Skip to main content

Advertisement

Periodic and subharmonic solutions for a class of second-order p-Laplacian Hamiltonian systems

Article metrics

  • 1396 Accesses

  • 2 Citations

Abstract

In this paper, the periodic and subharmonic solutions are investigated for a class of second-order non-autonomous ordinary differential equations with a p-Laplacian. With the perturbation technique and the dual least action principle, some existence results are given of solutions to the convex p-Laplacian systems.

1 Introduction

In this paper, we consider the second-order ordinary differential equations with a p-Laplacian

d d t Φ p ( x ˙ ( t ) ) +F ( t , x ( t ) ) =0,a.e. tR,
(1)

where p>1, x=( x 1 , x 2 ,, x N ), Φ p (x)= | x | p 2 x. Here || stands the Euclidean norm in R N . In what follows we always suppose that F:R× R N R, (t,x)F(t,x) is measurable and T-periodic in t for every x R N , continuously differentiable and convex in x for a.e. tR. When p=2, (1) reduces to the second-order Hamiltonian system

x ¨ (t)+F ( t , x ( t ) ) =0,a.e. tR.
(2)

Since F is T-periodic in t, it is natural to seek T-periodic solutions of (1) and (2). Also, F is kT-periodic for kN, one can search the kT-periodic solutions, which is called subharmonic solutions. By a subharmonic solution, it means a kT periodic solution with k2 an integer, that is, the minimal period is strictly greater than T. When k=1, it is a periodic solution or harmonic. Clearly, a solution x of (1) over [0,kT] verifying x(0)=x(kT) and x ˙ (0)= x ˙ (kT) can be extended by kT-periodicity over to give a kT-periodic solution. So, it is fine to study the periodic boundary value of problem of (1) over [0,kT] to present conditions for the existence of the periodic and subharmonic solutions of (1). For the study of the subharmonic solutions, we refer to [1]–[5] for a few examples.

The dual action principle was firstly introduced by Clarke [6] and developed by Clarke and Ekeland [7]–[11], which is from the spirit of optimal control theory and convex analysis; see [12]. Following this use and the direct variational method, the periodic solutions are obtained for the Hamiltonian system of (2) in [2], [5], [13], [14] and the references therein. Mawhin and Willem [2] presented the existence results of solutions to the more general systems by using such a principle and the perturbation technique argument. In 2007, Tian and Ge [15] generalized the existence results to the p-Laplacian system (1). By transforming the variable, they found a first-order Hamiltonian system equivalent to the second-order p-Laplacian system (1). Then by applying the Clarke duality, the perturbation technique, and the least action principle, they obtained the existence result of the periodic solution. The authors [16] also discussed the multi-point boundary value problem of a second-order differential equation with a p-Laplacian.

Motivated by the work listed above, we aim to discuss the periodic and subharmonic solutions to the p-Laplacian systems of (1). The proofs are depending on the dual least action principle and the perturbation arguments. To the best of our knowledge, there is no paper discussing the subharmonics solutions of p-Laplacian system by using the dual least action principle. This paper is a first try. The existence of the periodic solutions obtained in this paper slightly improves the result in [15]. New arguments are considered for the posterior estimates of the periodic solutions and the existence of subharmonic solutions are presented, which extend those in [2], [5].

Other existence results for periodic and subharmonic solutions of the p-Laplacian differential equation using other variational methods can be found in [4], [17]–[24].

The paper is organized as follows. In Section 2, we establish the variational structure of (1) and transfer the existence of the solution into the existence of a critical point of the corresponding functional. The dual action is mainly discussed here. In Section 3, sufficient conditions are presented to guarantee the existence of the periodic solution of (1). We also estimate the prior bounds of all the periodic solutions of (1). The existence of the subharmonic solutions are given in the last section.

2 Preliminary and dual action

Let k1 be any integer and p>1 a constant. The Sobolev space W k T 1 , p (0,kT; R N ) is the space of functions x:[0,kT] R N with x L p (0,kT; R N ) having a weak derivative x ˙ L p (0,kT; R N ) and x(0)=x(kT). The norm over W k T 1 , p is defined by

x W k T 1 , p = ( 0 k T | x ( t ) | p d t + 0 k T | x ˙ ( t ) | p d t ) 1 p .

