A new method for the boundedness of semilinear Duffing equations at resonance

We introduce a new method for the boundedness problem of 
semilinear Duffing equations at resonance. In particular, it can 
be used to study a class of semilinear equations at resonance 
without the polynomial-like growth condition. As an application, 
we prove the boundedness of all the solutions for the equation 
$\ddot{x}+n^2x+g(x)+\psi(x)=p(t)$ 
 under the Lazer-Leach condition on $g$ and $p$, where $n\in \mathbb{N^+}$, $p(t)$ and $\psi(x)$ are periodic and $g(x)$ is bounded.

1. Introduction and the main results. The study of semilinear equations at resonance has a long history. The interest in this model is motivated both by its connections to application and by a remarkable richness of the related dynamical systems. It is well known that the linear equation x + n 2 x = sin nt, n ∈ N + has no bounded solutions, whereẍ = d 2 x/dt 2 . Another interesting example was constructed by Ding [5], who proved that each solution of the equation x + n 2 x + arctan x = 4 cos nt, n ∈ N + is unbounded. Due to these resonance phenomenons, the existence of bounded solutions and the boundedness of all the solutions for semilinear equation at resonance are very delicate. In 1969, Lazer and Leach [9] studied the following semilinear equations: x + n 2 x + g(x) = p(t), n ∈ N + ,

ZHIGUO WANG, YIQIAN WANG AND DAXIONG PIAO
where p(t + 2π) = p(t) and g is continuous and bounded. They proved that if 2π 0 p(t)e −int dt < 2(lim inf then (1) has at least one 2π-periodic solution. Moreover, they obtained that each solution of (1) is unbounded if 2π 0 p(t)e −int dt ≥ 2(sup g − inf g).
Thus if then condition (2) is sufficient and necessary for the existence of bounded solutions. For this reason, (2) is called Lazer-Leach condition.
In 1996, Alonso and Ortega [1] studied the following equation: x + n 2 x + g(x) + ψ(x) = p(t), n ∈ N + , where g and p are as same as above and the perturbation ψ(x) will be small at infinity in the following sense: where Ψ(x) = x 0 ψ(x)dx. They proved that each solution with large initial condition is unbounded if Other conditions for the existence of bounded and unbounded solutions are described in [1,2,6,8,15,16] and their references.
The pioneering work on the boundedness of (1) was due to Ortega [19]. He proved a variant of Moser's small twist theorem, by which he obtained the boundedness for the equationẍ + n 2 x + h L (x) = p(t), p(t) ∈ C 5 (R/2πZ), where L > 0 and h L (x) is a piecewise linear function of the form and p(t) satisfies 2π 0 p(t)e −int dt < 4L.
With Ortega's small twist theorem, he showed that the Lazer-Leach condition (2) is sufficient for the boundedness of (1). Moreover, if (3) holds true, then Lazer-Leach's result [9] implies that (2) is also necessary for the boundedness.
All the above mentioned results concern the semilinear equations at resonance with a polynomial-like growth potential, that is, the potential, say g(x, t), is bounded and satisfies lim |x|→+∞ x m D m x g(x, t) = 0 (6) for some m > 0.
In this paper, we study the boundedness of the equation where g(x) is a polynomial-like function, p(t + 2π) = p(t) and ψ(x + T ) = ψ(x) satisfying T 0 ψ(x)dx = 0. 1 It is easy to see that usually ψ(x) does not satisfy (6). The most typical example of (7) may beẍ + n 2 x + g(x) + sin x = p(t), n ∈ N + . So we consider the equation with a mixed potential including not only a polynomial-like growth term but also a periodic term.
We will prove that the Lazer-Leach condition on g and p is sufficient for the boundedness of (7). In other words, the periodic term ψ does not play any role in the boundedness. More precisely, we prove that: Theorem 1.1. Assume g(x) ∈ C m1 (R), ψ(x) ∈ C m1 (R/T Z) and p(t) ∈ C m2 (R/2π Z) with m 1 = 18, m 2 = 14. Suppose the following conditions hold true: (A 1 ) g(±∞) = lim x→±∞ g(x) exist and are finite, Then under the following Lazer-Leach condition: every solution of (7) is bounded.
On the other hand, under the assumptions in Theorem 1.1, g(x) is bounded and then Alonso-Ortega's result, cf. Proposition 3.4 in [1] is applicable and it implies the existence of unbounded solutions for (7), also see [11]. Therefore we obtain the following conclusion: 1 It is no loss of generality to assume T 0 ψ(x)dx = 0. In fact, if T 0 ψ(x)dx = 0, then we can

