- Research
- Open Access
- Published:

# Three periodic solutions for a class of ordinary *p*-Hamiltonian systems

*Boundary Value Problems*
**volume 2014**, Article number: 150 (2014)

## Abstract

We study the *p*-Hamiltonian systems -{({|{u}^{\prime}|}^{p-2}{u}^{\prime})}^{\prime}+A(t){|u|}^{p-2}u=\mathrm{\nabla}F(t,u)+\lambda \mathrm{\nabla}G(t,u), u(0)-u(T)={u}^{\prime}(0)-{u}^{\prime}(T)=0. Three periodic solutions are obtained by using a three critical points theorem.

## Introduction

Consider the *p*-Hamiltonian systems

where p>1, T>0, \lambda \in (-\mathrm{\infty},+\mathrm{\infty}), F:[0,T]\times {\mathbf{R}}^{N}\to \mathbf{R} is a function such that F(\cdot ,x) is continuous in [0,T] for all x\in {\mathbf{R}}^{N} and F(\cdot ,x) is a {C}^{1}-function in {\mathbf{R}}^{N} for almost every t\in [0,T], and G:[0,T]\times {\mathbf{R}}^{N}\to \mathbf{R} is measurable in [0,T] and {C}^{1}\in {\mathbf{R}}^{N}. A={({a}_{ij}(t))}_{N\times N} is symmetric, A\in C([0,T],{\mathbf{R}}^{N\times N}), and there exists a positive constant {\lambda}_{1} such that (A(t){|x|}^{p-2}x,x)\ge {\lambda}_{1}^{p}{|x|}^{p} for all x\in {\mathbf{R}}^{N} and t\in [0,T], that is, A(t) is positive definite for all t\in [0,T].

In recent years, the three critical points theorem of Ricceri [[1]] has widely been used to solve differential equations; see [[2]–[4]] and references therein.

In [[5]], Li *et al.* have studied the three periodic solutions for *p*-Hamiltonian systems

Their technical approach is based on two general three critical points theorems obtained by Averna and Bonanno [[6]] and Ricceri [[4]].

In [[7]], Shang and Zhang obtained three solutions for a perturbed Dirichlet boundary value problem involving the *p*-Laplacian by using the following Theorem A. In this paper, we generalize the results in [[7]] on problem (1.1).

### Theorem A

*Let* *X* *be a separable and reflexive real Banach space*, *and let*\varphi ,\psi :X\to \mathbf{R}*be two continuously Gâteaux differentiable functionals*. *Assume that* *ψ* *is sequentially weakly lower semicontinuous and even that* *ϕ* *is sequentially weakly continuous and odd*, *and that*, *for some*b>0*and for each*\lambda \in [-b,b], *the functional*\psi +\lambda \varphi*satisfies the Palais*-*Smale condition and*

*Finally*, *assume that there exists*k>0*such that*

*Then*, *for every*b>0, *there exist an open interval*\mathrm{\Lambda}\subset [-b,b]*and a positive real number σ*, *such that for every*\lambda \in \mathrm{\Lambda}, *the equation*

*admits at least three solutions whose norms are smaller than* *σ*.

## Proofs of theorems

First, we give some notations and definitions. Let

and is endowed with the norm

Let {\phi}_{\lambda}:{W}_{T}^{1,p}\to \mathbf{R} be defined by the energy functional

where \psi (u)=\frac{1}{p}{\parallel u\parallel}^{p}-{\int}_{0}^{T}F(t,u(t))\phantom{\rule{0.2em}{0ex}}dt, \varphi (u)={\int}_{0}^{T}G(t,u(t))\phantom{\rule{0.2em}{0ex}}dt.

Then {\phi}_{\lambda}\in {C}^{\prime}({W}_{T}^{1,p},\mathbf{R}) and one can check that

for all u,v\in {W}_{T}^{1,p}. It is well known that the *T*-periodic solutions of problem (1.1) correspond to the critical points of {\phi}_{\lambda}.

