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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)+λG(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, AC([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 [[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

{ ( | 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 [[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

[[1], [7]]

Let X be a separable and reflexive real Banach space, and letϕ,ψ:XRbe 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 someb>0and for eachλ[b,b], the functionalψ+λϕsatisfies the Palais-Smale condition and

lim x [ ψ ( x ) + λ ϕ ( x ) ] =+.
(3)

Finally, assume that there existsk>0such that

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

Then, for everyb>0, there exist an open intervalΛ[b,b]and a positive real number σ, such that for everyλΛ, 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

W T 1 , p = { u : [ 0 , T ] R N u  is absolutely continuous , u ( 0 ) = u ( T ) , u L p ( 0 , T ; R N ) }
(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 eachu 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 somec>0and1q<p;

(H5) F(t,)is even andG(t,)is odd for a.e. t[0,T].

Then, for everyb>0, there exist an open intervalΛ[b,b]and a positive real number σ, such that for everyλΛ, 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

| F ( t , x ) | C ε +ε | x | p 1 ,for every x R N , a.e. t[0,T],
(11)
| F ( t , x ) | C ε + ε p | x | p ,for every x R N , a.e. t[0,T],
(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

| G ( t , x ) | c|x|+ c p | x | q ,for every x R N , a.e. t[0,T].
(13)

For λR, 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

φ λ ( u n )C, φ λ ( u n )0as n.
(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

u n u 0 , weakly in  W T 1 , p , u n u 0 , strongly in  L p [ 0 , T ] , u n ( x ) u 0 ( x ) , a.e.  t [ 0 , T ] .
(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)

We already know that

φ λ ( u n ) , u n u 0 0,as n.
(19)

By (2.4) and (H4) we have

0 T ( F ( t , u n ( t ) ) , u n ( t ) u 0 ( t ) ) d t 0 , as  n , 0 T ( G ( t , u n ( t ) ) , u n ( t ) u 0 ( t ) ) d t 0 , as  n .
(20)

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

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 , as  n .
(21)

This together with the weak convergence of u n u 0 in W T 1 , p implies that

u n u 0 ,strongly in  W T 1 , p .
(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

| F ( t , x ) | >L|x|,for 0<|x|δ, and a.e. t[0,T].
(24)

We choose a function 0v 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

| F ( t , x ) | ε | x | p 1 ,if 0<ρ=|x|< ρ 0 (ε).
(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 λΛ, 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

F(t,x)2 B 1 | x | p p B 2 ,for x R N , a.e. t[0,T].
(31)

Then, for everyb>0, there exist an open intervalΛ[b,b]and a positive real number σ, such that for everyλΛ, 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 λR, 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 λΛ, 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.

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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

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  • p-Hamiltonian systems
  • three periodic solutions
  • three critical points theorem
  • 34K13
  • 34B15
  • 58E30