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Periodic solutions for pLaplacian Rayleigh equations with singularities
Boundary Value Problems volume 2016, Article number: 96 (2016)
Abstract
In this paper, the problem of existence of periodic solutions is studied for pLaplacian Rayleigh equations with a singularity at \(x=0\). By using the topological degree theory, some new results are obtained.
Introduction
In recent years, the periodic problem for some types of singular equations has attracted much attention of many researchers because the singular nonlinearity possesses a significant role in many practical situations. For example, the differential equation
described the motion of a piston in a cylinder closed at one extremity. The singular term \(\frac{1}{x}\) in the equation models the restoring force which is caused by a compressed perfect gas (see [1] and the references therein). The interest in studying the equations with a singularity began with some work of Forbat and Huaux [2]. Later, the interest in such problem was renewed by Gordon in [3, 4], and Lazer and Solimini in [5]. For the recent developments on the study of this problem, here, we refer the reader to [6–16], and we notice that the equations studied previously were either of the type of Duffing equations [7–11, 14–16] or of the type of Liénard equations [11, 12, 17, 18]. For example, Jebelean and Mawhin in [1] considered the problem of the existence of positive periodic solutions for the following pLaplacian Liénard equations with a singularity:
and
where \(p>1\) is a constant, \(f: [0,+\infty)\rightarrow\mathbb{R}\) is an arbitrary continuous function, \(h: \mathbb{R}\rightarrow\mathbb{R}\) is a Tperiodic function with \(h\in L^{\infty}[0,T]\), \(g:(0,+\infty )\rightarrow(0,+\infty)\) is continuous, and singular at \(x=0\), this means that \(g(x)\) is unbounded as \(x\rightarrow0^{+}\). The crucial condition imposed on \(g(x)\) is that \(g(x)\rightarrow+\infty\) as \(x\rightarrow0^{+}\), i.e., equation (1.1) is of attractive type, and equation (1.2) is of repulsive type. Zhang in [17] studied the problem of periodic solutions of the Liénard equation with a repulsive singularity at \(x=0\),
In [18], Wang further studied the existence of positive periodic solutions for a delay Liénard equation with a repulsive singularity at \(x=0\),
The methods in [1] for equation (1.2), in [17] for equation (1.3) and in [18] for equation (1.4) were all based on topological degree theory, and the upper and lower solutions techniques was used in [1] for equation (1.1). But as far as we are aware of, few results appeared on the existence of periodic solutions for pLaplacian Rayleigh equation with a singularity.
Motivated by this, in this paper, we study the existence of positive Tperiodic solutions for pLaplacian Rayleigh equation with a singularity of the form
and
where \(p>1\) is a constant, \(f:\mathbb{R}\rightarrow\mathbb{R}\) is an arbitrary continuous function, \(g_{1},g_{2}:(0,+\infty)\rightarrow\mathbb {R}\) are all continuous and \(g_{1}(x)\) is unbounded as \(x\rightarrow 0^{+}\), \(h: \mathbb{R}\rightarrow\mathbb{R}\) is a Tperiodic continuous function. Clearly, equation (1.5) and equation (1.6) are all singular at \(x=0\). By using ManásevichMawhin’s continuation theorem, some new results are obtained.
The interesting thing is that the singular term in equation (1.6) (or in equation (1.5)) is not required to have \(g_{1}(x)\rightarrow +\infty\) (or \(g_{1}(x)\rightarrow\infty\)) as \(x\rightarrow0^{+}\). For example, let
where \(\mu\ge1\) is a constant. It is easy to verify that \(g_{1}(x)\) does not approach +∞ as \(x\rightarrow0^{+}\). Furthermore, if \(x\in C^{1}(\mathbb{R},\mathbb{R})\) with Tperiodic, then the first order derivative term \(f(x)x'\) in equations (1.1)(1.4) satisfies \(\int_{0}^{T}f(x(t))x'(t)\,dt=0\), which is crucial for obtaining an a priori bounds of all the possible Tperiodic solutions for equation (1.1)equation (1.4). However, the first order derivative term in equation (1.5) and equation (1.6) is \(f(x')\); generally, \(\int_{0}^{T}f(x'(t))\,dt=0\) does not hold. This means that the method for estimating an a priori bounds of all the possible Tperiodic solutions to equation (1.5) and equation (1.6) is different from the corresponding ones in [1, 17, 18].
Preliminary lemmas
The following two lemmas (Lemma 2.1 and Lemma 2.2) are all consequences of Theorem 3.1 in [19].
Lemma 2.1
Assume that there exist constants \(0<\eta _{0}<\eta_{1}\), \(M_{2}>0\), such that the following conditions hold.

