Abstract
We investigate a second-order periodic system with singular potential and resonance. Utilizing the main integral method and fixed point theorems, we establish the existence and multiplicity of periodic solutions with respect to time under certain assumptions on the unbounded or oscillatory term.
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1 Introduction and main result
Many scholars have investigated the singular second-order equation
with functions \(V(x)\), \(g(x)\), and \(p(t)\) satisfying certain restrictions. The multiplicity and existence of periodic solutions for Eq. (1.1) are discussed by utilizing the topology degree theory [1]. Qian and Torres [2] use the Poincaré–Birkhoff twist theorem to find the dynamical features of Eq. (1.1). Jiang [3] employs invariant curves and Moser’s small twist theorem to discuss the boundedness of solutions for Eq. (1.1). The unbounded and periodic solutions of Eq. (1.1) may coexist [4]. Assuming that \(g(x)=0\) and \(p(t+\pi )=p(t)\), Capietto et al. [5] consider Eq. (1.1) with
where \(x_{+}=\max \{x,0\}\), \(x_{-}=\max \{-x,0\}\), and \(\nu >0\) is an integer. Using the Moser twist theorem and the Lazer–Leach assumption
Capietto et al. [5] investigated the boundedness of solutions and quasi-periodic solutions for Eq. (1.1). Following the ideas in [5], Jiang [3] and Liu [6] and Wang and Jiang [7] discussed the boundedness of solutions for Eq. (1.1) under different conditions.
To clearly understand the objective of our work, we recall the main result about Eq. (1.1) in Lazer and Leach [8]. Suppose that \(g(x)\) is smooth and bounded and that \(V(x)\) satisfies the conditions
where the domain of \(V(x)\) is \((a,+\infty )\), \(m>0\) is an integer, and the constant a belongs to \(\in (-\infty,0)\). Let
Applying the analysis of phase plane and topological degree methods, it is proved in [8] that Eq.‘(1.1) possesses at least one 2π-periodic solution, provided that there is \(g_{0}\in [g_{*}^{-},g_{*}^{+}]\) \((g_{*}^{-}=\liminf_{\rho \rightarrow +\infty}g_{*}(\rho ), g_{*}^{+}=\limsup_{\rho \rightarrow +\infty}g_{*}(\rho ) )\) (i.e.. \(p_{*}\) has a regular value of \(g_{0}\)) and the number of zeros of \(p_{*}-g_{0}\) in \([0,2\pi /m]\) is not 2.
In 2019, Ma [9] considered the periodic solution of Eq. (1.1). Suppose that \(V'(x)=n^{2}x, p(t)\in C^{2}(\mathbb{R}/2\pi \mathbb{Z})\), and \(g(x)\in C^{1}(\mathbb{R})\) with the restrictions
Ma [9] obtained that Eq. (1.1) possesses at least one 2π solution, provided that
and Eq. (1.1) possesses an unbounded sequence of 2π-periodic solutions if
We observe that the function \(V(x)\) is globally defined in \(\mathbb{R}\) and the function \(g(x)\) is unbounded or oscillatory in [9]. Note that in [8], \(V(x)\) possesses a repulsive singularity at a, and \(g(x)\) is bounded. Also, in [5], \(V(x)\) possesses a repulsive singularity at −1, and \(g(x)=0\).
When the function \(V(x)\) is of the form (1.2), a natural question is to find restrictions imposed on unbounded function \(g(x)\) to make Eq. (1.1) have at least one π-periodic solution and possess an unbounded sequence of π-periodic solutions. The objective of this work is to handle this problem. Precisely speaking, we investigate the existence and multiplicity of periodic solutions of the problem
where \(\mathbb{S}^{1}=\mathbb{R}/\pi \mathbb{Z}\) and \(\lim_{x\rightarrow +\infty}x^{k-\frac{1}{2}}g^{(k)}(x)=0\), \(k=0,1\). Here we state that the function \(g(x)\) considered in our work is different from those in [5, 8] and is the same as that in [9]. The novelty of our work is that the function \(V(x)\) is of the the form (1.2), which is different from those in [8, 9].
The auxiliary equation of Eq. (1.1) takes the form
The Hamiltonian function of (1.4) has the expression
where \(G(x)=\int _{0}^{{x}}g(s)\,ds\). For \(H>0\), we denote by \(\tau (H)\) the least positive period of the orbit \(\Gamma _{H}:H_{0}(x,y)=H, \bar{p}=\int _{0}^{\pi}p(s)e^{is}\,ds\). Set
Equation (1.4) possesses the following autonomous Hamiltonian system:
Now we state the main result of our work.