Here, we recall that

|x|= i = 1 N | x i | 2 , x L p = ( 0 k T | x ( t ) | p d t ) 1 p , x = max t [ 0 , k T ] |x(t)|.

Consider the spaces X and Y defined by

X = { u = ( u 1 , u 2 ) : [ 0 , k T ] R 2 N , u 1 W k T 1 , p ( 0 , k T ; R N ) , u 2 W k T 1 , q ( 0 , k T ; R N ) } , Y = { v = ( v 1 , v 2 ) : [ 0 , k T ] R 2 N , v 1 W k T 1 , q ( 0 , k T ; R N ) , v 2 W k T 1 , p ( 0 , k T ; R N ) } ,

with the norm u X = u 1 W k T 1 , p + u 2 W k T 1 , q and v Y = v 1 W k T 1 , q + v 2 W k T 1 , p , respectively. Here q is a constant such that 1 p + 1 q =1. It is easy to verify that X and Y are reflexive Banach spaces and X =Y.

For any h=( h 1 , h 2 ) L 1 (0,kT; R N × R N ), the mean value is defined by

h ¯ = 1 k T 0 k T h(t)dt.

Similarly, we can define h ¯ 1 and h ¯ 2 . We denote by Y ˜ the subspace of Y by

Y ˜ ={vY: v ¯ =0}.

We easily find the following inequality.

Lemma 2.1

For every u 1 W k T 1 , p and u ¯ 1 =0, we have

u 1 L p kT u ˙ 1 L p .

For every u 2 W k T 1 , q and u ¯ 2 =0, we have

u 2 L q kT u ˙ 2 L q .

Let u 1 =x, α u 2 = Φ p ( x ˙ ), then the second-order p-Laplacian system (1) can be changed to the first-order ones,

{ u ˙ 1 = Φ q ( α u 2 ( t ) ) , u ˙ 2 = 1 α F ( t , u 1 ( t ) ) ,
(3)

where α>0 is a parameter. Define H i :[0,kT]× R N R, i=1,2, and H:[0,kT]× R 2 N R by

H 1 ( t , u 1 ) = 1 α F ( t , u 1 ) , H 2 ( t , u 2 ) = α q 1 q | u 2 | q , H ( t , u ) = H 1 ( t , u 1 ) + H 2 ( t , u 2 )

for u=( u 1 , u 2 ) R 2 N . Then the system (3) can be written by

J u ˙ (t)+H ( t , u ( t ) ) =0,
(4)

where J= ( 0 N I N I N 0 N ) is the symplectic matrix. Obviously, J 2 = I 2 N and (Ju,v)=(u,Jv) for all u,v R 2 N .

By the kT-periodic property, we have the Hamiltonian action of (4) on X given by

ψ(u)= 0 k T ( 1 2 ( J u ˙ ( t ) , u ( t ) ) H ( t , u ( t ) ) ) dt,
(5)

where (,) is the inner product in R 2 N . Because the first part of ψ is indefinite, the dual least process is applied to discuss the existence of the periodic solutions of (4).

Let Γ 0 the set of all convex lower semi-continuous functions from R 2 N to (,] whose effective domain (not equal to ∞) is nonempty. Then the Fenchel transform H (t,) of H(t,) Γ 0 is defined by

H (t,v)= sup u R 2 N ( ( v , u ) H ( t , u ) ) .
(6)

Similarly, we also can define

F (t, v 1 )= sup x R N ( ( v 1 , x ) F ( t , x ) ) .

For more properties of the Fenchel transform, we refer to [2], [12]. By Theorem 2.2 and Proposition 2.4 in [2], we have the following conclusion.

Lemma 2.2

Suppose the following condition holds.

(A0):There exist positive constants α, δ, and positive functionsβ,γ L q (0,kT;[0,+))such that

δ p | x | p β(t)F(t,x) α p | x | p +γ(t)

holds for a.e. t[0,kT]andx R N .

Then

1 q α q p | v 1 | q γ(t) F (t, v 1 ) 1 q δ q p | v 1 | q +β(t),a.e. t[0,kT].