(1)
For each \(\lambda\in(0,1]\), each possible positive Tperiodic solution x to the equation
$$ \bigl(\biglu'\bigr^{p2}u' \bigr)'+ \lambda f \bigl(u' \bigr)\lambda g_{1}(u)+\lambda g_{2}(t,u)=\lambda h(t) $$satisfies the inequalities \(\eta_{0}< x(t)<\eta_{1}\) and \(x'(t)< M_{2}\) for all \(t\in[0,T]\).

(2)
Each possible solution c to the equation
$$ g_{1}(c)g_{2}(c)+\bar{h}=0 $$satisfies the inequality \(\eta_{0}< c<\eta_{1}\).

(3)
It holds
$$ \bigl(g_{1}(\eta_{0})g_{2}( \eta_{0})+\bar{h} \bigr) \bigl(g_{1}(\eta _{1})g_{2}(\eta_{1})+\bar{h} \bigr)< 0. $$
Then equation (1.5) has at least one Tperiodic solution u such that \(\eta_{0}< u(t)<\eta_{1}\) for all \(t\in[0,T]\).
Lemma 2.2
Assume that there exist constants \(0<\eta _{0}<\eta_{1}\), \(M_{2}>0\), such that the following conditions hold:

(1)
For each \(\lambda\in(0,1]\), each possible positive Tperiodic solution x to the equation
$$ \bigl(\biglu'\bigr^{p2}u' \bigr)'+ \lambda f \bigl(u' \bigr)+\lambda g_{1}(u)\lambda g_{2}(u)=\lambda h(t) $$satisfies the inequalities \(\eta_{0}< x(t)<\eta_{1}\) and \(x'(t)< M_{2}\) for all \(t\in[0,T]\).

(2)
Each possible solution c to the equation
$$ g_{1}(c)g_{2}(c)\bar{h}=0 $$satisfies the inequality \(\eta_{0}< c<\eta_{1}\).

(3)
It holds
$$ \bigl(g_{1}(\eta_{0})g_{2}( \eta_{0})\bar{h} \bigr) \bigl(g_{1}(\eta _{1})g_{2}(\eta_{1})\bar{h} \bigr)< 0. $$
Then equation (1.6) has at least one Tperiodic solution u such that \(\eta_{0}< u(t)<\eta_{1}\) for all \(t\in[0,T]\).
In order to study the existence of positive periodic solutions to equation (1.5) and equation (1.6), we list the following assumptions:
(H_{1}) there are positive constants \(m_{0}\) and \(m_{1}\) with \(m_{0}< m_{1}\) such that
and
where \(h_{\infty}=\max_{t\in[0,T]}h(t)\);
(H_{2}) \(g_{1}(x)\ge0\) for all \(x\in(0,+\infty)\), and \(\int _{0}^{1}g_{1}(s)\,ds=+\infty\);
(H_{3}) \(f(0)=0\);
(H_{4}) there are constants n, \(\sigma_{0}\) and \(\sigma_{1}\) with \(n>1\), \(0<\sigma_{0}\le\sigma_{1}\) such that
and
Now, we embed equation (1.5) and equation (1.6) into the following two equations family with a parameter \(\lambda\in(0,1]\), respectively:
and
Lemma 2.3
Assume that assumptions (H_{1}) and (H_{3}) hold, let \(m_{0}\) and \(m_{1}\) be positive constants determined in assumption (H_{1}). Then the following conclusions hold:

(1)
for each possible positive Tperiodic solution \(u(t)\) of equation (2.3) there exists \(\tau\in[0,T]\) such that
$$ m_{0}< u(\tau)< m_{1}; $$(2.5) 
(2)
each possible solution c to the equation
$$ g_{1}(c)g_{2}(c)+\bar{h}=0 $$satisfies the inequality \(m_{0}< c< m_{1}\);

(3)
\(g_{1}(u)g_{2}(u)+\bar{h}>0\) for all \(u\in(0,m_{0}]\), and \(g_{1}(u)g_{2}(u)+\bar{h}<0\) for all \(u\in[m_{1},+\infty)\).
Proof
(1) Suppose that \(u(t)\) be an arbitrary positive Tperiodic solution to equation (2.3), then
Let \(t_{0}\) and \(t_{1}\) be the maximum point and the minimum point of \(u(t)\) on \([0,T]\), respectively, then \(u'(t_{0})=0\) and \(u'(t_{1})=0\). We can prove that
In fact, if (2.7) does not hold, then \((u'^{p2}u')' _{t=t_{0}}>0\). By the continuity of \((u'(t)^{p2}u'(t))'\) for \(t\in [t_{0},t_{0}+T]\), we see that there is a constant \(\delta\in(0,T)\) such that \((u'(t)^{p2}u'(t))'>0\) for \(t\in(t_{0},t_{0}+\delta)\), and then \(u'(t)^{p2}u'(t)>u'(t)^{p2}u'(t) _{t=t_{0}}=0\) for \(t\in (t_{0},t_{0}+\delta)\), i.e., \(u'(t)>0\) for \(t\in(t_{0},t_{0}+\delta )\), which results in \(u(t)>u(t_{0})\) for \(t\in(t_{0},t_{0}+\delta)\). So \(u(t_{0})<\max_{t\in[t_{0},t_{0}+T]}u(t)=\max_{t\in[0,T]}u(t)\), which contradicts the fact that \(t_{0}\) is the maximum point of \(u(t)\) on \([0,T]\). This contradiction implies that (2.7) holds. Similarly, we have
It follows from (2.6) and (2.7) that
By using assumption (H_{3}), we have \(f(u'(t_{0}))=f(0)=0\), which together with (2.9) yields
and, by using condition (2.1) in assumption (H_{1}), we have
Similarly, condition (2.2) in assumption (H_{1}), together (2.6) and (2.8), implies that
Without loss of generality, suppose \(u(t_{0})>m_{1}\), then by (2.10) and (2.11), we obtain from the intermediate value property of the continuous function \(u(t)\) that (2.5) holds.
(2) Conclusion (2), as well as conclusion (3), follows directly from assumption (H_{1}). □
Similar to the proof of Lemma 2.3, we obtain the following result.
Lemma 2.4
Assume that assumptions (H_{1}) and (H_{3}) hold, let \(m_{0}\) and \(m_{1}\) be positive constants determined in assumption (H_{1}). Then the following conclusions hold:

(1)
each possible positive Tperiodic solution \(u(t)\) to equation (2.4) satisfies
$$ m_{0}< u(t)< m_{1} \quad\textit{for all } t\in[0,T]; $$ 
(2)
each possible solution c to the equation
$$ g_{1}(c)g_{2}(c)\bar{h}=0 $$satisfies the inequality \(m_{0}< c< m_{1}\);

