Boundedness of solutions to a second-order periodic system with p-Laplacian and unbounded perturbation terms

Abstract

The second-order periodic system with p-Laplacian and unbounded time-dependent perturbation terms is investigated. Using the principle integral method, it is shown that under certain assumptions on the unbounded and periodic terms, all solutions to the equation possess boundedness.

1 Introduction and main result

Consider the following second-order differential equation

$$ (\varphi _{p}(x'))'+a\varphi _{p}(x^{+})-b\varphi _{p}(x^{-})+z(t)f(x)=e(t), $$

(1.1)

where $\varphi _{p}(s)=|s|^{p-2}s$ with constant $p>2$. Variable $x\in \mathbb{R}$, $t\in \mathbb{R}$, $x^{+}=max(x,0)$, $x^{-}=max(-x,0)$. a and b are positive constants ($a \neq b$) satisfying $a^{-\frac{1}{p}}+b^{-\frac{1}{p}}=2\omega ^{-1}$, ω is an irrational number, $f(x)=o(|x|)$, $z(t)$ and $e(t)$ are $2\pi _{p}$ periodic functions with $\pi _{p}=\frac{2\pi (p-1)^{\frac{1}{p}}}{psin\frac{\pi}{p}}$.

When $p=2$, Eq. (1.1) is turned into

$$ x''+ax^{+}-bx^{-}+z(t)f(x)=e(t),\quad \pi _{p}=\pi . $$

(1.2)

Provided that $z(t)f(x)=0$, $e(t)=1+\gamma h(t)$ in which $h(t)$ is a suitable function, investigating the boundedness of solutions to Eq. (1.2) is very complicated. Ortega [1] proves that every solution to Eq. (1.2) is bounded if $h\in C^{4}(\mathbb{S}^{1})$, where $\mathbb{S}^{1}=\mathbb{R}/2\pi \mathbb{Z}$, and γ is sufficiently small. Under certain conditions on the initial data, Alonso and Ortega [2] obtain that there exists a function $e(t)$ to ensure that all solutions to Eq. (1.2) are unbounded. Ambrosio [3] establishes the boundedness to solutions to fractional relativistic Schrödinger equations. A differential inclusion system involving the $p(t)$-Laplacian is investigated in [4]. Giacomoni et al. [5] utilize the bifurcation theory to discuss the multiplicity for a strongly singular quasi-linear problem. The asymptotic properties of solutions for a second-order nonlinear discrete equation of the Emden-Fowler type are acquired in [6]. Under appropriate restrictions, Jiao et al. [7] discuss the boundedness of all solutions to Eq. (1.2) (see also [8–10]).

For $p\geq 2$, when $a^{-\frac{1}{p}}+b^{-\frac{1}{p}}=2\omega ^{-1}$, where $\omega ^{-1}$ is an irrational number, Yang [11] investigates Eq. (1.1) and obtains that all the solution to Eq. (1.1) are bounded under certain assumptions. Liu [12] discusses the bounded condition for Eq. (1.1) provided that f is smooth and $\lim \limits _{x\rightarrow \pm \infty}f(x)$ is finite. Ma [13] discusses the bounded condition for Eq. (1.1) provided that f is unbounded and $z(t)=1$.

When $p=2$, without the assumption that $\lim \limits _{x\rightarrow \pm \infty}f(x)$ is finite, Zhang [14] has acquired the conditions to ensure that each solution of Eq. (1.1) is bounded. In this work, we will extend the result in [14] to the case $p>2$ under the following assumptions:

$(A_{1}): z(t), e(t)\in C^{6}(\mathbb{S}^{1})$, where $\mathbb{S}^{1}=\mathbb{R}/2\pi _{p}\mathbb{Z}$.

$(A_{2})$: If $f(x)\in C^{6}(\mathbb{R}\setminus \{0\})\cap \mathbb{C}^{0}( \mathbb{\mathbb{R}})$, then there are two positive constants C and $\frac{1}{p-1}<\gamma <1$, such that $|x^{k}f^{(k)}(x)|\leq C|x|^{\gamma}$, provided that $x\in \mathbb{R}\setminus \{0\}$ and $0\leq k\leq 6$.

$(A_{3})$: There exist positive constants $\beta _{1}$ and $\beta _{2}$ such that $p\beta _{1}>q\beta _{2}>0$, where positive constants p and q satisfy $\frac{1}{p}+\frac{1}{q}=1$ and

$$\begin{aligned} xf(x)\geq \beta _{1}|x|^{\gamma +1},\ \ x^{2}f'(x)\leq \beta _{2}|x|^{ \gamma +1}, \quad x\in \mathbb{R}\setminus \{0\}. \end{aligned}$$

Here, we mention that condition $(A_{1})$ does not require $z(t)=1$, namely, condition $(A_{1})$ is different from $z(t)=1$ in Ma [13]. Now, we state our main conclusion.

Theorem 1.1

Assume that $p>2$ and $(A_{1})-(A_{3})$ hold and $\hat{z}=\frac{1}{2\pi _{p}}\int _{0}^{2\pi _{p}}z(t)dt\neq 0$. Then every solution of Eq. (1.1) is bounded, namely, $\sup \limits _{t\in \mathbb{R}}(|x(t)|+|x'(t)|)<\infty$.

We set $F(x)=\int _{0}^{x}f(s)ds$. In this work, we utilize c and C to denote any positive constants (not concerning their quantity). k, l, m and n are nonnegative integers.

The structure of this work is the following: Sect. 2 presents action-angle variables, exchanging time and angle variables, and several lemmas. Section 3 provides the proof of Theorem 1.1.