As A(t) is positive definite for all t\in [0,T], we have Lemma 2.1.

### Lemma 2.1

*For each*u\in {W}_{T}^{1,p},

*where*{\parallel u\parallel}_{{L}^{p}}={\int}_{0}^{T}{|u(t)|}^{p}\phantom{\rule{0.2em}{0ex}}dt.

### Theorem 2.1

*Suppose that* *F* *and* *G* *satisfy the following conditions*:

(H1) {lim}_{|x|\to \mathrm{\infty}}\frac{|\mathrm{\nabla}F(t,x)|}{{|x|}^{p-1}}=0, *for a*.*e*. t\in [0,T];

(H2) {lim}_{|x|\to 0}\frac{|\mathrm{\nabla}F(t,x)|}{{|x|}^{p-1}}=0, *for a*.*e*. t\in [0,T];

(H3) {lim}_{|x|\to 0}\frac{F(t,x)}{{|x|}^{p}}=\mathrm{\infty}, *for a*.*e*. t\in [0,T];

(H4) |\mathrm{\nabla}G(t,x)|\le c(1+{|x|}^{q-1}), \mathrm{\forall}x\in {\mathbf{R}}^{N}, *a*.*e*. t\in [0,T], *for some*c>0*and*1\le q<p;

(H5) F(t,\cdot )*is even and*G(t,\cdot )*is odd for a*.*e*. t\in [0,T].

*Then*, *for every*b>0, *there exist an open interval*\mathrm{\Lambda}\subset [-b,b]*and a positive real number σ*, *such that for every*\lambda \in \mathrm{\Lambda}, *problem* (1.1) *admits at least three solutions whose norms are smaller than* *σ*.

### Proof

By (H1) and (H2), given \epsilon >0, we may find a constant {C}_{\epsilon}>0 such that

and so the functional \psi (u) is continuously Gâteaux differentiable functional and sequentially weakly continuous in the space {W}_{T}^{1,p}. Also, by (H4), we know \varphi (u) is sequentially weakly continuous. According to (H4), we get

For \mathrm{\forall}\lambda \in \mathbf{R}, from the inequality (2.5) and (2.6), we deduce that

Since p>q, *ε* small enough, we have

Now, we prove that {\phi}_{\lambda} satisfies the (PS) condition.

Suppose \{{u}_{n}\} is a (PS) sequence of {\phi}_{\lambda}, that is, there exists C>0 such that

Assume that \parallel {u}_{n}\parallel \to \mathrm{\infty}. By (2.7), which contradicts {\phi}_{\lambda}({u}_{n})\to C. Thus \{{u}_{n}\} is bounded. We may assume that there exists {u}_{0}\in {W}_{T}^{1,p} satisfying

Observe that

We already know that

By (2.4) and (H4) we have

Using this, (2.8), and (2.9) we obtain

This together with the weak convergence of {u}_{n}\rightharpoonup {u}_{0} in {W}_{T}^{1,p} implies that

Hence, {\phi}_{\lambda} satisfies the (PS) condition. Next, we want to prove that

Owing to the assumption (H3), we can find \delta >0, for L>0, such that

We choose a function 0\ne v\in {C}_{0}^{\mathrm{\infty}}([0,T]), put L>{\parallel v\parallel}^{p}/(p{\int}_{0}^{T}{|v|}^{p}\phantom{\rule{0.2em}{0ex}}dt), and we take \epsilon >0 small. Then we obtain

Thus (2.10) holds.

