# 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 $$a^{-\frac{1}{p}}+b^{-\frac{1}{p}}=2\omega ^{-1}$$, Ï‰ 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.Â (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.Â (1.2) is very complicated. Ortega [1] proves that every solution to Eq.Â (1.2) is bounded if $$h\in C^{4}(\mathbb{S}^{1})$$, where $$\mathbb{S}^{1}=\mathbb{R}/2\pi \mathbb{Z}$$, and Î³ 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.Â (1.2) are unbounded. Ambrosio [3] establishes the boundedness to solutions to fractional relativistic SchrÃ¶dinger 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.Â (1.2) (see also [8â€“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.Â (1.1) and obtains that all the solution to Eq.Â (1.1) are bounded under certain assumptions. Liu [12] discusses the bounded condition for Eq.Â (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.Â (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.Â (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.Â (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.Â 2 presents action-angle variables, exchanging time and angle variables, and several lemmas. SectionÂ 3 provides the proof of TheoremÂ 1.1.

## 2 Preliminaries

In this part, we provide several lemmas that help prove TheoremÂ 1.1. Throughout Sect.Â 2, we assume that the hypotheses of TheoremÂ 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.Â (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 [15â€“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.â€ƒâ–¡

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 LemmaÂ 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 LemmaÂ 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)â€“(2.7) completes the proof of (2.8).â€ƒâ–¡

### 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,Footnote 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 TheoremÂ 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, Î¸, 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.Â (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 Î¸, 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.â€ƒâ–¡

### 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 Î¸. 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 LemmaÂ 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.â€ƒâ–¡

Next, we eliminate the new time variable Î¸ 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 LemmaÂ 2.2. The proof of LemmaÂ 2.8 is finished.â€ƒâ–¡

## 3 Proof of main result

Without loss of generality, we only need to prove TheoremÂ 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 LemmaÂ 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 LemmaÂ 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 LemmaÂ 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 LemmaÂ 3.1.â€ƒâ–¡

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 Ï‘ 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 LemmaÂ 3.2 is completed.â€ƒâ–¡

From Lemmas 3.1â€“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 PoincarÃ© 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 PoincarÃ¨ 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 Î“ 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 Moserâ€™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.Â (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.Â (2.1) starting from inside of $$\psi ^{-1}(\Gamma _{\delta})$$ is contained inside of $$\psi ^{-1}(\Gamma _{\delta})$$. Thus, the solution of Eq.Â (2.1) is bounded. The proof of TheoremÂ 1.1 is finished.

## Data availability

No datasets were generated or analysed during the current study.

## Notes

1. $$C^{4}$$ is four times continuously differentiable functions in $$\mathbb{R}$$ or $$\mathbb{S}^{1}$$, and $$C^{6}$$ is six times continuously differentiable functions in $$\mathbb{R}$$ or $$\mathbb{S}^{1}$$.

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## Acknowledgements

The authors are very grateful to the reviewers for their valuable suggestions and comments, which lead to the meaningful improvement of this work.

## Funding

This work is supported by National Natural Science Foundation of China (No. 12361042) and the 14th Five Year Key Discipline of Xinjiang Autonomous Region (78756342).

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Xing, X., Wang, H. & Lai, S. Boundedness of solutions to a second-order periodic system with p-Laplacian and unbounded perturbation terms. Bound Value Probl 2024, 103 (2024). https://doi.org/10.1186/s13661-024-01911-w