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ZHIGUO WANG, YIQIAN WANG AND DAXIONG PIAO Corollary 1. Assume g(x), ψ(x) and p(t) satisfy the conditions in Theorem 1.1. (8) is sufficient and necessary for the boundedness of (7). Remark 1. When n in (7) is replaced by a Diophantine irrational, a similar result has been obtained by [7].
The new ingredients in our proof are as follows. Instead of applying Ortega's small twist theorem, we use a rotation transformation in Subsection 3.1 to deal with resonance ( also see [24]). With such a transformation, the linear term disappears in the new Hamiltonian (and a sublinear one is obtained), and one will not meet the difficulty of resonance any more. For example, when ψ(x) = 0, Moser's small twist theorem is directly applicable, see [25].
For the case ψ(x) = 0, however, a new difficulty appears due to the lack of the polynomial-like growth condition. In fact, the estimates on the derivatives of the perturbations are very poor, see Lemma 2.4. Hence the perturbation in the sublinear system can not be reduced to be small enough in C 4 -topology by the standard method, see Remark 4. To overcome the difficulty for the case ψ(x) = 0, our observation is that although with the method of solving homological equations, the new perturbation does not become smaller than the old one, the smoothness of the 'troublesome' term in the new perturbation of the Hamiltonian on some variable (that is, the old angle variable) become better, see Subsection 4.3. Thus repeatedly solving homological equations will leads to a Hamiltonian whose 'troublesome' perturbation term possesses a much better regularity on the angle variable. Then the proof can be finished by following the standard computations of Dieckerhoff-Zehnder and Moser's theorem, see Subsection 4.5.
It is worth to note that the periodic assumption on ψ(x) is not necessary. In fact we can show the boundedness holds when ψ(x) = φ(x 1+δ ) with φ(x) periodic and δ > 0 small enough. Moreover, ψ(x) can be replaced by a function ψ(x, t) which is periodic on both x and t, see [24]. Thus we show that the classical polynomial-like growth conditions can be considerably weakened. For more references, one can see [10], [22].
The paper is organized as follows. In Section 2, we state some preliminary estimates. In Section 3, we introduce a rotation transformation and then make canonical transformations such that all non-oscillating terms are transformed into normal form possessing desirable properties. The main difficulty in this paper lies in how to deal with oscillating terms caused by ψ(x). For this purpose, in Section 4 we make canonical transformations to improve estimates on the derivatives of oscillating terms and subsequently change the system into a nearly integrable one. Thus Theorem 1.1 is proved by Moser's twist theorem in Section 5. The proof of some lemmas can be found in the Appendix.
In the remain part of this paper, we suppose g(x), ψ(x) and p(t) satisfy the conditions in Theorem 1.1.
Let y =ẋ/n, equation (7) is equivalent to a Hamiltonian system with Hamiltonian where G(x) = is transformed into where x = x(I, θ) = 2 n I 1 2 cos nθ for simplicity.

ZHIGUO WANG, YIQIAN WANG AND DAXIONG PIAO
Since ∂ I H > 1/2 when I is sufficiently large, we can solve H(I, θ, t) = h for I as following: where R(h, t, θ) is determined implicitly by the equation It is clear that h → +∞ if and only if I → +∞. From Arnold's transformation on exchanging the roles of angle and time variables, it is well known that the new is equivalent to the original one, see [3,10,11,24], etc. We present some estimates on R(h, t, θ) in (11).
The proof is given in the Appendix. Moreover, from the identity (12), R has the following form by Taylor's formula: (13) yields that Therefore, the Hamiltonian is and the following estimates hold: Lemma 2.6. For h large enough, θ, t ∈ S 1 , k + j ≤ m 1 − 1, and l ≤ m 2 , it holds that: The proof is given in the Appendix.
Remark 3. From Lemmas 2.1, 2.3, 2.4 and 2.6, it shows that f 1 , f 2 and R 01 satisfy the polynomial-like growth condition (6) with variable h, while − 1 n Ψ(x) and R 02 do not satisfy the polynomial-like growth condition due to the oscillating property of the periodic function Ψ(x) which is rather different from the previous works.
3. The normal forms for non-oscillating terms. In this section, we first introduce a rotation transformation to eliminate the linear part of the Hamiltonian which help us to obtain a sublinear function, then obtain the normal form for nonoscillating terms by canonical transformations.

A rotation transformation.
Inspired by the techniques in KAM theory (for example [26]), we construct a rotation transformation Intuitively, we adopt a rotating coordinate system with an angle speed 1.
Under Φ 1 , the Hamiltonian I is transformed into I 1 as following Proof. It is obtained from Lemma 2.6.