From (H2), \mathrm{\forall}\epsilon >0, \mathrm{\exists}{\rho}_{0}(\epsilon )>0 such that

Thus

Choose \epsilon ={\lambda}_{1}/2, one has

Hence, there exists k>0 such that

So we have

The condition (H5) implies *ψ* is even and *ϕ* is odd. All the assumptions of Theorem A are verified. Thus, for every b>0 there exist an open interval \mathrm{\Lambda}\subset [-b,b] and a positive real number *σ*, such that for every \lambda \in \mathrm{\Lambda}, problem (1.1) admits at least three weak solutions in {W}_{T}^{1,p} whose norms are smaller than *σ*. □

### Theorem 2.2

*If* *F* *and* *G* *satisfy assumptions* (H1)-(H2), (H4)-(H5), *and the following condition* (H3′):

(H3′): *there is a constant*{B}_{1}=sup\{1/{\int}_{0}^{T}{|u(t)|}^{p}\phantom{\rule{0.2em}{0ex}}dt:\parallel u\parallel =1\}, {B}_{2}\ge 0, *such that*

*Then*, *for every*b>0, *there exist an open interval*\mathrm{\Lambda}\subset [-b,b]*and a positive real number σ*, *such that for every*\lambda \in \mathrm{\Lambda}, *problem* (1.1) *admits at least three solutions whose norms are smaller than* *σ*.

### Proof

The proof is similar to the one of Theorem 2.1. So we give only a sketch of it. By the proof of Theorem 2.1, the functional *ψ* and *ϕ* are sequentially weakly lower semicontinuous and continuously Gâteaux differentiable in {W}_{T}^{1,p}, *ψ* is even and *ϕ* is odd. For every \lambda \in \mathbf{R}, the functional \psi +\lambda \varphi satisfies the (PS) condition and

To this end, we choose a function v\in {W}_{T}^{1,p} with \parallel v\parallel =1. By condition (H3), a simple calculation shows that, as s\to \mathrm{\infty},

Then (2.11) implies that \psi (sv)<0 for s>0 large enough. So, we choose large enough, {s}_{0}>0, let {u}_{1}={s}_{0}v, such that \psi ({u}_{1})<0. Thus, we get

By the proof of Theorem 2.1 we know that there exists k>0, such that

According to Theorem A, for every b>0 there exist an open interval \mathrm{\Lambda}\subset [-b,b] and a positive real number *σ*, such that for every \lambda \in \mathrm{\Lambda}, problem (1.1) admits at least three weak solutions in {W}_{T}^{1,p} whose norms are smaller than *σ*. □

## References

Bonanno G: Some remarks on a three critical points theorem.

*Nonlinear Anal.*2003, 54: 651-665. 10.1016/S0362-546X(03)00092-0Afrouzi GA, Heidarkhani S: Three solutions for a Dirichlet boundary value problem involving the

*p*-Laplacian.*Nonlinear Anal.*2007, 66: 2281-2288. 10.1016/j.na.2006.03.019Ricceri B: On a three critical points theorem.

*Arch. Math.*2000, 75: 220-226. 10.1007/s000130050496Ricceri B: A three critical points theorem revisited.

*Nonlinear Anal.*2009, 70: 3084-3089. 10.1016/j.na.2008.04.010Li C, Ou Z-Q, Tang C: Three periodic solutions for

*p*-Hamiltonian systems.*Nonlinear Anal.*2011, 74: 1596-1606. 10.1016/j.na.2010.10.030Averna D, Bonanno G: A three critical point theorems and its applications to the ordinary Dirichlet problem.

*Topol. Methods Nonlinear Anal.*2003, 22: 93-103.Shang X, Zhang J: Three solutions for a perturbed Dirichlet boundary value problem involving the

*p*-Laplacian.*Nonlinear Anal.*2010, 72: 1417-1422. 10.1016/j.na.2009.08.025

## Acknowledgements

Supported by the Natural Science Foundation of Shanxi Province (No. 2012011004-1) of China.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Competing interests

The author declares that they have no competing interests.

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## About this article

### Cite this article

Meng, Q. Three periodic solutions for a class of ordinary *p*-Hamiltonian systems.
*Bound Value Probl* **2014, **150 (2014). https://doi.org/10.1186/s13661-014-0150-2

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/s13661-014-0150-2

- p-Hamiltonian systems
- three periodic solutions
- three critical points theorem

- 34K13
- 34B15
- 58E30