(3)
\(g_{1}(u)g_{2}(u)\bar{h}>0\) for all \(u\in(0,m_{0}]\), and \(g_{1}(u)g_{2}(u)\bar{h}<0\) for all \(u\in[m_{1},+\infty)\).
Main results
Theorem 3.1
Assume that assumptions (H_{1}), (H_{2}), and (H_{4}) hold, then equation (1.5) has at least one positive Tperiodic solution.
Proof
First of all, we will show that there exist \(M_{1}\), \(M_{2}\) with \(M_{1}>m_{1}\) and \(M_{2}>0\) such that each positive Tperiodic solution \(u(t)\) of equation (2.3) satisfies the inequalities
In fact, if u is a positive Tperiodic solution of equation (2.3), then
It is easy to see that assumption (H_{3}) can be deduced from (H_{4}), so by using Lemma 2.3, we see that there is a point \(\tau\in [0,T]\) such that
Multiplying (3.2) with \(u'(t)\) and integrating over the interval \([0,T]\), we have
It follows from assumption (H_{4}) that
which together with (3.4) yields
i.e.,
and then by (3.4), we get
Let \(G=\max_{x\in[0,M_{1}]}g_{2}(x)\), then it follows from (3.2) and the fact of \(g_{1}\in C((0,\infty),[0,+\infty))\) that
Furthermore, by integrating (3.2) over the interval \([0,T]\), we have
Substituting it into (3.7), and by using assumption (H_{3}), we have
which together with (3.5) yields
Since \(\max_{t\in[0,T]}u'(t)^{p1}\le\int _{0}^{T}(u'(t)^{p2}u'(t))'\,dt\), it follows from (3.8) that
i.e.,
where \(M_{2}=1+2^{\frac{1}{p1}} [\sigma_{1}\sigma_{0}^{1}T^{\frac {1}{n}} (\int_{0}^{T}h(t)^{\frac{n}{n1}}\,dt )^{\frac {n1}{n}}+TG+\int_{0}^{T}h(t)\,dt ]^{\frac{1}{p1}}\).
Below, we will show that there exists a constant \(M_{0}\in (0,m_{0})\), such that
Let τ be determined as in Lemma 2.3. Multiplying (3.2) by \(u'(t)\) and integrating over the interval \([\tau,t]\) (or \([t,\tau]\)), we get
Set \(y(t)=u'(t)^{p2}u'(t)\), then \(y(t)\) is absolutely continuous and \(u'(t)=y(t)^{q2}y(t)\), where \(q\in(1,+\infty)\) with \(\frac {1}{p}+\frac{1}{q}=1\). So
Substituting it into (3.12), we get
which yields the estimate
From (3.9), we get
which gives
with
From (H_{2}) there exists \(M_{0}\in(0,m_{0})\) such that
Therefore, if there is a \(t^{*}\in[\tau,\tau+T]\) such that \(u(t^{*})\le M_{0}\), then from (3.14) we get
which contradicts (3.13). This contradiction shows that \(u(t)>M_{0}\) for all \(t\in[0,T]\). So (3.11) holds. Let \(\eta_{0}\in(0, M_{0})\) and \(\eta_{1}\in(M_{1}, +\infty)\) be two constants, then from (3.6), (3.10), and (3.11), we see that each possible positive Tperiodic solution u to equation (2.3) satisfies
This implies that condition (1) of Lemma 2.1 is satisfied. We can deduce from conclusion (2) of Lemma 2.3 that each possible solution c to the equation
satisfies the inequality \(\eta_{0}< c<\eta_{1}\), and from conclusion (3) of Lemma 2.3, we obtain
and
which results in
So condition (3) of Lemma 2.1 holds. By using Lemma 2.1, we see that equation (1.5) has at least one positive Tperiodic solution. The proof is complete. □
Theorem 3.2
Assume that assumptions (H_{1}) and (H_{3}) hold, then equation (1.6) has at least one positive Tperiodic solution.
Proof
Suppose that \(u(t)\) be an arbitrary Tperiodic solution to equation (2.4), then
By using Lemma 2.4, we see that
where \(m_{0}\) and \(m_{1}\) are constants determined in assumption (H_{1}). Multiplying equation (3.15) with \((u'^{p2}u')'\) and integrating over the interval \([0,T]\), we have
