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

## Abstract

We study the p-Hamiltonian systems $− ( | u ′ | p − 2 u ′ ) ′ +A(t) | u | p − 2 u=∇F(t,u)+λ∇G(t,u)$, $u(0)−u(T)= u ′ (0)− u ′ (T)=0$. Three periodic solutions are obtained by using a three critical points theorem.

## Introduction

Consider the p-Hamiltonian systems

${ − ( | u ′ | p − 2 u ′ ) ′ + A ( t ) | u | p − 2 u = ∇ F ( t , u ) + λ ∇ G ( t , u ) , u ( 0 ) − u ( T ) = u ′ ( 0 ) − u ′ ( T ) = 0 ,$
(1)

where $p>1$, $T>0$, $λ∈(−∞,+∞)$, $F:[0,T]× R N →R$ is a function such that $F(⋅,x)$ is continuous in $[0,T]$ for all $x∈ R N$ and $F(⋅,x)$ is a $C 1$-function in $R N$ for almost every $t∈[0,T]$, and $G:[0,T]× R N →R$ is measurable in $[0,T]$ and $C 1 ∈ R N$. $A= ( a i j ( t ) ) N × N$ is symmetric, $A∈C([0,T], R N × N )$, and there exists a positive constant $λ 1$ such that $(A(t) | x | p − 2 x,x)≥ λ 1 p | x | p$ for all $x∈ R N$ and $t∈[0,T]$, that is, $A(t)$ is positive definite for all $t∈[0,T]$.

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

${ − ( | u ′ | p − 2 u ′ ) ′ + A ( t ) | u | p − 2 u = λ ∇ F ( t , u ) + μ ∇ G ( t , u ) , u ( 0 ) − u ( T ) = u ′ ( 0 ) − u ′ ( T ) = 0 .$
(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$ϕ,ψ:X→R$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$λ∈[−b,b]$, the functional$ψ+λϕ$satisfies the Palais-Smale condition and

$lim ∥ x ∥ → ∞ [ ψ ( x ) + λ ϕ ( x ) ] =+∞.$
(3)

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

$inf x ∈ X ψ(x)< inf | ϕ ( x ) | < k ψ(u).$
(4)

Then, for every$b>0$, there exist an open interval$Λ⊂[−b,b]$and a positive real number σ, such that for every$λ∈Λ$, the equation

$ψ ′ (x)+λ ϕ ′ (x)=0$
(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

$∥u∥= ( ∫ 0 T | u ′ ( t ) | p d t + ∫ 0 T ( A ( t ) | u ( t ) | p − 2 u ( t ) , u ( t ) ) d t ) 1 p .$
(7)

Let $φ λ : W T 1 , p →R$ be defined by the energy functional

$φ λ (u)=ψ(u)+λϕ(u),$
(8)

where $ψ(u)= 1 p ∥ u ∥ p − ∫ 0 T F(t,u(t))dt$, $ϕ(u)= ∫ 0 T G(t,u(t))dt$.

Then $φ λ ∈ C ′ ( W T 1 , p ,R)$ and one can check that

$〈 φ λ ′ ( u ) , v 〉 = ∫ 0 T [ ( | u ′ ( t ) | p − 2 u ′ ( t ) , v ′ ( t ) ) − ( ∇ F ( t , u ( t ) ) , v ( t ) ) − λ ( ∇ G ( t , u ( t ) ) , v ( t ) ) ] d t ,$
(9)

for all $u,v∈ W T 1 , p$. It is well known that the T-periodic solutions of problem (1.1) correspond to the critical points of $φ λ$.

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

### Lemma 2.1

For each$u∈ W T 1 , p$,

$λ 1 ∥ u ∥ L p ≤∥u∥,$
(10)

where$∥ u ∥ L p = ∫ 0 T | u ( t ) | p dt$.

### Theorem 2.1

Suppose that F and G satisfy the following conditions:

(H1) $lim | x | → ∞ | ∇ F ( t , x ) | | x | p − 1 =0$, for a.e. $t∈[0,T]$;

(H2) $lim | x | → 0 | ∇ F ( t , x ) | | x | p − 1 =0$, for a.e. $t∈[0,T]$;

(H3) $lim | x | → 0 F ( t , x ) | x | p =∞$, for a.e. $t∈[0,T]$;

(H4) $|∇G(t,x)|≤c(1+ | x | q − 1 )$, $∀x∈ R N$, a.e. $t∈[0,T]$, for some$c>0$and$1≤q;

(H5) $F(t,⋅)$is even and$G(t,⋅)$is odd for a.e. $t∈[0,T]$.

Then, for every$b>0$, there exist an open interval$Λ⊂[−b,b]$and a positive real number σ, such that for every$λ∈Λ$, problem (1.1) admits at least three solutions whose norms are smaller than σ.