3.2.
The normal form with f 1 (h 1 , θ). We make a canonical transformation Φ 2 : with the generating function S 2 (h 2 , θ) determined by Under Φ 2 , the Hamiltonian I 1 is transformed into I 2 as following It is clear that (16) implies Thus, I 2 is rewritten by and Moreover, for k + j ≤ m 1 − 1, it holds that: The proof is given in the Appendix.

Thus we have
and for k + j ≤ m 1 − 1, Moreover, the map Φ 3 satisfies Proof. From (8), Lemmas 2.2 and 3.3, (21) and (22) holds. The rest of the proof is similar to the one for lemma 3.2.
Remark 4. For the case Ψ(x) = 0, the boundedness of the system with Hamiltonian I 3 can be obtained by a standard method, that is, by solving a series of homological equations on generating functions, see [25]. At the first sight, it seems plausible that this method is still valid if Ψ(x) = 0. However, it is not true, due to the fact that the perturbation of I 3 does not satisfy the polynomial-like growth condition.
To show this, let us study a simple but similar case. Consider the following Hamiltonian , where α is defined as above satisfying (21) and P satisfies Suppose Φ 0 is a canonical transformation of the form: Thus S 0 satisfies the similar inequalities as P does. But in the new Hamiltonian expressed in (ρ, τ, t), there is a term of the formP = ∂[f2] ∂t (h, τ ) · ∂S0 ∂ρ . From (18) and estimates on S 0 , we can only haveP = O(1) and no better estimates, saỹ 2 ) as we expect, can be obtained. Thus the new perturbation may be of the same order as the old one! In conclusion, we need to find some kind of new transformations, as shown in Section 4.

4.1.
A canonical transformation for Ψ(x). In this subsection, we will make a transformation to deal with Ψ(x). Recall all the transformations we have done before this section: and then For convenience, we denote f 3 (h 3 , t 3 , θ) = 1 n Ψ(x). The following proposition shows that the average [ f 3 ](h 3 , t 3 ) possesses an estimate better than the one for f 3 (or f 3 ) itself, which is important for us to obtain the boundedness.

Proposition 1.
For h 3 large enough, θ, t 3 ∈ S 1 and any > 0, it holds that Proof. The proposition is a special case of Lemma 2.7 in [24].
Remark 5. In Lemma 4.1 we will prove a better result than Proposition 1 by using the theory of oscillatory integral [20]. It is worthy to note that the function of Proposition 1 and that of Lemma 4.1 is similar except that the latter can help us to reduce the smooth requirement of the original system. In other words, Proposition 1 itself is enough for us to obtain the boundedness with a higher smoothness condition on g(x), ψ(x), p(t). moreover, The proof is involved, hence we give it in the Appendix. Now we make a transformation Φ 4 : (h 4 , t 4 , θ) → (h 3 , t 3 , θ) implicitly given by with the generating function S 4 (h 4 , t 3 , θ) determined by

ZHIGUO WANG, YIQIAN WANG AND DAXIONG PIAO
Under Φ 4 , the Hamiltonian I 3 is transformed into Then we have the following estimates.
The oscillating terms in I 4 include −[ f 3 ], R 42 and R 43 while the "worst" term among them is R 43 . For simplicity, without causing confusion, we still denote the sum −[ f 3 ] + R 42 + R 43 by R 43 , i.e.

4.2.
A canonical transformation for the perturbation R 41 . Before dealing with the oscillating term R 43 , we first reduce the non-oscillating term R 41 to be small enough by a canonical transformation.
Let Φ 5 : (h 5 , t 5 , θ) → (h 4 , t 4 , θ) be implicitly given by with the generating function S 5 (h 5 , t 4 , θ) determined by Under Φ 5 , the Hamiltonian I 4 is transformed into where R 5 (h 5 , t 5 , θ) Lemma 4.3. Assume h 5 large enough, θ, t 5 ∈ S 1 . Then it holds that for k + j ≤ Proof. Following Lemma 4.2 and similar to the proof of Lemma 3.2, the estimates are obtained by a direct computation.

ZHIGUO WANG, YIQIAN WANG AND DAXIONG PIAO
4.3. Improvement of estimates on derivatives of the oscillating terms. We improve the estimates on the oscillating term R 5 in this subsection. It will help us to obtain a nearly integrable superlinear system in Subsection 4.5.
To prove Lemma 4.4, we give the following iteration lemma firstly.
From (47), we find that
Appendix A. The theory of oscillatory integral. We need the following theorem on oscillatory integrals in finitely smooth topology later.