Let \(G_{1}=\max_{0\le x\le m_{1}}g_{2}(x)\), \(G_{2}=\max_{m_{0}\le x\le m_{1}}g_{1}(x)\). Take \(y(t)=u'(t)^{p2}u'(t)\), then \(u'(t)=y(t)^{q2}y(t)\), where \(q\in(1,+\infty)\) is a constant with \(\frac{1}{p}+\frac{1}{q}=1\), then
So, it follows from (3.16) and (3.17) that
which results in
Since
it follows from (3.18) that
i.e.,
Let \(M_{2}= T^{\frac{1}{p1}} (G_{1}+G_{2}+h_{\infty} )^{\frac {1}{2(p1)}}+1\), and \(u(t)\) be an arbitrary Tperiodic solution to equation (2.4). Then from (3.16) and (3.19), we see that
This implies that condition (1) of Lemma 2.2 is satisfied, and it is easy to see from assumption (H_{1}) that conditions (2)(3) in Lemma 2.2 also hold. By using Lemma 2.2, we see that equation (1.6) possesses a Tperiodic solution \(u(t)\) such that \(m_{0}\le u(t)\le m_{1}\) for all \(t\in[0,T]\). The proof is complete. □
Now, if the singular restoring force term \(g_{1}(x)\) in equation (1.5) (or in equation (1.6)) satisfies
and \(g_{2}(x)\) satisfies
then assumption (H_{1}) holds. Thus, by applying Theorem 3.1 and Theorem 3.2, respectively, we can obtain the following results.
Corollary 3.1
Assume that (3.20), (3.21), and assumptions (H_{2}) and (H_{4}) hold, then equation (1.5) has at least one positive Tperiodic solution.
Corollary 3.2
Assume that (3.20), (3.21), and assumption (H_{3}) hold, then equation (1.6) has at least one positive Tperiodic solution.
Example 3.1
Considering the following equation:
Corresponding to equation (1.5), \(f(u)=u^{3}\), \(g_{1}(u)\) can be regarded as \(g_{1}(u)=\frac{1}{u^{2}}\sin\frac{1}{u}\), \(g_{2}(u)=u^{2}2\), and \(h(t)=\cos t\). Since
it follows that assumption (H_{2}) holds. By simple calculating, we can chose \(\sigma_{0}=\sigma_{1}=1\), \(n=4\), \(m_{0}=\frac{1}{2}\), and \(m_{1}=2\) such that verifying assumptions (H_{1}) and (H_{4}). Thus, by using Theorem 3.1, we see that equation (3.22) has at least one positive 2πperiodic solution.
Example 3.2
Consider the following equation:
Corresponding to equation (1.6), \(f(u)=u^{4}\), \(g_{1}(u)=\frac {1}{u^{\frac{1}{2}}}\frac{1}{u^{\frac{1}{4}}}\), \(g_{2}(u)=u^{3}\), and \(h(t)=\cos t\). It is easy to see that conditions (3.20) and (3.21), and assumption (H_{3}) are all satisfied. By using Corollary 3.2, we see that equation (3.23) has at least one positive 2πperiodic solution.
Remark 3.1
The first order derivative term in equation (3.23) is \((x'(t))^{4}\), and then \(\int_{0}^{T}(x'(t))^{4}\,dt\neq0\) for all Tperiodic continuous function x, generally. This implies that the methods used for studying equation (1.1) in [1] is not valid for equation (3.23).
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Acknowledgements
The work is sponsored by the National Natural Science Foundation of China (11271197). The authors are grateful to anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.
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Lu, S., Zhong, T. & Chen, L. Periodic solutions for pLaplacian Rayleigh equations with singularities. Bound Value Probl 2016, 96 (2016). https://doi.org/10.1186/s1366101606058
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Keywords
 Rayleigh equation
 topological degree
 singularity
 periodic solution