2 Preliminaries

In this part, we provide several lemmas that help prove Theorem 1.1. Throughout Sect. 2, we assume that the hypotheses of Theorem 1.1 always hold.

2.1 Action-angle coordinates

Let $x'=-\omega \varphi _{q}(y)$, then $y=-\omega ^{1-p}\varphi _{p}(x')$, and the equivalent form of Eq. (1.1) is the following:

$$ x'=-\omega \varphi _{q}(y),\quad \quad y'=\omega [a_{1}\varphi _{p}(x^{+})-b_{1} \varphi _{p}(x^{-})]+\omega ^{1-p}[z(t)f(x)-e(t)] $$

with the Hamiltonian function

$$ H(x,y,t)=\frac{\omega}{q}|y|^{q}+\frac{\omega}{p}(a_{1}|x^{+}|^{p}+b_{1}|x^{-}|^{p})+ \omega ^{1-p} (z(t)F(x)-e(t)x), $$

(2.1)

where $a_{1}=\omega ^{-p}a$, $b_{1}=\omega ^{-p}b$, $a_{1}$ and $b_{1}$ satisfy $a_{1}^{-\frac{1}{p}}+b_{1}^{-\frac{1}{p}}=2$.

Let $sin_{p}(t)$ satisfy the problem

$$ (\varphi _{p}(C'(t)))'+\varphi _{p}(C(t))=0,\quad C(0)=0,\quad C'(0)=1. $$

From the conclusions in [15–17], we confirm that $sin_{p}(t)$ is a $2\pi _{p}$-periodic $C^{2}$ odd function with $sin_{p}(\pi _{p}-t)=sin_{p}(t)$ for $t\in [0,\frac{\pi _{p}}{2}]$ and $sin_{p}(2\pi _{p}-t)=-sin_{p}(t)$ for $t\in [\pi _{p},2\pi _{p}]$. Moreover, for $t\in [0,\frac{\pi _{p}}{2}]$ and $sin_{p}'(t)>0$, $sin_{p}(t)\in (0,(p-1)^{\frac{1}{p}})$ is implicitly given by

$$ \int _{0}^{sin_{p}(t)}\frac{ds}{(1-\frac{s^{p}}{p-1})^{\frac{1}{p}}}=t. $$

Suppose that $v(t)$ satisfies the initial problem

$$ (\varphi _{p}(x'(t)))'+a_{1}\varphi _{p}(x^{+})-b_{1}\varphi _{p}(x^{-})=0, \quad x(0)=(p-1)^{\frac{1}{p}},\quad x'(0)=0. $$

Letting $\varphi _{p}(v')=u$ and $q=p/(p-1)>1$ yields

$$ \frac{|u|^{q}}{q}+\frac{a_{1}|v^{+}|^{p}+b_{1}|v^{-}|^{p}}{p}= \frac{a_{1}}{q}. $$

(2.2)

Using (2.2), we obtain that the action-angle coordinate transformation $\psi _{0}$: $x= (d_{1}r)^{\frac{1}{p}}v(\theta )$, $y=(d_{1}r)^{\frac{1}{q}}u( \theta )$ with $d_{1}=pa_{1}^{-1}$. $\psi _{0}$ is a symplectic transformation since its value of the Jacobian determinant is 1. Under $\psi _{0}$, Hamiltonian function (2.1) is transformed into

$$ h(r,\theta ,t)=\omega r+\omega ^{1-p}z(t)F((d_{1}r)^{\frac{1}{p}}v( \theta ))-\omega ^{1-p}e(t)(d_{1}r)^{\frac{1}{p}}v(\theta )\in \mathbb{C}^{1,1,6}(\mathbb{R^{+}}\times \mathbb{S}^{1}\times \mathbb{S}^{1}). $$

(2.3)

Let $\Xi =\{\theta \in \mathbb{S}^{1}:v(\theta )=0\}$. When $\theta \in \mathbb{S}^{1}\backslash \Xi$ ($t\in \mathbb{S}^{1}$ is a parameter), we have $h(r,t,\theta )\in \mathbb{C}^{6}$ with respect to r.

2.2 Lemmas

Utilizing the ideas in [13, 14, 18], from conditions $(A_{2})$ and $(A_{3})$, we obtain the following conclusions.

Lemma 2.1

For $r\gg 1, k\leq 6$, it holds that

$$\begin{aligned}& |\partial _{r}^{k}F((d_{1}r)^{\frac{1}{p}}v(\theta ))|\leq Cr^{-k+ \frac{\gamma +1}{p}},\\& |\partial _{r}^{k}f((d_{1}r)^{\frac{1}{p}}v(\theta ))|\leq Cr^{-k+ \frac{\gamma}{p}}, \end{aligned}$$

in which $\theta \in \mathbb{S}^{1}$ provided that $k=1$; $\theta \in \mathbb{S}^{1}\setminus \Xi$ if $k\geq 2$.