Theorem 1.1
Suppose that \(g(x)\in C^{1}(\mathbb{R})\), \(p(t)\in C^{2}(\mathbb{R}/\pi \mathbb{Z})\), and
Then
(i) Problem (1.3) possesses at least one π-periodic solution if
(ii) Problem (1.3) possesses an unbounded sequence of π-periodic solutions if
In Sect. 2, we present several lemmas, and in Sect. 3, we provide the proof of Theorem 1.1.
2 Preliminaries
2.1 Action-angle coordinates
To use action-angle variables, we write the auxiliary equations
with the Hamiltonian function
Let \(T_{0}(h)\) denote the time period of the integral curve \(\Gamma _{h}\) of (2.1) with \(H_{1}(x,y)=h\). Denote by \(I=I_{0}(h)\) the area enclosed by the closed curve \(\Gamma _{h}\) for \(h>0\). Assume that \(-1<-\alpha _{h}<0<\beta _{h}\) satisfy \(V(-\alpha _{h})=V(\beta _{h})=h\) and
Thus we have
For conciseness in the following discussions, we always use c or C to represent positive constants.
We acquire \(c\sqrt{h}< I_{-}(h)< C\sqrt{h}\). Let \(h=h_{0}(I)\) be the inverse function of \(I=I_{0}(h)\). Using the expression of \(V(x)\) and the definition of \(I_{0}(h)\), we derive that
Lemma 2.1
[3] For \(n=0,1,2\), we have
where
For \((x,y)\in (-1,+\infty )\times \mathbb{R}\), we define the transformation \(\Psi _{1}:(x,y)\rightarrow (\theta,I)\) by
and
Note that the first equation in (1.3) possesses the Hamiltonian system
associated with
In the new variables \((\theta,I)\), we write the Hamiltonian systems of (1.3) and (1.4) in the forms
and
respectively, where
and
For \(x>0\), using (2.4) gives rise to
Using formulas (2.5)–(2.9), we want to obtain estimates of the function \(G(x(I,\theta ))\). We need the following lemma.
Lemma 2.2
[3] For sufficiently large I, if \(x>0\), then
For sufficiently large I, if \(x<0\), then
To transform the first equation in problem (1.3) into a nearly integrable equation, we introduce the transformation \(\Psi _{2}:(I,\theta,t)\rightarrow (H,\varphi,\eta )\),
that is, the time and energy are the new angular and action variables, respectively.
As \(I\rightarrow +\infty \), from (1.6), (2.2), and Lemma 2.2 we conclude that \(H/I\rightarrow 1\) and \(\frac{\partial H}{\partial I}>0\). Applying the implicit function theorem, we derive that there exists a function \(R_{2}(H,t,\theta )\) belonging to \(C^{2}\) with \(|R_{2}|< H/2\pi \) such that the function
satisfies (2.7). Thus, under the transform \(\Psi _{2}\), the Hamiltonian function (2.7) is transformed into the Hamiltonian function (2.10).
Note that the inverse function of \(I_{0}\) is \(h_{0}\). Using (2.7), we acquire
Utilizing (1.6), Lemma 2.2, and \(|R_{2}|< H/2\pi \) gives rise to
where (here and further) \(\epsilon (H)\) stands for a nonnegative function satisfying \(\lim_{H\rightarrow +\infty}\epsilon (H)=0\).
Similarly, we can prove that there is a function \(R_{1}(H,\theta )\) in space \(C^{2}\) with \(|R_{1}|< H/2\pi \) such that
satisfies
Thus we have
and
Let \(\Xi =\{\theta \in \mathbb{S}^{1}: \sin (\frac{T_{0}(h)}{\pi} \theta -\frac{T_{-}(h)}{2} )=0\}\). Obviously, by Lemma 2.1 the measure of Ξ is zero.
Lemma 2.3
If the function \(R(H,t,\theta )\) belongs to \(C^{2}\) and \(|R(H,t,\theta )|\leq \epsilon (H) H\), provided that \(h=\frac{H}{\pi}+R(H,t,\theta )\), then
for \(k=0\) and \(\theta \in \mathbb{S}^{1}\) and for \(k=1\) and \(\theta \in \mathbb{S}^{1}\setminus \Xi \).
