Periodic solutions for a singular damped differential equation
- Jing Li^{1},
- Shengjun Li^{2} and
- Ziheng Zhang^{3}Email author
https://doi.org/10.1186/s13661-014-0269-1
© Li et al.; licensee Springer 2015
Received: 20 October 2014
Accepted: 17 December 2014
Published: 10 January 2015
Abstract
Based on a variational approach, we prove that a second-order singular damped differential equation has at least one periodic solution when some reasonable assumptions are satisfied.
Keywords
MSC
1 Introduction
Recently, Eq. (1.1) has also been investigated by several authors; see, for instance, [12, 13] (application of Leray-Schauder alternative principle) and [14] (using Schauder’s fixed point theorem). In general cases, it is very difficult or impossible to apply variational methods to Eq. (1.1) when \(\int_{0}^{T}q(t)\,dt>0\). In this paper, we consider the case \(\int_{0}^{T}q(t)\,dt=0\) and under some reasonable assumptions, we establish the corresponding variational framework of T-periodic solutions for Eq. (1.1) on an appropriate Sobolev space and give a new criterion to guarantee the existence of at least one nontrivial T-periodic solution of Eq. (1.1) using a variant of the mountain pass theorem. We refer the reader to [15–17] for the details about variational methods.
- (H1)
\(q, g\in C(\mathbb{R}/{T\mathbb{Z}})\) with \(\int_{0}^{T} q(t)\,dt=0\);
- (H2)\(f\in C((0,\infty),\mathbb{R})\) has a repulsive singularity at \(u=0\), i.e.,$$\lim_{u\rightarrow0^{+}}f(u)=-\infty; $$
- (H3)
\(\lim_{u\rightarrow0^{+}}F(u)=+\infty\), where \(F(u)= \int_{1}^{u} f(s)\,ds\);
- (H4)
\(M=\sup\{f(s): 0< s<+\infty\}\) is bounded;
- (H5)\(\lim_{u\rightarrow+\infty} (F(u)- \bar{g} u) =+\infty\), where \(\bar{g}\) is defined by$$\bar{g}\stackrel{\mathrm{def}}{=} \frac{1}{\int_{0}^{T} \exp (\int_{0}^{t}q(s)\,ds )\,dt}\int_{0}^{T} g(t)\exp \biggl(\int_{0}^{T}q(s)\,ds \biggr)\,dt. $$
Theorem 1.1
Assume that (H1)-(H5) are satisfied. Then Eq. (1.1) has at least one nontrivial T-periodic solution.
Remark 1
The remaining part of this paper is organized as follows. Some preliminaries are presented in Section 2. In Section 3, the proof of Theorem 1.1 is given.
2 Preliminary results
Lemma 2.1
[15, Proposition 1.3] (Wirtinger’s inequality)
In order to obtain the existence of T-periodic solutions of Eq. (2.1), the following version of the mountain pass theorem will be used in our argument.
Lemma 2.2
[15, Theorem 4.10]
3 Proof of Theorem 1.1
In this section, we give the proof of Theorem 1.1.
Proof
The proof will be divided into four steps.
Step 1. \(\Phi_{\lambda}\) satisfies the (PS)-condition.
(ii) Assume that the first possibility occurs, i.e., \(m_{n}\rightarrow-\infty\) as \(n\rightarrow+\infty\). We replace \(M_{n}\) by \(-m_{n}\) in the preceding arguments, and we also get a contradiction.
Therefore, \(\Phi_{\lambda}\) satisfies the (PS) condition. This completes the proof of the claim.
Step 3. We show that there exists \(\lambda_{0}\in(0,1)\) with the property that, for every \(\lambda\in(0,\lambda_{0})\), any solution u of Eq. (2.1) satisfying \(\Phi_{\lambda}(u)\geq-m\) is such that \(\min u\geq\lambda_{0}\), and hence u is a solution of Eq. (1.1).
- (i)
\(\lambda_{n}\leq\frac{1}{n}\);
- (ii)
\(u_{n}\) is a solution of Eq. (2.1) with \(\lambda =\lambda_{n}\);
- (iii)
\(\Phi_{\lambda_{n}}(u_{n})\geq-m\);
- (iv)
\(\min u_{n}<\frac{1}{n}\).
Step 4. We prove that \(\Phi_{\lambda}\) has a mountain pass geometry for \(\lambda\leq\lambda_{0}\).
Since \(\inf_{u\in\partial\Omega}\Phi_{\lambda }(u)\geq -m\), it follows from Step 3 that \(u_{\lambda}\) is a solution of Eq. (1.1). Now the proof is finished. □
Declarations
Acknowledgements
The authors would like to thank the anonymous referee for his/her valuable comments, which have improved the correctness and presentation of the manuscript. This work is supported by the National Natural Science Foundation of China (Grant No. 11101304).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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