Lemma 2.2

Let

$$ \bar{F}(r)=\int _{0}^{2\pi _{p}}F((d_{1}\omega ^{-1}r)^{\frac{1}{p}}v( \theta ))d\theta . $$

(2.4)

For $r\gg 1$, the following conclusions hold

$$\begin{aligned}& |\bar{F}^{(k)}(r)|\leq Cr^{-k+\frac{\gamma +1}{p}},\quad k\leq 6,\\& \quad \bar{F}'(r)\geq cr^{-1+\frac{\gamma +1}{p}} \end{aligned}$$

and

$$ \bar{F}''(r) \leq -Cr^{-2+\frac{\gamma +1}{p}}. $$

Proof

For the sake of simplicity, we write $x=(d_{1}\omega ^{-1}r)^{\frac{1}{p}}v(\theta )$. Using (2.4) and noticing that $\Xi \bigcap [0,2\pi _{p}]$ is a finite set, we have

$$ \bar{F}'(r)=\frac{1}{pr}\int _{[0,2\pi _{p}]\setminus \Xi}f(x)xd \theta . $$

Using condition (A3) yields

$$ \bar{F}'(r)=\frac{1}{pr}\int _{[0,2\pi _{p}]\setminus \Xi}f(x)xd \theta \geq \frac{\beta _{1}}{pr}\int _{[0,2\pi _{p}]\setminus \Xi}|x|^{ \gamma +1}d\theta =cr^{-1+\frac{\gamma +1}{p}}. $$

Differentiating (2.4) with respect to variable r, from the above analysis and condition (A3), we have

$$\begin{aligned} & \bar{F}''(r)=\frac{1}{p^{2}r^{2}}\int _{[0,2\pi _{p}]\setminus \Xi}f(x)x^{2}d \theta -\frac{1}{qr}\bar{F}'(r) \\ &\,\;\;\quad \quad \quad \leq \frac{\beta _{2}}{p^{2}r^{2}}\int _{[0,2 \pi _{p}]\setminus \Xi}|x|^{\gamma +1}d\theta -\frac{1}{qr}\bar{F}'(r) \\ &\,\;\;\quad \quad \quad \leq \frac{\beta _{2}}{pr\beta _{1}}\bar{F}'(r)- \frac{1}{qr}\bar{F}'(r) \\ &\,\;\;\quad \quad \quad = \Big(\frac{\beta _{2}}{p\beta _{1}}- \frac{1}{q}\Big)\frac{\bar{F}'(r)}{r} \\ &\,\;\;\quad \quad \quad \leq \Big(\frac{\beta _{2}}{p\beta _{1}}- \frac{1}{q}\Big)cr^{-2+\frac{\gamma +1}{p}}, \end{aligned}$$

which finishes the proof. □

From Lemmas 2.1 and 2.2. combined with condition $(A_{1})$, we obtain that the following conclusion holds.

Lemma 2.3

Let $h_{1}(r,\theta ,t)=\omega ^{1-p}z(t)F((d_{1}r)^{\frac{1}{p}}v( \theta ))-\omega ^{1-p}e(t)(d_{1}r)^{\frac{1}{p}}v(\theta )$. For $r\gg 1,t\in \mathbb{S}^{1}$ then

$$ |\partial _{r}^{k}\partial _{t}^{l}h_{1}(r,\theta ,t)|\leq c r^{-k+ \frac{\gamma +1}{p}}, $$

(2.5)

in which $\theta \in \mathbb{S}^{1}$ provided that $k=1$; $\theta \in \mathbb{S}^{1}\setminus \Xi$ if $k\geq 2$.

Let

$$ g(r,\theta ,t)=r^{-\frac{1}{p}}h_{1}(r,\theta ,t). $$

(2.6)

From Lemma 2.3, for $r\gg 1$, we have

$$ |\partial _{r}^{k}\partial _{t}^{l}g(r,\theta ,t)|\leq cr^{-k+ \frac{\gamma}{p}}, \quad k+l\leq 6. $$

(2.7)

Lemma 2.4

For $r\gg 1, k+l\leq 6$, then

$$\begin{aligned} \left \{ \textstyle\begin{array}{l} 0< cr\leq h(r,t,\theta )< Cr, \\ \partial _{r}h(r,t,\theta )>\frac{\omega}{2}, \\ |\partial _{r}^{k}\partial _{t}^{l}h(r,t,\theta )|\leq Cr^{-k+1}, \end{array}\displaystyle \right . \end{aligned}$$

(2.8)

in which $\theta \in \mathbb{S}^{1}$ provided that $k=1$; $\theta \in \mathbb{S}^{1}\setminus \Xi$ if $k\geq 2$.

Proof

From (2.3) and Lemma 2.1, we obtain

$$ \lim _{r\rightarrow +\infty}\frac{h}{r}=\omega >0, $$

and for $r\gg 1$,

$$ \frac{\partial h}{\partial r}=\omega +\omega ^{1-p}z(t)\partial _{r}F((d_{1}r)^{ \frac{1}{p}}v(\theta ))-\frac{d_{1}}{p}\omega ^{1-p}e(t)(d_{1}r)^{ \frac{1}{p}-1}v(\theta )>\frac{\omega}{2}, $$

which together with (2.5)–(2.7) completes the proof of (2.8). □

Lemma 2.5

[15] Provided that function $f(x,t)$ satisfies

$$ |\partial _{x}^{k}\partial _{t}^{l}f(x,t)|\leq Cx^{-k}|f(x,t)| $$

for all sufficiently large $x>0$ and all $k,l:k+l\leq N$, where $N\in \mathbb{N}$. Suppose that

$$ \partial _{x}f(x,t)\geq c x^{-1}f(x,t)>0 $$

for all sufficiently large $x>0$. Then, the inverse function $g(y,t)$ of f in x satisfies

$$ |\partial _{y}^{k}\partial _{t}^{l}g(y,t)|\leq Cy^{-k}g(y,t) $$

for all $K+l\leq N$ and all sufficiently large y.

Using Lemmas 2.3 and 2.4, for $h\gg 1, t\in \mathbb{S}^{1}$, we have

$$ |\partial _{h}^{k}\partial _{t}^{l}r(h,t,\theta )|\leq Ch^{-k+1}, \quad k+l\leq 6,\quad \theta \in \mathbb{S}^{1}\setminus \Xi . $$

(2.9)

Thus, we write (2.3) as

$$ h(r,\theta ,t)=\omega r +r^{\frac{1}{p}}g(r,\theta ,t),\quad r=r(h,t, \theta ). $$

(2.10)

In fact,¹$v(t)\in C^{2}(\mathbb{S}^{1})$ does not belong to $C^{4}(\mathbb{S}^{1})$. We exchange the time and angle variables to prove Theorem 1.1.