Proof
From (1.6), for any \(\epsilon >0\), there exist constants \(M_{k}>0\) such that
Since \(\lim_{H\rightarrow +\infty}h(H,t,\theta )\rightarrow + \infty \) uniformly for t and \(\theta \setminus \Xi \), there exist positive numbers \(M_{k}>0\) such that for \(H>M_{k}\),
Thus, for \(|\sqrt{2h}\sin (\frac{T_{0}(h)}{\pi}\theta - \frac{T_{-}(h)}{2} ) |>M_{k}\), we have
and for \(|\sqrt{2h}\sin (\frac{T_{0}(h)}{\pi}\theta - \frac{T_{-}(h)}{2} ) |\leq M_{k}\), we have
which ends the proof. □
For \(r\in \mathbb{R}\), when \(H\gg 1\), we write \(u=u(H,t,\theta )\in C^{2}(r,\epsilon )\) if \(|\partial _{H}^{k}u|\leq \epsilon (H)H^{r-k}\) for \(k=0,1,2\) and \(|\partial _{H}^{k}\partial _{t}u|\leq CH^{\frac{1}{2}-k}\) for \(k=0,1\).
Lemma 2.4
Let \(h=\frac{H}{\pi}+u(H,t,\theta )\) with \(u\in C(\frac{3}{4},\epsilon )\). Then
for \(k=0,1\) and \(\theta \in \mathbb{S}^{1}\setminus \Xi \).
Proof
When \(x>0\), by a direct computation we have
and
From (2.3), Lemma 2.1, and Lemma 2.3 we obtain (2.15). When \(x<0\), using Lemma 2.2, we acquire that (2.15) holds. The proof is finished. □
Lemma 2.5
For all \(\epsilon >0\), t, and \(\theta \in \mathbb{S}^{1}\setminus \Xi \), as \(H\rightarrow +\infty \), we have
Proof
(i) When \(k+l=0\), the conclusion follows from (1.6), (2.2), (2.11), and Lemma 2.2.
(ii) When \(k+l=1\), define
For \(|\triangle |\geq \frac{1}{2}\) and \(H\gg 1\), we get
Using Lemma 2.2 and (1.6) yields
Applying Lemma 2.2 gives rise to
Thus \(|\partial _{H}R_{2}(H,t,\theta )|\leq \epsilon (H) H^{\frac{3}{4}-1}\) and \(|\partial _{t}R_{2}(H,t,\theta )|\leq C H^{\frac{1}{2}}\).
(iii) When \(k+l=2\), differentiating both sides of (2.16) with respect to H and t, respectively, we acquire
Differentiating both sides of (2.17) about t gives
A direct computation yields
From Lemmas 2.1–2.5 we have \(| \partial _{H}\Delta |<\epsilon (H) H^{-\frac{5}{4}}\) and \(|\partial _{H}R_{2}(H,t,\theta )|<\epsilon (H)H^{\frac{3}{4}-1}\).
Similarly to the above estimates, we have
The proof is finished. □
Using (2.14) and (2.15), similarly to the proof of Lemma 2.5, we acquire the conclusion.
Lemma 2.6
For all \(\epsilon >0\), t, and \(\theta \in \mathbb{S}^{1}\setminus \Xi \), as \(H\rightarrow +\infty \), we have
Next we rewrite (2.5) with new variables as a nearly integrable system. To handle this process, we apply
to express
Thus we obtain
where
Lemma 2.7
For all \(\epsilon >0\), t, and \(\theta \in \mathbb{S}^{1}\setminus \Xi \), as \(H\rightarrow +\infty \), the function \(R(H,t,\theta )\) possesses the property
Proof
(i) When \(k+l=0\), using Lemmas 2.1 and 2.5 and (2.18), we obtain (2.20).
(ii) When \(k=1\) and \(l=0\),
and
We derive (2.20) from Lemmas 2.2 and 2.4–2.6.