### Proof

By (H1) and (H2), given $ε>0$, we may find a constant $C ε >0$ such that

(11)
(12)

and so the functional $ψ(u)$ is continuously Gâteaux differentiable functional and sequentially weakly continuous in the space $W T 1 , p$. Also, by (H4), we know $ϕ(u)$ is sequentially weakly continuous. According to (H4), we get

(13)

For $∀λ∈R$, from the inequality (2.5) and (2.6), we deduce that

$ψ ( u ) + λ ϕ ( u ) ≥ 1 p ∥ u ∥ p − ∫ 0 T ( C ε + ε p | u ( t ) | p ) d t − λ ∫ 0 T ( c | u ( t ) | + c q | u ( t ) | q ) d t ≥ 1 p ( 1 − ε λ 1 ) ∥ u ∥ p − c λ q λ 1 T p − q q ∥ u ∥ q − c λ λ 1 T p − 1 p ∥ u ∥ − ε T .$
(14)

Since $p>q$, ε small enough, we have

$lim ∥ u ∥ → ∞ [ ψ ( u ) + λ ϕ ( u ) ] =+∞.$
(15)

Now, we prove that $φ λ$ satisfies the (PS) condition.

Suppose ${ u n }$ is a (PS) sequence of $φ λ$, that is, there exists $C>0$ such that

(16)

Assume that $∥ u n ∥→∞$. By (2.7), which contradicts $φ λ ( u n )→C$. Thus ${ u n }$ is bounded. We may assume that there exists $u 0 ∈ W T 1 , p$ satisfying

(17)

Observe that

$〈 φ λ ′ ( u n ) , u n − u 0 〉 = ∫ 0 T [ ( | u n ′ ( t ) | p − 2 u n ′ ( t ) , u n ′ ( t ) − u 0 ′ ( t ) ) + ( A ( t ) | u n ( t ) | p − 2 u n ( t ) , u n ( t ) − u 0 ( t ) ) ] d t − ∫ 0 T ( ( ∇ F ( t , u n ( t ) ) , u n ( t ) − u 0 ( t ) ) ) d t − λ ∫ 0 T ( ∇ G ( t , u n ( t ) ) , u n ( t ) − u 0 ( t ) ) d t .$
(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, $φ λ$ satisfies the (PS) condition. Next, we want to prove that

$inf u ∈ W T 1 , p ψ(u)<0.$
(23)

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

(24)

We choose a function $0≠v∈ C 0 ∞ ([0,T])$, put $L> ∥ v ∥ p /(p ∫ 0 T | v | p dt)$, and we take $ε>0$ small. Then we obtain

$ψ ( ε v ) = 1 p ∥ ε v ∥ p − ∫ 0 T F ( t , ε v ( t ) ) d t ≤ ε p p ∥ v ∥ p − L ε p ∫ 0 T | v ( t ) | p d t < 0 .$
(25)

Thus (2.10) holds.

From (H2), $∀ε>0$, $∃ ρ 0 (ε)>0$ such that

(26)

Thus

$∫ 0 T F ( t , u ( t ) ) dt≤ ε p ∫ 0 T | u ( t ) | p dt≤ ε p λ 1 ∥ u ∥ p .$
(27)

Choose $ε= λ 1 /2$, one has

$ψ ( u ) = 1 p ∥ u ∥ p − ε p λ 1 ∥ u ∥ p = 1 2 p ∥ u ∥ p > 0 .$
(28)

Hence, there exists $k>0$ such that

$inf | ϕ ( u ) | < k ψ(u)=0.$
(29)

So we have

$inf u ∈ W T 1 , p ψ(u)< inf | ϕ ( u ) | < k ψ(u).$
(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 $Λ⊂[−b,b]$ and a positive real number σ, such that for every $λ∈Λ$, 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$B 1 =sup{1/ ∫ 0 T | u ( t ) | p dt:∥u∥=1}$, $B 2 ≥0$, such that

(31)

Then, for every$b>0$, there exist an open interval$Λ⊂[−b,b]$and a positive real number σ, such that for every$λ∈Λ$, 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 $λ∈R$, the functional $ψ+λϕ$ satisfies the (PS) condition and

$lim ∥ u ∥ → ∞ (ψ+λϕ)=+∞.$
(32)

To this end, we choose a function $v∈ W T 1 , p$ with $∥v∥=1$. By condition (H3), a simple calculation shows that, as $s→∞$,

$ψ ( s v ) = 1 p ∥ s v ∥ p − ∫ 0 T F ( t , s v ( t ) ) d t ≤ s p p ∥ v ∥ p − 2 s p B 1 p ∫ 0 T | v ( t ) | p d t + B 2 T ≤ − s p p + B 2 T → − ∞ .$
(33)

Then (2.11) implies that $ψ(sv)<0$ for $s>0$ large enough. So, we choose large enough, $s 0 >0$, let $u 1 = s 0 v$, such that $ψ( u 1 )<0$. Thus, we get

$inf u ∈ W T 1 , p ψ(u)<0.$
(34)

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

$inf u ∈ W T 1 , p ψ(u)< inf | ϕ ( u ) | < k ψ(u).$
(35)

According to Theorem A, for every $b>0$ there exist an open interval $Λ⊂[−b,b]$ and a positive real number σ, such that for every $λ∈Λ$, 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. 