2.3 Exchange of time and angle variables

Based on the conclusions in [15], the identity $rd\theta -hdt=-(hdt-rd\theta )$ guarantees that if we can solve $r=r(h,t,\theta )$ from (2.3) as a function of h, t, θ, then

$$ \frac{dh}{d\theta}=-\frac{\partial r}{\partial t}r(h,t,\theta ), \quad \quad \frac{dt}{d\theta}=\frac{\partial r}{\partial h}r(h,t, \theta ), $$

(2.11)

i.e., Eq. (2.11) is a Hamiltonian system with a Hamiltonian function $r=r(h,t,\theta )$ in which action, angle, and time variables are h, t, and θ, respectively. The following lemma gives a more detailed description of r in (2.10) according to the magnitude of h.

Lemma 2.6

Provided that $h\gg 1, \theta \in \mathbb{S}^{1}\setminus \Xi$, $t\in \mathbb{S}^{1}$, it holds that

$$ r(h,t,\theta )=\omega ^{-1}h-\omega ^{-p}z(t)F((d_{1}\omega ^{-1}h)^{ \frac{1}{p}}v(\theta ))+R(h,t,\theta ), $$

(2.12)

where

$$ |\partial _{h}^{k}\partial _{t}^{l}R(h,t,\theta )|\leq Ch^{-k+max\{ \gamma ,\frac{1}{p}\}},\quad k+l\leq 6. $$

(2.13)

Proof

Using the identity (2.10) yields

$$ r=\omega ^{-1}h-\omega ^{-1}r^{\frac{1}{p}}g(r,t,\theta ). $$

(2.14)

Utilizing the identity (2.6) and the Taylor formula, we obtain that function $g=g(r,\theta ,t)$ satisfies

$$\begin{aligned} & g(r,\theta ,t)=g(\omega ^{-1}h-\omega ^{-1}r^{\frac{1}{p}}g, \theta ,t) \\ &\quad \quad = g(\omega ^{-1}h,\theta ,t)+R_{0}(h,t,\theta ) \\ &\quad \quad =(\omega ^{-1}h)^{-\frac{1}{p}}\omega ^{1-p}z(t)F((d_{1} \omega ^{-1}h)^{\frac{1}{p}}v(\theta ))-d_{1}^{\frac{1}{p}}\omega ^{1-p} e(t)v(\theta )+R_{0}(h,t,\theta ), \end{aligned}$$

(2.15)

in which $R_{0}(h,t,\theta )=-\int _{0}^{1}g_{r}^{\prime }(\omega ^{-1}h-s\omega ^{-1}r^{ \frac{1}{p}}g,\theta ,t))\omega ^{-1}r^{\frac{1}{p}}gds$.

Substituting (2.14) into (2.15), we have

$$\begin{aligned} & r=\omega ^{-1}h-\omega ^{-1}g(r,t,\theta )(\omega ^{-1}h)^{ \frac{1}{p}}(1-h^{-1}r^{\frac{1}{p}}g)^{\frac{1}{p}} \\ &\quad =\omega ^{-1}h-\omega ^{-1}g(r,t,\theta )(\omega ^{-1}h)^{ \frac{1}{p}} \\ &\quad \quad \quad \quad \quad \quad +\frac{1}{p}\omega ^{-1}g(r,t, \theta ) (\omega ^{-1}h)^{\frac{1}{p}}\int _{0}^{1}(1-sh^{-1}r^{ \frac{1}{p}}g)^{\frac{1}{p}-1}h^{-1}r^{\frac{1}{p}}gds \\ &\quad =\omega ^{-1}h-\omega ^{-p}z(t)F((d_{1}\omega ^{-1}h)^{ \frac{1}{p}}v(\theta ))+R_{1}(h,t,\theta )+R_{2}(h,t,\theta )+R_{3}(h,t, \theta ), \end{aligned}$$

where

$$\begin{aligned} &R_{1}(h,t,\theta )=\omega ^{-(2+\frac{1}{p})}h^{\frac{1}{p}}\int _{0}^{1}g_{r}( \omega ^{-1}h-s\omega ^{-1}r^{\frac{1}{p}}g,\theta ,t)r^{\frac{1}{p}}gds, \\ &R_{2}(h,t,\theta )=\frac{1}{p}\omega ^{-(1+\frac{1}{p})}h^{ \frac{1}{p}-1}\int _{0}^{1}(1-sh^{-1}r^{\frac{1}{p}}g)^{\frac{1}{p}-1}r^{ \frac{1}{p}}g^{2}ds, \\ &R_{3}(h,t,\theta )=d_{1}^{\frac{1}{p}}\omega ^{-(p+\frac{1}{p})}v( \theta )e(t)h^{\frac{1}{p}}. \end{aligned}$$

Direct computation gives

$$ \partial _{h}^{k}\partial _{t}^{l}r^{\frac{1}{p}}(h,t,\theta )=\sum r^{ \frac{1}{p}-m}\partial _{h}^{k_{1}}\partial _{t}^{l_{1}}r(h,t,\theta ) \partial _{h}^{k_{2}}\partial _{t}^{l_{2}}r(h,t,\theta )\cdot \cdot \cdot \partial _{h}^{k_{m}}\partial _{t}^{l_{m}}r(h,t,\theta ) $$

with $1\leq m\leq k+l$, $k_{1}+k_{2}\cdot \cdot \cdot +k_{m}=k$ and $l_{1}+l_{2}+\cdots +l_{m}=l$. Using (2.9) yields

$$ |\partial _{h}^{k}\partial _{t}^{l}r^{\frac{1}{p}}(h,t,\theta )|\leq Ch^{-k+ \frac{1}{p}}. $$