(iii) When \(k=0\) and \(l=1\),
and
We obtain inequality (2.20) from Lemmas 2.1–2.2 and 2.4–2.6. □
Now we rewrite (2.5) with the variables \(H,t,\theta \). Utilizing (2.19) yields
where
For the new perturbation R̃, from Lemmas 2.1–2.2 and 2.7 we have
Lemma 2.8
If (1.6) holds, then
Proof
Using (2.12) and (2.13), we get
Applying (1.5), (2.2), and the definition of \(h_{0}(I)\) gives rise to
Thus
For \(k=0\), we obtain (2.23) from Lemmas 2.1, 2.4, and 2.6. Note that
Using Lemmas 2.1, 2.4, and 2.6, we get \(H^{\frac{5}{4}}\tau '(H)\leq \epsilon (H)\). Thus we obtain
Hence (2.23) holds for \(k=1\). The proof is finished. □
2.2 Canonical transformations
Lemma 2.9
If (1.6) holds, then there exists a canonical transform
associated with \(T(\rho,\theta +\pi )=T(\rho,\theta )\) such that the transformed system of (2.21) takes the form
where
and
For the new perturbation R̂ and for all \(\epsilon >0\), t, and \(\theta \in \mathbb{S}^{1}\setminus \Xi \), if \(H\rightarrow +\infty \) and \(k+l\leq 1\), then
Proof
Define \(\Psi _{3}\) implicitly by
where the function \(S=S(H,\tau,\theta )\) will be determined later. Using \(\Psi _{3}\), (2.21) becomes
Now we choose
Therefore \(\rho =H\). Assuming that \(T(H,\theta )=\partial _{H}S(H,\theta )\), we know that \(\Psi _{3}\) takes the form
and the function Ĩ reads as
where
By a direct computation, (2.26) is derived from Lemmas 2.1, 2.2, and 2.6–2.8. □
3 Proof of main result
Now we introduce a small parameter \(\delta >0\) satisfying
where a and b such that \(b>a>0\) do not depend on \(\delta >0\).
In the new variables \((v,\tau )\), (2.25) takes the form
where
Denote \(\hat{R}(v,\tau,\theta,\delta )=\delta ^{2}\hat{R}(\delta ^{-2}v, \tau,\theta )\). From (2.22) we derive that for \(k+l\leq 1\),
Because of
we write system (3.1) in the form
Let \((v(\theta,v_{0},\tau _{0}),\tau (\theta,v_{0},\tau _{0}))\) denote the solution of (3.3) associated with the initial data
Utilizing (3.2), we conclude that if \(\delta \ll 1\), then a solution of (3.3) exists in \([0,2\pi ]\) for any \((v_{0},\tau _{0})\in [a,b]\times [0,\pi ]\). Moreover,
Assume that the solution \((v(\theta,v_{0},\tau _{0}),\tau (\theta,v_{0},\tau _{0}))\) is of the form
Then the Poincaré map of (3.3), represented by P, possesses the expression
Since \((v(\theta,v_{0},\tau _{0}),\tau (\theta,v_{0},\tau _{0}))\) is a solution of (3.3), we acquire
From (3.2) and (3.4) we derive that
uniformly in \(\theta \in \mathbb{S}^{1}\setminus \Xi \).
Lemma 2.9, (3.2), (3.3), and (3.4) yield
Thus the Poincaré map P of (3.3) takes the form
where
and
Applying arguments similar to those in [5], we obtain the following estimates:
When \(x<0\), we have that
When \(x>0\), from the definition of θ it follows that
Thus from (3.5) we obtain
Similarly, we derive from (3.6) that
We conclude that the Poincaré map P reads as
Using
and noticing (1.7), we acquire
Let \(\varpi >0\) and
Since \(\sqrt{H}\Gamma '(H)\rightarrow 0\) as \(H\rightarrow +\infty \), there exists a number \(\bar{H}>\frac{1}{b}>0\) satisfying
for \(H\geq \bar{H}\). Utilizing (3.11), we can choose a sequence \(\{H_{m}^{1} \}_{m=1}^{\infty}\) with \(\bar{H}\leq H_{m}^{1}\rightarrow +\infty \) such that
and
Take \(\delta _{1m}=(bH_{m}^{1})^{-\frac{1}{2}}\). Then we have \(\delta _{1m}\rightarrow 0\) as \(m\rightarrow +\infty \). It follows from (3.13) and (3.14) that
and hence \([\delta _{1m}^{-2}b^{-1},\delta _{1m}^{-2}b^{-1}+(a^{-1}-b^{-1}) \delta _{1m}^{-1}]\subset [\delta _{1m}^{-2}b^{-1},\delta _{1m}^{-2}a^{-1}] \subset [H_{m}^{1},H_{m+1}^{1}]\).
For any \(H\in [\delta _{1m}^{-2}b^{-1},\delta _{1m}^{-2}b^{-1}+(a^{-1}-b^{-1}) \delta _{1m}^{-1}]\), we claim that
Indeed, suppose that there is \(H_{m}^{1*} \in [\delta _{1m}^{-2}b^{-1},\delta _{1m}^{-2}b^{-1}+(a^{-1}-b^{-1}) \delta _{1m}^{-1}]\) such that
Using (3.12), we have
which is a contradiction. Thus (3.15) holds.