Furthermore, we have F (t,) C 1 ( R N )and

| F (t, v 1 )| ( p δ ( | v 1 | + β ( t ) + γ ( t ) ) + 1 ) q 1 ,a.e. t[0,kT].

From the definition of H 2 and Lemma 2.2, we have the following result.

Corollary 2.3

H 2 (t, v 2 )= 1 p α | v 2 | p .

Easily we find H (t,v)= 1 α F (t, v 1 )+ H 2 (t, v 2 ). So when F satisfies the condition (A0), H (t,v) is continuously differentiable in v for a.e. t[0,kT] and

H (t,v)=(v,u)H(t,u)v=H(t,u)u= H (t,v).
(7)

Let v=Ju and by duality, we have

ψ ( u ) = 0 k T ( 1 2 ( J u ˙ ( t ) , u ( t ) ) H ( t , u ( t ) ) ) d t = 0 k T ( 1 2 ( v ˙ ( t ) , u ( t ) ) H ( t , u ( t ) ) ) d t = 0 k T ( 1 2 ( v ˙ ( t ) , u ( t ) ) + ( v ˙ ( t ) , u ( t ) ) H ( t , u ( t ) ) ) d t = 0 k T ( 1 2 ( J v ˙ ( t ) , v ( t ) ) + H ( t , v ˙ ( t ) ) ) d t .

So the dual action can be defined on Y by

X(v)= 0 k T ( 1 2 ( J v ˙ ( t ) , v ( t ) ) + H ( t , v ˙ ( t ) ) ) dt.
(8)

X is the functional we needed since the critical points of X on Y coincide with the solutions of (4). Because X(v+c)=X(v), it suffices to find a critical point in Y ˜ . Similarly to the discussion of the related lemmas in reference [15], we have the following results.

Lemma 2.4

Suppose the condition (A0) holds. ThenXis continuously differential on Y ˜ . For anyh Y ˜ , we have

X ( v ) , h = 0 k T ( 1 2 ( J v ˙ ( t ) , h ( t ) ) + ( H ( t , v ˙ ( t ) ) 1 2 J v ( t ) , h ˙ ( t ) ) ) dt.
(9)

Lemma 2.5

Suppose the condition (A0) holds. Ifv Y ˜ is a critical point ofX, then the functionu(t)= H (t, v ˙ (t))is the kT-periodic solution of (4) in X.

Lemma 2.6

For everyuX, we have

0 k T ( J u ˙ ( t ) , u ( t ) ) dt 2 k T p u ˙ 1 L p p 2 k T q u ˙ 2 L q q .

For everyvY, we have

0 k T ( J v ˙ ( t ) , v ( t ) ) dt 2 k T q v ˙ 1 L q q 2 k T p v ˙ 2 L p p .

Remark 2.1

There are some extended versions of the inequalities in Lemmas 2.1 and 2.6; see [24], [25].

3 Periodic solutions

In this section, we discuss the kT-periodic solution of (4). Here we note that the H(t,u) is T-periodic in t for each u R 2 N .

Theorem 3.1

Suppose the following conditions are satisfied.

(A1):There existsl L p q (0,kT; R N )such that for allx R N and a.e. t[0,kT], one has

F(t,x) ( l ( t ) , Φ p ( x ) ) .

(A2):There existsα(0, ( k T ) max { p , q } / q ), γ L max { p , q } (0,kT; R N )such that for allx R N and a.e. t[0,kT], one has

F(t,x) α 2 p | x | p +γ(t).

(A3): 0 k T F(t,x)dt+as|x|+, x R N .

Then the problem (4) has at least one solutionu=( u 1 , u 2 )Xsuch that u 1 is the kT-periodic solutions of (1) and

v(t)=J [ u ( t ) 1 k T 0 k T u ( s ) d s ]

minimizes the dual action

X: Y ˜ (,],v 0 k T ( 1 2 ( J v ˙ ( t ) , v ( t ) ) + H ( t , v ˙ ( t ) ) ) dt.

Proof

From the discussion in Section 2, we can see that if the problem (4) has one solution u=( u 1 , u 2 )X, then u 1 W k T 1 , p is for the kT-periodic solutions of (1). Now we need to prove the problem (4) has at least one kT-periodic solution in X. The proof is divided into three parts.