Similarly, we acquire

$$\begin{aligned}& |\partial _{h}^{k}\partial _{t}^{l}g(h,t,\theta )|\leq Ch^{-k+ \frac{\gamma}{p}},\\& |\partial _{h}^{k}\partial _{t}^{l}g_{r}(\omega ^{-1}h-s\omega ^{-1}r^{ \frac{1}{p}}g,t,\theta )|\leq C^{-k-1+\frac{\gamma}{p}}. \end{aligned}$$

Using $p>2$ and the expression of $R_{1}$ yields

$$ |\partial _{h}^{k}\partial _{t}^{l}R_{1}(h,t,\theta )|\leq Ch^{-k-1+ \frac{2+2\gamma}{p}}\leq C h^{-k+\gamma}. $$

Analogously, we obtain

$$\begin{aligned}& |\partial _{h}^{k}\partial _{t}^{l}R_{2}(h,t,\theta )|\leq Ch^{-k+ \gamma},\\& |\partial _{h}^{k}\partial _{t}^{l}R_{3}(h,t,\theta )|\leq Ch^{-k+ \frac{1}{p}}. \end{aligned}$$

Letting $R(h,t,\theta )=R_{1}(h,t,\theta )+R_{2}(h,t,\theta )+R_{3}(h,t, \theta )$, we obtain that inequality (2.13) holds. □

2.4 Canonical transformation

In this part, two lemmas are established to make sure that the Poincare map of the new system is close to a twist map.

Lemma 2.7

There exists a canonical transformation $\psi _{1}$ of the form: $\psi _{1}:(\lambda ,\varphi )\rightarrow (h,t)$

$$ h=\lambda +U(\lambda ,t,\theta ),\quad \varphi =t+V(\lambda ,t, \theta ), $$

where U and V are $2\pi _{p}$ periodic about θ. Under $\psi _{1}$, the Hamiltonian function (2.12) is transformed into

$$ r_{1}(\lambda ,\varphi ,\theta )=\omega ^{-1}\lambda -\omega ^{-p} \hat{z}F((d_{1} \lambda \omega ^{-1})^{\frac{1}{p}}v(\theta ))+ \bar{R}_{1}(\lambda ,\varphi ,\theta ). $$

(2.16)

Moreover, for $\lambda \gg 1, \theta \in \mathbb{S}^{1}\setminus \Xi$, $t\in \mathbb{S}^{1}$, it holds that

$$ |\partial _{\lambda}^{k}\partial _{\varphi}^{l}\bar{R}_{1}(\lambda , \varphi ,\theta )|\leq C\lambda ^{-k+max\{\gamma ,\frac{\gamma +1}{p} \}},\quad k+l\leq 5. $$

(2.17)

Proof

We make a transformation $\psi _{1}:(\lambda ,\varphi )\rightarrow (h,t)$ implicitly given by

$$ h=\lambda +\partial _{t}S_{1}(\lambda ,t,\theta ),\quad \varphi =t+ \partial _{\lambda}S_{1}(\lambda ,t,\theta ) $$

(2.18)

with

$$ S_{1}(\lambda ,t,\theta )=\int _{0}^{t}\omega ^{1-p}z_{1}(t)F\Big((d_{1} \lambda \omega ^{-1})^{\frac{1}{p}}v(\theta )\Big)dt. $$

Under $\psi _{1}$, Hamiltonian (2.12) becomes

$$\begin{aligned} &r_{1}(\lambda ,\varphi ,\theta )=\omega ^{-1}(\lambda +\partial _{t}S_{1})- \omega ^{-p}\hat{z}F\Big((d_{1}\omega ^{-1}(\lambda +\partial _{t}S_{1}))^{ \frac{1}{p}}v(\theta )\Big)+\partial _{\theta}S_{1} \\ &\quad \quad \quad \quad \quad \quad -\omega ^{-p}z_{1}(t)F\Big((d_{1} \omega ^{-1}(\lambda +\partial _{t}S_{1}))^{\frac{1}{p}}v(\theta ) \Big)+R(\lambda +\partial _{t}S_{1},t,\theta ) \\ &\quad \quad \quad \quad =\omega ^{-1}\lambda -\omega ^{-p}\hat{z}F \Big((d_{1}\lambda \omega ^{-1})^{\frac{1}{p}}v(\theta )\Big)+R_{4}( \lambda ,\varphi ,\theta ) \\ &\quad \quad \quad \quad \quad \quad \quad +R_{5}(\lambda ,\varphi , \theta )+R_{6}(\lambda ,\varphi ,\theta )+R_{7}(\lambda ,\varphi , \theta ), \end{aligned}$$

where $z_{1}(t)=z(t)-\hat{z}$ and

$$\begin{aligned} &R_{4}=-\omega ^{p}\hat{z}\int _{0}^{1}\partial _{\lambda}F\Big((d_{1} \omega ^{-1}(\lambda +\mu \partial _{t}S_{1}))^{\frac{1}{p}}v(\theta ) \Big)\partial _{t}S_{1}d\mu \\ &\quad =-\frac{\hat{z}\omega ^{p}}{p}\int _{0}^{1}f\Big((d_{1} \omega ^{-1}(\lambda +\mu \partial _{t}S_{1}))^{\frac{1}{p}}v(\theta ) \Big)v(\theta )d_{1}\omega ^{-1} (\lambda +\mu \partial _{t}S_{1})^{- \frac{1}{q}}\partial _{t}S_{1}d\mu , \end{aligned}$$

