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

## Abstract

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

## Introduction

Consider the p-Hamiltonian systems

$\left\{\begin{array}{l}-{\left({|{u}^{\prime }|}^{p-2}{u}^{\prime }\right)}^{\prime }+A\left(t\right){|u|}^{p-2}u=\mathrm{\nabla }F\left(t,u\right)+\lambda \mathrm{\nabla }G\left(t,u\right),\\ u\left(0\right)-u\left(T\right)={u}^{\prime }\left(0\right)-{u}^{\prime }\left(T\right)=0,\end{array}$
(1)

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

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

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

$\left\{\begin{array}{l}-{\left({|{u}^{\prime }|}^{p-2}{u}^{\prime }\right)}^{\prime }+A\left(t\right){|u|}^{p-2}u=\lambda \mathrm{\nabla }F\left(t,u\right)+\mu \mathrm{\nabla }G\left(t,u\right),\\ u\left(0\right)-u\left(T\right)={u}^{\prime }\left(0\right)-{u}^{\prime }\left(T\right)=0.\end{array}$
(2)

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

In [], 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 [] 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$\lambda \in \left[-b,b\right]$, the functional$\psi +\lambda \varphi$satisfies the Palais-Smale condition and

$\underset{\parallel x\parallel \to \mathrm{\infty }}{lim}\left[\psi \left(x\right)+\lambda \varphi \left(x\right)\right]=+\mathrm{\infty }.$
(3)

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

$\underset{x\in X}{inf}\psi \left(x\right)<\underset{|\varphi \left(x\right)|
(4)

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

${\psi }^{\prime }\left(x\right)+\lambda {\varphi }^{\prime }\left(x\right)=0$
(5)

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

## Proofs of theorems

First, we give some notations and definitions. Let

(6)

and is endowed with the norm

$\parallel u\parallel ={\left({\int }_{0}^{T}{|{u}^{\prime }\left(t\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dt+{\int }_{0}^{T}\left(A\left(t\right){|u\left(t\right)|}^{p-2}u\left(t\right),u\left(t\right)\right)\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{p}}.$
(7)

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

${\phi }_{\lambda }\left(u\right)=\psi \left(u\right)+\lambda \varphi \left(u\right),$
(8)

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

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

$\begin{array}{rl}〈{\phi }_{\lambda }^{\prime }\left(u\right),v〉=& {\int }_{0}^{T}\left[\left({|{u}^{\prime }\left(t\right)|}^{p-2}{u}^{\prime }\left(t\right),{v}^{\prime }\left(t\right)\right)-\left(\mathrm{\nabla }F\left(t,u\left(t\right)\right),v\left(t\right)\right)\\ -\lambda \left(\mathrm{\nabla }G\left(t,u\left(t\right)\right),v\left(t\right)\right)\right]\phantom{\rule{0.2em}{0ex}}dt,\end{array}$
(9)

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

### Lemma 2.1

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

${\lambda }_{1}{\parallel u\parallel }_{{L}^{p}}\le \parallel u\parallel ,$
(10)

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

### Theorem 2.1

Suppose that F and G satisfy the following conditions:

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

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

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

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

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

Then, for every$b>0$, there exist an open interval$\mathrm{\Lambda }\subset \left[-b,b\right]$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

(11)
(12)

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

(13)

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

$\begin{array}{rcl}\psi \left(u\right)+\lambda \varphi \left(u\right)& \ge & \frac{1}{p}{\parallel u\parallel }^{p}-{\int }_{0}^{T}\left({C}_{\epsilon }+\frac{\epsilon }{p}{|u\left(t\right)|}^{p}\right)\phantom{\rule{0.2em}{0ex}}dt-\lambda {\int }_{0}^{T}\left(c|u\left(t\right)|+\frac{c}{q}{|u\left(t\right)|}^{q}\right)\phantom{\rule{0.2em}{0ex}}dt\\ \ge & \frac{1}{p}\left(1-\frac{\epsilon }{{\lambda }_{1}}\right){\parallel u\parallel }^{p}-\frac{c\lambda }{q{\lambda }_{1}}{T}^{\frac{p-q}{q}}{\parallel u\parallel }^{q}-\frac{c\lambda }{{\lambda }_{1}}{T}^{\frac{p-1}{p}}\parallel u\parallel -\epsilon T.\end{array}$
(14)

Since $p>q$, ε small enough, we have

$\underset{\parallel u\parallel \to \mathrm{\infty }}{lim}\left[\psi \left(u\right)+\lambda \varphi \left(u\right)\right]=+\mathrm{\infty }.$
(15)

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

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

(16)

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

(17)