Step 1: Existence of a solution for the perturbed problem.

Choose ϵ 0 >0 such that

0<α+ ϵ 0 <min { ( k T ) 1 p , ( k T ) 1 } ,

and for any 0<ϵ< ϵ 0 , let

H 1 ϵ (t, u 1 )= H 1 (t, u 1 )+ ϵ p | u 1 | p .

Clearly, H 1 ϵ (t, u 1 ) is strictly convex and continuously differentiable in u 1 for a.e. t[0,kT]. From (A1) and (A2), we have

1 α |l(t)| | u 1 | p 1 + ϵ p | u 1 | p H 1 ϵ (t, u 1 ) α + ϵ p | u 1 | p + 1 α γ(t).

Because the function g(s)= ϵ 2 p s p 1 α |l(t)| s p 1 , s>0, attains its minimum at s=2(p1)|l(t)|/αϵ, we have

1 α p ( 2 ( p 1 ) α ϵ ) p 1 |l(t) | p + ϵ 2 p | u 1 | p H 1 ϵ (t, u 1 ) α + ϵ p | u 1 | p + 1 α γ(t).

Let H ϵ (t,u)= H 1 ϵ (t, u 1 )+ H 2 (t, u 2 ). Easily we find H ϵ (t,v)= H 1 ϵ (t, v 1 )+ H 2 (t, v 2 ), where v=( v 1 , v 2 )Y. From Lemma 2.4 and Lemma 2.5, we find that the perturbed dual action

X ϵ (v)= 0 k T ( 1 2 ( J v ˙ ( t ) , v ( t ) ) + H ϵ ( t , v ˙ ( t ) ) ) dt
(10)

is continuously differentiable on Y ˜ and if v ϵ Y ˜ is a critical point of X ϵ , the function u ϵ defined by

u ϵ (t)= H ϵ ( t , v ˙ ϵ ( t ) )

is a solution of

{ J u ˙ ( t ) + H ϵ ( t , u ( t ) ) = 0 , u ( 0 ) = u ( k T ) ,
(11)

i.e.,

{ u ˙ 1 ϵ ( t ) Φ q ( α u 2 ϵ ( t ) ) = 0 , u ˙ 2 ϵ ( t ) + ϵ Φ p ( u 1 ϵ ( t ) ) + 1 α F ( t , u 1 ϵ ( t ) ) = 0 , u ϵ ( 0 ) = u ϵ ( k T ) ,
(12)

where we rewrite u ϵ =( u 1 ϵ , u 2 ϵ )X. Meanwhile, from Lemma 2.2, we have

H ϵ ( t , v ˙ ( t ) ) 1 q ( α + ϵ ) q / p | v ˙ 1 (t) | q + 1 p α 1 | v ˙ 2 (t) | p 1 α γ(t),

which together with (10) and Lemma 2.6 implies that

X ϵ ( v ) 1 q ( ( α + ϵ ) q / p k T ) 0 k T | v ˙ 1 ( t ) | q d t + 1 p ( α 1 k T ) 0 k T | v ˙ 2 ( t ) | p d t 0 k T 1 α γ ( t ) d t δ 10 v ˙ 1 L q q + δ 20 v ˙ 2 L p p γ 0
(13)

with δ 10 >0, δ 20 >0. Thus every minimizing sequence of X ϵ on Y ˜ is bounded by (13) and Lemma 2.1. From the continuity of H ϵ and the definition of H ϵ , we can see that the second term of X ϵ is weakly lower semi-continuous on Y ˜ . Meanwhile the first part of X ϵ is weakly continuous. So X ϵ is weakly lower semi-continuous. This implies that X ϵ has a minimum at some v ϵ Y ˜ . So (11) has a solution u ϵ = H ϵ (t, v ˙ ϵ (t)).

Step 2: Estimation of u ϵ .

It follows from a similar discussion to Lemma 2.2 that

| H 1 (t, u 1 )| ( p α q p ( | u 1 | + | l ( t ) | | u 1 | p 1 + γ ( t ) ) ) p 1 +1.