(2.19)

$$\begin{aligned} &R_{5}=-\int _{0}^{1}\omega ^{-p}z_{1}(t)\partial _{d_{1}\omega ^{-1} \lambda}F\Big((d_{1}\omega ^{-1} (\lambda +\mu \partial _{t}S_{1}))^{ \frac{1}{p}}v(\theta )\Big)d_{1}\omega ^{-1}\partial _{t}S_{1}d\mu , \\ &R_{6}=\partial _{\theta}S_{1}(\lambda ,t,\theta ), \\ &R_{7}=R(\lambda +\partial _{t}S_{1},t,\theta ). \end{aligned}$$

From Lemma 2.1, for $\lambda \gg 1$ and $k+l\leq 6$, we have

$$ |\partial _{\lambda}^{k}\partial _{t}^{l}S_{1}(\lambda ,t,\theta )| \leq C\lambda ^{-k+\frac{\gamma +1}{p}}, $$

(2.20)

which together with (2.18) yields

$$\begin{aligned} \left \{ \textstyle\begin{array}{l} \frac{1}{2}< \partial _{\varphi}t(\lambda ,\varphi ,\theta )< \frac{3}{2},\quad |\partial _{\lambda}t(\lambda ,\varphi ,\theta )|< \lambda ^{-2+\frac{\gamma +1}{p}}, \\ |\partial _{\lambda}h(\lambda ,\varphi ,\theta )|\leq C,\quad | \partial _{\varphi}h(\lambda ,\varphi ,\theta )|\leq C\lambda ^{ \frac{\gamma +1}{p}}. \end{array}\displaystyle \right . \end{aligned}$$

(2.21)

For $2\leq k+l\leq 5$, utilizing direct calculations gives rise to

$$ |\partial _{\lambda}^{k}\partial _{\varphi}^{l}h(\lambda ,\varphi , \theta )|\leq C \lambda ^{-k+\frac{\gamma +1}{p}},\quad |\partial _{ \lambda}^{k}\partial _{\varphi}^{l}t(\lambda ,\varphi ,\theta )|\leq C \lambda ^{-k-1+\frac{\gamma +1}{p}}. $$

(2.22)

First, we prove $|\partial _{\lambda}^{k}\partial _{\varphi}^{l}R_{4}|\leq C \lambda ^{-k+ \frac{\gamma +1}{p}}$. Direct computation gives

$$ \partial _{\lambda}^{k}\partial _{\varphi}^{l}\partial _{t}S_{1}( \lambda ,t,\theta )=\sum \partial _{\lambda}^{m}\partial _{t}^{n+1}S_{1}( \lambda ,t,\theta ) \partial _{\lambda}^{k_{1}}\partial _{\varphi}^{l_{1}} t\partial _{\lambda}^{k_{2}}\partial _{\varphi}^{l_{2}}t\cdot \cdot \cdot \partial _{\lambda}^{k_{n}}\partial _{\varphi}^{l_{n}}t $$

with $1\leq m+n\leq k+l$, $m+k_{1}+k_{2}\cdot \cdot \cdot +k_{m}=k$ and $l_{1}+l_{2}+\cdots +l_{n}=l$. Using (2.20), (2.21), and (2.22) yields

$$ |\partial _{\lambda}^{k}\partial _{\varphi}^{l}\partial _{t}S_{1}| \leq C\lambda ^{-k+\frac{\gamma +1}{p}}. $$

In the same way, we obtain

$$ |\partial _{\lambda}^{k}\partial _{\varphi}^{l}(\lambda +\mu \partial _{t}S_{1})^{-\frac{1}{q}}|\leq C\lambda ^{-k-\frac{1}{q}} $$

and

$$ \Bigg|\partial _{\lambda}^{k}\partial _{\varphi}^{l}f\Big((d_{1} \omega ^{-1}(\lambda +\mu \partial _{t}S_{1}))^{\frac{1}{p}}v(\theta ) \Big)\Bigg|\leq C\lambda ^{-k+\frac{\gamma}{p}}. $$

Noticing $0<\frac{1}{p-1}<\gamma <1$, from (2.19), we have $|\partial _{\lambda}^{k}\partial _{\varphi}^{l}R_{4}|\leq C \lambda ^{-k+ \frac{\gamma}{p}}$. Similarly, we obtain

$$ |\partial _{\lambda}^{k}\partial _{\varphi}^{l}R_{i}|\leq C \lambda ^{-k+ \frac{\gamma +1}{p}},\quad i=5,6. $$

Applying (2.13), (2.21), and (2.22) gives rise to

$$ |\partial _{\lambda}^{k}\partial _{\varphi}^{l}R_{7}|\leq C \lambda ^{-k+max \{\gamma ,\frac{1}{p}\}}. $$

Set $\bar{R}_{1}(\lambda ,\varphi ,\theta )=R_{4}(\lambda ,\varphi , \theta )+R_{5}(\lambda ,\varphi ,\theta ) +R_{6}(\lambda ,\varphi , \theta )+R_{7}(\lambda ,\varphi ,\theta )$. Hence, inequality (2.17) holds. □

Next, we eliminate the new time variable θ at the first time by constructing the transformation.