Observe that

$\begin{array}{r}〈{\phi }_{\lambda }^{\prime }\left({u}_{n}\right),{u}_{n}-{u}_{0}〉\\ \phantom{\rule{1em}{0ex}}={\int }_{0}^{T}\left[\left({|{u}_{n}^{\prime }\left(t\right)|}^{p-2}{u}_{n}^{\prime }\left(t\right),{u}_{n}^{\prime }\left(t\right)-{u}_{0}^{\prime }\left(t\right)\right)+\left(A\left(t\right){|{u}_{n}\left(t\right)|}^{p-2}{u}_{n}\left(t\right),{u}_{n}\left(t\right)-{u}_{0}\left(t\right)\right)\right]\phantom{\rule{0.2em}{0ex}}dt\\ \phantom{\rule{2em}{0ex}}-{\int }_{0}^{T}\left(\left(\mathrm{\nabla }F\left(t,{u}_{n}\left(t\right)\right),{u}_{n}\left(t\right)-{u}_{0}\left(t\right)\right)\right)\phantom{\rule{0.2em}{0ex}}dt\\ \phantom{\rule{2em}{0ex}}-\lambda {\int }_{0}^{T}\left(\mathrm{\nabla }G\left(t,{u}_{n}\left(t\right)\right),{u}_{n}\left(t\right)-{u}_{0}\left(t\right)\right)\phantom{\rule{0.2em}{0ex}}dt.\end{array}$
(18)

(19)

By (2.4) and (H4) we have

(20)

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

(21)

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

(22)

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

$\underset{u\in {W}_{T}^{1,p}}{inf}\psi \left(u\right)<0.$
(23)

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

(24)

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

$\begin{array}{rcl}\psi \left(\epsilon v\right)& =& \frac{1}{p}{\parallel \epsilon v\parallel }^{p}-{\int }_{0}^{T}F\left(t,\epsilon v\left(t\right)\right)\phantom{\rule{0.2em}{0ex}}dt\\ \le & \frac{{\epsilon }^{p}}{p}{\parallel v\parallel }^{p}-L{\epsilon }^{p}{\int }_{0}^{T}{|v\left(t\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dt<0.\end{array}$
(25)

Thus (2.10) holds.

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

(26)

Thus

${\int }_{0}^{T}F\left(t,u\left(t\right)\right)\phantom{\rule{0.2em}{0ex}}dt\le \frac{\epsilon }{p}{\int }_{0}^{T}{|u\left(t\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dt\le \frac{\epsilon }{p{\lambda }_{1}}{\parallel u\parallel }^{p}.$
(27)

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

$\begin{array}{rcl}\psi \left(u\right)& =& \frac{1}{p}{\parallel u\parallel }^{p}-\frac{\epsilon }{p{\lambda }_{1}}{\parallel u\parallel }^{p}\\ =& \frac{1}{2p}{\parallel u\parallel }^{p}>0.\end{array}$
(28)

Hence, there exists $k>0$ such that

$\underset{|\varphi \left(u\right)|
(29)

So we have

$\underset{u\in {W}_{T}^{1,p}}{inf}\psi \left(u\right)<\underset{|\varphi \left(u\right)|
(30)

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 $\mathrm{\Lambda }\subset \left[-b,b\right]$ 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${B}_{1}=sup\left\{1/{\int }_{0}^{T}{|u\left(t\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dt:\parallel u\parallel =1\right\}$, ${B}_{2}\ge 0$, such that

(31)

Then, for every$b>0$, there exist an open interval$\mathrm{\Lambda }\subset \left[-b,b\right]$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

$\underset{\parallel u\parallel \to \mathrm{\infty }}{lim}\left(\psi +\lambda \varphi \right)=+\mathrm{\infty }.$
(32)

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 $s\to \mathrm{\infty }$,

$\begin{array}{rcl}\psi \left(sv\right)& =& \frac{1}{p}{\parallel sv\parallel }^{p}-{\int }_{0}^{T}F\left(t,sv\left(t\right)\right)\phantom{\rule{0.2em}{0ex}}dt\\ \le & \frac{{s}^{p}}{p}{\parallel v\parallel }^{p}-2\frac{{s}^{p}{B}_{1}}{p}{\int }_{0}^{T}{|v\left(t\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dt+{B}_{2}T\\ \le & -\frac{{s}^{p}}{p}+{B}_{2}T\to -\mathrm{\infty }.\end{array}$
(33)

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

$\underset{u\in {W}_{T}^{1,p}}{inf}\psi \left(u\right)<0.$
(34)

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

$\underset{u\in {W}_{T}^{1,p}}{inf}\psi \left(u\right)<\underset{|\varphi \left(u\right)|
(35)

According to Theorem A, for every $b>0$ there exist an open interval $\mathrm{\Lambda }\subset \left[-b,b\right]$ 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. □

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## Acknowledgements

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

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Correspondence to Qiong Meng. 