So for any u 1 R N , H 1 (t, u 1 ) L q [0,kT]. Then the function

H ¯ 1 : R N R, u 1 0 k T H 1 (t, u 1 )dt

is continuously differentiable. By (A3), H ¯ 1 has a minimum at some point u ¯ 1 R N such that

0 k T H 1 (t, u ¯ 1 )dt=0.

Easily, H 2 (t, u 2 )= Φ p (α u 2 ) and

0 k T H 2 (t,0)dt=0.

So, let u ¯ =( u ¯ 1 ,0) R 2 N , and then v ˙ (t)=H(t, u ¯ ) has a unique solution ω Y ˜ . By duality, we also have

H (t, ω ˙ )=( ω ˙ , u ¯ )H(t, u ¯ ),

and H (, ω ˙ ()) L 1 [0,kT]. From the inequality H(t, u ¯ ) H ϵ (t, u ¯ ), we have H ϵ (t, ω ˙ ) H (t, ω ˙ ), and from (13), we obtain

δ 10 v ˙ 1 ϵ L q q + δ 20 v ˙ 2 ϵ L p p γ 0 X ϵ ( v ϵ ) X ϵ (ω)X(ω)= c 1 <+.

By Lemma 2.1, we have v ϵ Y c 2 , where c 1 and c 2 are constants independent of ϵ, as well as the following constants c i , i=3,4,5. Furthermore, from u ˙ ϵ =J v ˙ ϵ , we have

u ˙ 1 ϵ L p + u ˙ 2 ϵ L q = v ˙ 1 ϵ L q + v ˙ 2 ϵ L p v ϵ Y c 2 ,

and from 0 k T u ˜ ϵ (t)dt=0, i.e.

| u ˜ ϵ (t)|| 0 k T u ˜ ˙ ϵ (t)dt| ( k T ) 1 / q u ˜ ˙ 1 ϵ L p + ( k T ) 1 / p u ˜ ˙ 2 ϵ L q ,

we have u ˜ ϵ X c 3 .

Meanwhile, by the convexity of F(t,), we have

H ( t , u ¯ ϵ / 2 ) 1 2 H ( t , u ϵ ( t ) ) + 1 2 H ( t , u ˜ ϵ ( t ) ) 1 2 H ( t , u ϵ ( t ) ) 1 2 H ( t , 0 ) + 1 2 H ( t , 0 ) + α 2 p | u ˜ 1 ϵ ( t ) | p + 1 2 α γ ( t ) + α q 1 2 q | u ˜ 2 ϵ ( t ) | q 1 2 ( H ( t , u ϵ ( t ) ) , u ϵ ( t ) ) + α 2 p | u ˜ 1 ϵ ( t ) | p + α q 1 2 q | u ˜ 2 ϵ ( t ) | q + 1 α γ ( t ) = 1 2 ( J u ˙ ϵ ( t ) , u ϵ ( t ) ) ϵ ( Φ p ( u 1 ϵ ( t ) ) , u 1 ϵ ( t ) ) + α 2 p | u ˜ 1 ϵ ( t ) | p + α q 1 2 q | u ˜ 2 ϵ ( t ) | q + 1 α γ ( t ) .

So,

0 k T F ( t , u ¯ ϵ / 2 ) d t 0 k T 1 2 ( J u ˙ ϵ ( t ) , u ϵ ( t ) ) d t + α 2 p u ˜ ϵ L p p + α q 1 2 q u ˜ 2 ϵ L q q + γ 0 1 2 u ˜ 1 ϵ L p u ˜ ˙ 2 ϵ L q + 1 2 u ˜ ˙ 1 ϵ L p u ˜ 2 ϵ L q + α 2 p u ˜ ϵ L p p + α q 1 2 q u ˜ 2 ϵ L q q + γ 0 c 4 .

This together with condition (A3) implies that | u ¯ ϵ | is bounded. Consequently,

u ϵ X u ¯ ϵ X + u ˜ ϵ X c 5 .

Step 3: Existence of a solution for the problem (4).

Since u ϵ X is bounded, there is a sequence ϵ n 0 (n) with ϵ n (0, ϵ 0 ) and a function uX such that

u ϵ n uas n.

Moreover, from v ˙ ϵ =J u ˙ ϵ , we have

v ϵ (t)=J ( u ϵ ( t ) u ¯ ϵ ) .