Lemma 2.8

There exists a canonical transformation $\psi _{2}: (\lambda ,\varphi )\rightarrow (\lambda ,\tau )$:

$$ \psi _{2}:\lambda =\lambda ,\quad \varphi =\tau +\partial _{\lambda}S_{2}( \lambda ,\theta )). $$

Under $\psi _{2}$, the Hamiltonian (2.16) is transformed into

$$ r_{2}(\lambda ,\tau ,\theta )=\omega ^{-1}\lambda -\omega ^{-p} \hat{z}\bar{F}(\lambda )+\bar{R}_{2}(\lambda ,\tau ,\theta ). $$

(2.23)

The new disturbance term $\bar{R}_{2}$ satisfies

$$ |\partial _{\lambda}^{k}\partial _{\tau}^{l}\bar{R}_{2}(\lambda , \tau ,\theta )|\leq C\lambda ^{-k+max\{\gamma ,\frac{\gamma +1}{p}\}} $$

(2.24)

for $k+l\leq 5, \lambda \gg 1$, $\theta \in \mathbb{S}^{1}\setminus \Xi$ and $t\in \mathbb{S}^{1}$.

Proof

We choose generating function

$$ S_{2}(\lambda ,\theta )=\int _{0}^{\theta}\omega ^{-p}\hat{z}[F((d_{1} \omega ^{-1}\lambda )^{\frac{1}{p}}v(\theta ))-\bar{F} (\lambda )]d \theta . $$

Under $\psi _{2}$, then the Hamiltonian (2.16) is transformed into

$$ r_{2}(\lambda ,\tau ,\theta )=r_{1}(\lambda ,\varphi ,\theta )+ \partial _{\theta}S_{2}=\omega ^{-1}\lambda -\omega ^{-p}\hat{z} \bar{F}(\lambda )+\bar{R}_{2}(\lambda ,\tau ,\theta ), $$

where

$$ \bar{R}_{2}(\lambda ,\tau ,\theta )=\bar{R}_{1}(\lambda ,\tau + \partial _{\lambda}S_{2},\theta ). $$

(2.25)

Thus, inequality (2.24) is obtained from (2.17), (2.23), (2.25) and Lemma 2.2. The proof of Lemma 2.8 is finished. □

3 Proof of main result

Without loss of generality, we only need to prove Theorem 1.1 for the case $\hat{e}>0$. For $\hat{e}<0$, the proof is similar. For given $0<\delta <1$, define transformation $\psi _{3}:(\lambda ,\tau )\rightarrow (v,\tau )$ by

$$ \bar{F}'(\lambda )=\delta v\omega ^{p}(\hat{z})^{-1},\quad \tau = \tau ,\quad 1\leq v \leq 4. $$

(3.1)

Due to $\lambda \rightarrow +\infty$, $\bar{F}'(\lambda )\rightarrow 0$, thus $\lambda \rightarrow +\infty \Leftrightarrow \delta \rightarrow 0$. For $\lambda =\lambda (\delta v)$, the following estimates hold.

Lemma 3.1

$c\delta ^{\frac{p}{\gamma +1-p}}\leq \lambda (\delta v)\leq C\delta ^{ \frac{p}{\gamma +1-p}}$, $|\partial _{v}^{k}\lambda (\delta v)| \leq C \lambda (\delta v)\quad k\leq 4$.

Proof

From Lemma 2.2 and (3.1), we have $c\delta ^{\frac{p}{\gamma +1-p}}\leq \lambda (\delta v)\leq C\delta ^{ \frac{p}{\gamma +1-p}}$.

Differentiating (3.1) with respect to v, we have $\bar{F}''(\lambda )=\omega ^{p}\delta \hat{z}^{-1}$. Using Lemma 2.2 yields

$$ |\partial _{v}\lambda |=| \frac{\omega ^{p}\delta \hat{z}^{-1}}{\bar{F}''(\lambda )}|=| \frac{\omega ^{p}\delta \hat{z}^{-1}\lambda}{\bar{F}''(\lambda )\lambda}| \leq |\frac{\delta \lambda}{\lambda ^{-1+\frac{\gamma +1}{p}}}|=| \frac{c\delta \lambda}{\bar{F}'(\lambda )}|= \frac{c\delta \lambda}{\delta v}\leq C\lambda . $$

Taking $k(k>1)$ order derivative about v on both sides of (3.1), we obtain

$$ \bar{F}''(\lambda )\partial _{v}^{k}\lambda +\sum _{s=2}^{s=k}\bar{F}^{(s+1)} \partial _{v}^{k_{1}}\lambda \partial _{v}^{k_{2}}\cdot \cdot \cdot \partial _{v}^{k_{s}}\lambda =0 $$

with $k_{1}+k_{2}+\cdots +k_{s}=k$. Thus,

$$ \partial _{v}^{k}\lambda =\sum _{s=2}^{s=k} \frac{\bar{F}^{(s+1)}\partial _{v}^{k_{1}} \lambda \partial _{v}^{k_{2}}\cdot \cdot \cdot \partial _{v}^{k_{s}}\lambda}{\bar{F}''(\lambda )}. $$

From Lemma 2.2, using the induction methods yields

$$\begin{aligned} |\partial _{v}^{k}\lambda |\leq C \lambda , \quad k=2,3,4, \end{aligned}$$

which completes the proof of Lemma 3.1. □

From the definition $\psi _{3}$, we have

$$ \frac{dv}{d\theta}=\delta ^{-1}\omega ^{-p}\hat{z}\bar{F}''(\lambda ) \frac{d\lambda}{d\theta}=\delta ^{-1}\omega ^{-p}\hat{z} \bar{F}''( \lambda )\partial _{\tau}\bar{R}_{2}(\lambda ,\tau ,\theta ). $$