Let
Taking \(B_{1}=\{(x,y):\Psi _{4}\Psi _{3}\Psi _{2}\Psi _{1}(x,y)\subset D_{1} \}\) and using the fixed point theorem in Ding [10], we derive that (1.7) in Theorem 1.1 is a consequence of (3.7)–(3.9) and (3.10)–(3.11).
Considering (1.8), we assume that
Let \(\hat{\omega}>0\) and
Since \(\sqrt{H}\Gamma '(H)\rightarrow 0\) as \(H\rightarrow +\infty \), there exists a number \(\hat{H}>\frac{1}{b}>0\) such that
for \(H\geq \hat{H}\). Combining (3.16), we can find a sequence \(\{H_{m}^{2} \}_{m=1}^{\infty}\) associated with \(\hat{H}\leq H_{m}^{2}\rightarrow +\infty \) satisfying
and
Taking \(\delta _{2m}=(bH_{m}^{2})^{-\frac{1}{2}}\), we get \(\delta _{2m}\rightarrow 0\) as \(m\rightarrow +\infty \). It follows from (3.18) that
and hence \(H\in [\delta _{2m}^{-2}b^{-1},\delta _{2m}^{-2}b^{-1}+(a^{-1}-b^{-1}) \delta _{2m}^{-1}]\subset [H_{m}^{2},H_{m+1}^{2}]\).
For any \(H\in [\delta _{2m}^{-2}b^{-1},\delta _{2m}^{-2}b^{-1}+(a^{-1}-b^{-1}) \delta _{2m}^{-1}]\), we claim that
Indeed, assume that there is \(H_{m}^{2*}\in [\delta _{2m}^{-2}b^{-1},\delta _{2m}^{-2}b^{-1}+(a^{-1}-b^{-1}) \delta _{2m}^{-1}]\) such that
Then using (3.17), (3.18), and (3.19) we get
which is a contradiction. Thus (3.20) is valid.
Let
and \(B_{2}=\{(x,y):\Psi _{4}\Psi _{3}\Psi _{2}\Psi _{1}(x,y)\subset D_{2} \}\).
Now we choose \(\{H_{1m_{k}}\}_{1}^{+\infty}\) and \(\{H_{2m_{k}}\}_{1}^{+\infty}\) such that \(H_{1m_{k}}< H_{1m_{k}+1}< H_{2m_{k}}< H_{2m_{k}+1}\) and let
Setting \(B_{3}=\{(x,y):\Psi _{4}\Psi _{3}\Psi _{2}\Psi _{1}(x,y)\subset D_{3} \}\) and using the twist theorem in Ding [11], we obtain that inequality (1.8) in Theorem 1.1 is a consequence of (3.10), (3.15), (3.7), and (3.20). The proof of Theorem 1.1 is finished.
To verify the given conditions and understand our main result, we give the following remark.
Remark 3.1
Using Lemmas 2.1, 2.4, and 2.6, Eq. (2.24) takes the form
Combining (2.14) with Lemma 2.4, we have \(\partial _{H}R_{1}=-g(x)\partial _{I}x+o(\frac{1}{\sqrt{H}})\). Thus we obtain
By the results in [5], \(x=\sqrt{\frac{2H}{\pi}}\sin \theta +O(1)\), \(\partial _{I}x=\sqrt{\frac{1}{2\pi H}}\sin \theta +O(\frac{1}{H})\), and
For \(g(x)\) and \(V(x)\) in problem (1.3), if \(g(x)\) satisfies (1.6) and \(p(t)\) satisfies (1.7), then we know that the equation
has at least one π-periodic solutions. Letting \(p(t)\) satisfy (1.8), we conclude that the equation
has an unbounded sequence of π-periodic solutions.
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Acknowledgements
The authors would like to thank the anonymous referees for their valuable and helpful comments, which led to a meaningful improvement of the paper.
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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Dr. Xing and Dr. Wang give all the computaions and derivations of the paper. Lai checks the whole paper and corrects the paper and gives some suggeations. All authors contribute equally in this works.
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Xing, X., Wang, L. & Lai, S. Existence and multiplicity of periodic solutions for a nonlinear resonance equation with singularities. Bound Value Probl 2023, 110 (2023). https://doi.org/10.1186/s13661-023-01799-y
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DOI: https://doi.org/10.1186/s13661-023-01799-y