So { v ϵ n } converges weakly to v(t)=J(u(t) u ¯ ). From (12), we have the integrated form

{ u 1 ϵ n ( t ) u 1 ϵ n ( 0 ) 0 t Φ q ( α u 2 ϵ n ( s ) ) d s = 0 , u 2 ϵ n ( t ) u 2 ϵ n ( 0 ) + ϵ n 0 t Φ p ( α u 1 ϵ n ( s ) ) d s + 0 t 1 α F ( s , u 1 ϵ n ( s ) ) d s = 0 .

Because u ϵ n converges weakly to u in X, u ϵ n converges uniformly to u in C k T (see Proposition 1.2 in [2]). So, let n and we have

{ u 1 ( t ) u 1 ( 0 ) 0 t Φ q ( α u 2 ( s ) ) d s = 0 , u 2 ( t ) u 2 ϵ n ( 0 ) + 1 α 0 t F ( s , u 1 ( s ) ) d s = 0 ,

that is, u is a solution of (4) in X.

Finally, we show v=J(u(t) u ¯ ) minimizes the dual action X on Y ˜ . Because v ϵ n is a minimum of X ϵ n and H ϵ n (t,v) H (t,v), we have

X ϵ n ( v ϵ n ) X ϵ n (h)X(h),h Y ˜ .

By the duality between u ϵ n and v ˙ ϵ n and the definition of H ϵ n (t,v), we have

X ϵ n ( v ϵ n ) = 0 k T ( 1 2 ( J v ˙ ϵ n ( s ) , v ϵ n ( s ) ) + ( v ˙ ϵ n ( s ) , u ϵ n ( s ) ) H ϵ n ( s , u ϵ n ( s ) ) ) d s = 0 k T ( 1 2 ( J v ˙ ϵ n ( s ) , v ϵ n ( s ) ) + ( u ϵ n ( s ) , v ˙ ϵ n ( s ) ) H ( s , u ϵ n ( s ) ) ϵ n p | u 1 ϵ n ( s ) | p ) d s .

Moreover, as v ϵ n converges weakly to v in Y ˜ , J v ˙ ϵ n converges weakly to Jv. Letting n we obtain, by v ˙ (t)=H(t,u(t)),

lim n X ϵ n ( v ϵ n ) = lim n 0 T ( 1 2 ( J v ˙ ϵ n ( s ) , v ϵ n ( s ) ) + ( u ϵ n ( s ) , v ˙ ϵ n ( s ) ) H ( s , u ϵ n ( s ) ) ϵ n p | u 1 ϵ n ( s ) | p ) d s = 0 T ( 1 2 ( J v ˙ ( s ) , v ( s ) ) + H ( s , v ˙ ( s ) ) ) d s = X ( v ) .

So X(v)X(h) for all h Y ˜ . The proof is complete. □

Remark 3.1

Theorem 3.1 still holds if condition (A1) is changed to:

(A4): There exists l L p q / ( p m + 1 ) (0,kT; R N ) such that for all x R N and a.e. t[0,kT], one has

F(t,x) ( l ( t ) , Φ m ( x ) ) ,

where m is an integer such that 2mp.

When the parameter α is smaller, we can obtain the prior bound for all the solutions of the p-Laplacian system (1).

Theorem 3.2

If there existα(0,min{ ( 2 k T ) p / q ,p ( 2 k T ) 1 }), β0, γ0, andδ>0such that

δ|x|βF(t,x) α 2 p | x | p +γ

for a.e. t[0,kT]andx R N , then each solution of (1) satisfies the inequalities

0 k T | x ˙ (t) | p dt p k T ( β + γ ) p 2 k T α ,
(14)
0 k T |x(t)|dt k T ( β + γ ) δ ( 1 2 k T α q / p ) .
(15)

Proof

We still set u 1 =x, α u 2 = Φ p ( x ˙ ), and the equalities (3) hold. Easily we find

1 q α 2 q p |F(t, u 1 ) | q ( F ( t , u 1 ) , u 1 ) +β+γ.