Introducing a new time variable ϑ by $\theta =-\vartheta$ yields

$$ \frac{dv}{d\vartheta}=l_{1}(v,\tau ,\vartheta ,\delta ),\quad \frac{d\tau}{d\vartheta}=-\omega ^{-1}+\delta v+l_{2}(v,\tau , \vartheta ,\delta ), $$

(3.2)

where

$$\begin{aligned}& l_{1}(v,\tau ,\vartheta ,\delta )=\delta ^{-1}\omega ^{-p}\hat{z} \bar{F}''(\lambda )\partial _{\tau}\bar{R}_{2}(\lambda ,\tau ,- \vartheta ),\\& l_{2}(v,\tau ,\vartheta ,\delta )=-\partial _{\lambda}\bar{R}_{2}( \lambda ,\tau ,-\vartheta ). \end{aligned}$$

Lemma 3.2

Provided that $p>2$, $\frac{1}{p-1}<\gamma <1$, $0<\delta \ll 1$, $k+l\leq 4$ and $\tau \in \mathbb{S}^{1}\setminus \Xi (i=1,2)$, it holds that

$$ |\partial _{v}^{k}\partial _{\tau}^{l}l_{i}(v,\tau ,\vartheta , \delta )|\leq C \delta ^{\sigma}, $$

(3.3)

where $\sigma =\frac{p}{\gamma +1-p}(-1+\gamma )>0$.

Proof

For $k=0$, we have

$$ |\partial _{\tau}^{l}l_{2}|=|\partial _{\lambda}\partial _{\tau}^{l} \bar{R}_{2}(\lambda ,\tau ,-\vartheta )|\leq C\lambda ^{-1+max\{ \frac{\gamma +1}{p},\gamma \}}\leq C\delta ^{\frac{p}{\gamma +1-p}(-1+max \{\frac{\gamma +1}{p},\gamma \})}\leq C\delta ^{\sigma}. $$

Using the assumption $\gamma >\frac{1}{p-1}$ derives $\frac{1+\gamma}{p}<\gamma$. We have $|\partial _{\tau}^{l}l_{2}|\leq C\delta ^{\sigma}$.

For $k>0$, we obtain

$$\begin{aligned} &|\partial _{v}^{k}\partial _{\tau}^{l}l_{2}|=|\partial _{v}^{k} \partial _{\tau}^{l}\partial _{\lambda}\bar{R}_{2}(\lambda ,\tau ,- \vartheta )| \\ &\quad \quad \quad \quad \leq C\lambda ^{-1+max\{\frac{\gamma +1}{p}, \gamma \}} \\ &\quad \quad \quad \quad \leq C\delta ^{\frac{p}{\gamma +1-p}(-1+max \{\frac{\gamma +1}{p},\gamma \})} \\ &\quad \quad \quad \quad \leq C\delta ^{\sigma}. \end{aligned}$$

For $l_{1}$, we have the same estimate. The proof of Lemma 3.2 is completed. □

From Lemmas 3.1–3.2 and (3.3), we see that the solutions of (3.2) with initial value $v(0)=v_{0}\in [1,2]$, $\tau (0)=\tau _{0}$ do exist for $0\leq \vartheta \leq 4\pi _{p}$ if $\delta \ll 1$. Integrating (3.2) from 0 to $2\pi _{p}$, we derive that Poincaré map P in (3.2) takes the following form

$$\begin{aligned} P:\left \{ \textstyle\begin{array}{l} \tau _{2\pi _{p}}=\tau _{0}-\omega ^{-1}2\pi _{p}+\delta (v_{0}+P_{2}(v_{0}, \tau _{0},\delta )), \\ v_{2\pi _{p}}=v_{0}+\delta P_{1}(v_{0},\tau _{0},\delta ), \end{array}\displaystyle \right . \end{aligned}$$

where $|\partial _{v_{0}}^{k}\partial _{\tau _{0}}^{l}P_{i}|\leq C\delta ^{ \sigma -1}$ for $k+l\leq 4$, $i=1,2$.

Since P is a Poincarè map in (3.2), it is an area-preserving, and thus it possesses the intersection property in the annulus $[1,2]\times \mathbb{S}^{1}$. Namely, if Γ is an embedded circle in $[1,2]\times \mathbb{S}^{1}$ homotopic to a circle v= constant, then $P(\Gamma )\cap \Gamma \neq \emptyset$ (see [18]). Now, we have verified that the mapping P satisfies all the conditions of Moser’s twist theorem. Hence, there exists an invariant curve $\Gamma _{\delta}$ of P surrounding $v_{0}=1$ if $\delta \ll 1$. The $\Gamma _{\delta}$ is located in ring domain $\{(v,\tau )|\delta < v<2\delta \}$. Note that $\delta \rightarrow 0\Leftrightarrow \lambda \rightarrow \infty$. The points $(\lambda ,\varphi ,\theta )$ satisfying $r_{1}(\lambda ,\varphi ,\theta )=r_{1}(\lambda ,\varphi ,\theta )|_{( \lambda ,\varphi )\in \Gamma _{\delta}}$ form an invariant torus $\mathbf{T}_{\delta}^{2}$ in the extended phase space $(\lambda ,\varphi ,\theta )$. Thus, $\psi ^{-1}(\Gamma _{\delta})$ is an invariant torus for Eq. (2.1) in $(x,y,t)\in \mathbb{R}^{2}\times \mathbb{S}^{1}$, which is far away from $(0,0)$, where $\psi =\psi _{1}\psi _{0}$. The solution of Eq. (2.1) starting from inside of $\psi ^{-1}(\Gamma _{\delta})$ is contained inside of $\psi ^{-1}(\Gamma _{\delta})$. Thus, the solution of Eq. (2.1) is bounded. The proof of Theorem 1.1 is finished.