Mather theory, more precisely, the theorem in [8], to prove this theorem.
2 Proof of theorem
2.1 Action-angle variables and some estimates
Observe that (1.4) is equivalent to the following Hamiltonian system:
x′=∂H∂y,y′=−∂H∂x(2.1)
with the Hamiltonian function
H(x,y,t)=12y2+V(x)+p(t)(α+1)|x|αx.
In order to introduce action and angle variables, we first consider the auxiliary
autonomous equation:
x′=y,y′=−V′(x),(2.2)
which is an integrable Hamiltonian system with Hamiltonian function
H1(x,y,t)=12y2+V(x).
The closed curves H1(x,y,t)=h>0 are just the integral curves of (2.2).
Denote by T0(h) the time period of the integral curve Γh of (2.2) defined by H1(x,y,t)=h and by I the area enclosed by the closed curve Γh for every h>0. Let −1<−αh<0<βh be such that V(−αh)=V(βh)=h. It is easy to see that
Similar in estimating in [5], we have the estimation of functions I− and T−.
Lemma 1We have
hn|dnT−(h)dhn|≤Ch−12
and
hn|dnI−(h)dhn|≤Ch12,
wheren=0,1,…,6, h→+∞. Note that here and below we always useC, C0orC0′to indicate some constants.
Remark 1 It follows from the definitions of T+(h), T−(h) and Lemma 1 that
limh→+∞T−(h)=0,limh→+∞T+(h)=π.
Thus the time period T0(h) is dominated by T+(h) when h is sufficiently large. From the relation between T−(h) and I−(h), we know I0(h) is dominated by I+(h) when h is sufficiently large.
Remark 2 It also follow from the definition of I(h), I−(h), I+(h) and Remark 1 that
|hndnI0(h)dhn|≤C0I0(h)for n≥1.
Remark 3 Note that h=h0(I0) is the inverse function of I0. By Remark 2, we have
|Indnh(I)dIn|≤C0h(I)for n≥1.
We now carry out the standard reduction to the action-angle variables. For this purpose,
we define the generating function S(x,I)=∫C2(h−V(s))ds, where C is the part of the closed curve Γh connecting the point on the y-axis and point (x,y).
Introduce a new action variable ρ∈[1,2] and a parameter ϵ>0 by H=ϵ−2ρ. Then H≫1⇔0<ϵ≪1. Under this transformation, the system (2.15) is changed into the form
Obviously, if ϵ≪1, the solution (t(θ,t0,ρ0),ρ(θ,t0,ρ0)) of (2.16) with the initial date (t0,ρ0)∈R×[1,2] is defined in the interval θ∈[0,2π] and ρ(θ,t0,ρ0)∈[12,3]. So the Poincaré map of (2.16) is well defined in the domain R×[1,2].
The Poincaré map of (2.16) has intersection property.
The proof is similar to the corresponding one in [6].
For convenience, we introduce the notation Ok(1) and ok(1). We say a function f(t,ρ,θ,ϵ)∈Ok(1) if f is smooth in (t,ρ) and for k1+k2≤k,
|∂k1+k2∂tk1∂ρk2f(t,ρ,θ,ϵ)|≤C
for some constant C>0 which is independent of the arguments t, ρ, θ, ϵ.
Similarly, we say f(t,ρ,θ,ϵ)∈ok(1) if f is smooth in (t,ρ) and for k1+k2≤k,
limϵ→0|∂k1+k2∂tk1∂ρk2f(t,ρ,θ,ϵ)|=0,
uniformly in (t,ρ,θ).
2.3 Poincaré map and twist theorems
We will use Ortega’s small twist theorem to prove that the Poincaré map P has an invariant closed curve, if ϵ is sufficiently small. Let us first recall the theorem in [7].
Lemma 8 (Ortega’s theorem)
LetA=S1×[a,b]be a finite cylinder with universal coverA=R×[a,b]. The coordinate inAis denoted by(τ,ν). Consider a map
f¯:A→S×R.
We assume that the map has the intersection property. Suppose thatf:A→R×R, (τ0,ν0)→(τ1,ν1)is a lift off¯and it has the form
Analogously, one may obtain, by a direct but cumbersome commutation that
|∂k+lΔ∂ρ0k∂t0l|≤C⋅ϵ−α,
which means that
xα+1(θ,ϵ−2ρ)−xα+1(θ,ϵ−2ρ0)∈ϵ−αO6(1).
The estimates for ∂x∂H(θ,ϵ−2ρ)xα(θ,ϵ−2ρ)−∂x∂H(θ,ϵ−2ρ0)xα(θ,ϵ−2ρ0) follow from a similar argument, we omit it here. Thus, Lemma 9 is proved. □
Now we turn to give an asymptotic expression of Poincaré map of (2.15), that is, we
study the behavior of the functions Σ1 and Σ2 at θ=π as ϵ→0. In order to estimate Σ1 and Σ2, we need introduce the following definition and lemma. Let
Θ+(I)=meas{θ∈[0,π],x(H0,θ)>0},Θ−(I)=T0−Θ+(I),
where H0=ϵ−2ρ0.
Lemma 10
Θ+(I)=π+ϵO6(1),Θ−(I)=ϵO6(1).
Proof This lemma was proved in [5], so we omit the details. □
For estimate Σ1 and Σ2, we need the estimates of x and xH.
The other assumptions of Ortega’s theorem are easily verified. Hence, there is an
invariant curve of P in the annulus (t0,ρ0)∈S1×[1,2] which imply that the boundedness of our original equation (1.4). Then Theorem 1 is
proved.
2.5 Proof of Theorem 2
We apply Aubry-Mather theory. By Theorem B in [8] and the monotone twist property of the Poincaré map P guaranteed by ∂Ψ1∂ρ0<0. It is straightforward to check that Theorem 2 is correct.
Remark 4 In [9], the authors study the multiplicity of positive periodic solutions of singular Duffing
equations
x″+g(x)=p(t),
where g(x) satisfies the semilinear condition at infinity and the time map satisfies an oscillation
condition, and prove that the given equation possesses infinitely many positive 2π-periodic solutions by using the Poincaré-Birkhoff theorem. By the methods and techniques
in [9], we can also prove the existence of 2π-periodic solutions of (1.4) where V(x) satisfies the sublinear condition.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
Thanks are given to referees whose comments and suggestions were very helpful for
revising our paper.
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