Keywords:boundedness of solutions; singularity; small twist theorem
1 Introduction and main result
In 1991, Levi  considered the following equation:
where satisfies some growth conditions and . 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 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  studied the following equation:
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 in (1.2) is smooth and bounded, so a natural question is to find sufficient conditions on such that all solutions of (1.2) are bounded when is unbounded. The purpose of this paper is to deal with this problem.
We suppose Lazer-Leach assumption hold:
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 . The proof of Theorem 1 is based on a small twist theorem due to Ortega . 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.
We will apply Aubry-Mather theory, more precisely, the theorem in , 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:
with the Hamiltonian function
In order to introduce action and angle variables, we first consider the auxiliary autonomous equation:
which is an integrable Hamiltonian system with Hamiltonian function
By direct computation, we get
We then have
Similar in estimating in , we have the estimation of functions and .
Lemma 1We have
We now carry out the standard reduction to the action-angle variables. For this purpose, we define the generating function , where C is the part of the closed curve connecting the point on the y-axis and point .
which is symplectic since
From the above discussion, we can easily get
Lemma 2 [, Lemma 2.2]
2.2 New action and angle variables
This means that if one can solve I from (2.6) as a function of H (θ and t as parameters), then
is also a Hamiltonian system with Hamiltonian function I and now the action, angle and time variables are H, t and θ.
From (2.6) and Lemma 1, we have
So, we assume that I can be written as
which implies that
As a consequence, R is implicitly defined by
By Remark 2, Lemma 2 and the estimate of R, we have
We suppose that
By direct calculation, we have
By (2.11) and (2.12), we have
By (2.13), we have
Analogously, one may obtain, by a direct but cumbersome commutation, the following estimates.
The system (2.7) is of the form
which is also Hamiltonian system with the new Hamiltonian function
Lemma 7 [, Lemma 5.1]
The Poincaré map of (2.16) has intersection property.
The proof is similar to the corresponding one in .
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 .
Lemma 8 (Ortega’s theorem)
Then the Poincaré map of (2.16) is
Moreover, we can prove that
Lemma 9The following estimates hold:
By Lemma 2 and (2.25), we have
Analogously, one may obtain, by a direct but cumbersome commutation that
which means that
Now we turn to give an asymptotic expression of Poincaré map of (2.15), that is, we study the behavior of the functions and at as . In order to estimate and , we need introduce the following definition and lemma. Let
Proof This lemma was proved in , so we omit the details. □
which yields that
Lemma 11The following estimates hold true:
Thus, Lemma 11 is proved. □
2.4 Proof of Theorem 1
The other assumptions of Ortega’s theorem are easily verified. Hence, there is an invariant curve of P in the annulus 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  and the monotone twist property of the Poincaré map P guaranteed by . It is straightforward to check that Theorem 2 is correct.
Remark 4 In , the authors study the multiplicity of positive periodic solutions of singular Duffing equations
where 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 , we can also prove the existence of 2π-periodic solutions of (1.4) where satisfies the sublinear condition.
The author declares that they have no competing interests.
Thanks are given to referees whose comments and suggestions were very helpful for revising our paper.
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