It follows from

( H ( t , u ) , u ) = 1 α ( F ( t , u 1 ) , u 1 ) + ( Φ q ( α u 2 ) , u 2 )

that

1 q α 1 2 q p |F(t, u 1 ) | q + ( Φ q ( α u 2 ) , u 2 ) ( H ( t , u ) , u ) + β α + γ α .

That is,

1 q α q p | u ˙ 2 (t) | q + α q 1 | u 2 | q ( J u ˙ ( t ) , u ( t ) ) + β α + γ α .

Integrating over [0,kT] and using Lemma 2.6, we have

1 q α q p u ˙ 2 L q q + α q 1 u 2 L q q 0 k T ( J u ˙ ( t ) , u ( t ) ) d t + k T ( β + γ ) α 2 k T p u ˙ 1 L p p + 2 k T q u ˙ 2 L q q + k T ( β + γ ) α = 2 k T p α q u 2 L q q + 2 k T q u ˙ 2 L q q + k T ( β + γ ) α .

So,

( 1 q α q p 2 k T q ) u ˙ 2 L q q + ( α q 1 2 k T p α q ) u 2 L q q k T ( β + γ ) α .

Hence

u 2 L q q p k T ( β + γ ) α q ( p 2 k T α ) , u ˙ 2 L q q q k T ( β + γ ) α 1 q / p 2 k T α

and

u ˙ 1 L p p = Φ q ( α u 2 ) L p p = α q u 2 L q q p k T ( β + γ ) p 2 k T α .

Meanwhile, by the convexity of F, we have

δ 0 k T | u 1 ( t ) | d t β k T 0 k T F ( t , u 1 ( t ) ) d t 0 k T ( F ( t , 0 ) + ( F ( t , u 1 ( t ) ) , u 1 ( t ) ) ) d t γ k T + α 0 k T ( H ( t , u ( t ) ) , u ( t ) ) d t α 0 k T ( Φ q ( α u 2 ) , u 2 ) d t = γ k T α 0 k T ( J u ˙ ( t ) , u ( t ) ) d t α q u 2 L q q γ k T + α 2 k T q u ˙ 2 L q q γ k T + 2 k 2 T 2 ( β + γ ) α q / p 2 k T ,

which completes the proof. □

4 Subharmonic solutions

Theorem 4.1

Assume thatF:R× R N Ris continuous. Suppose further that

F(t,x)+
(16)

and

F ( t , x ) | x | p 0
(17)

as|x|+uniformly intR. Then for eachkR{0}, the system (1) has a kT-periodic solution x k such that

x k +

and such that the minimal period T k of x k tends towhenk+.

Proof

Let c 1 = max t R |F(t,0)|. Then condition (16) implies that there exists a r>0 such that

F(t,x)1+ c 1

for all tR and x R N with |x|>r. By the convexity of F, we have

1 + c 1 F ( t , r | x | x ) r | x | F ( t , x ) + ( 1 r | x | ) F ( t , 0 ) r | x | F ( t , x ) + ( 1 r | x | ) c 1 ,

i.e.

F(t,x) 1 r |x|+ c 1

for all tR and x R N with |x|>r. Furthermore, the continuity of F implies that there exist positive constants δ, β such that

F(t,x)δ|x|β
(18)

for all tR and x R N .

By the condition (17), there exists α(0, ( 2 k T ) max { p , q } / q ) such that

F(t,x) α 2 p | x | p +γ
(19)

for all tR and x R N . Then from Theorem 3.1, we can see that the problem (4) has at least one solution u k =( u 1 k , u 2 k )X such that u 1 k is the kT-periodic solution of (1) and

v k (t)=J [ u k ( t ) 1 k T 0 k T u k ( s ) d s ]

minimizes the dual action

X: Y ˜ (,],v 0 k T ( 1 2 ( J v ˙ ( t ) , v ( t ) ) + H ( t , v ˙ ( t ) ) ) dt.

Next we estimate the upper bound of c k =X( v k ). For any h=( h 1 , h 2 ) Y ˜ , we have

c k = X ( v k ) X ( h ) = 0 k T ( 1 2 ( J h ˙ ( t ) , h ( t ) ) + H ( t , h ˙ ( t ) ) ) d t = 0 k T ( 1 2 ( J h ˙ ( t ) , h ( t ) ) + 1 α F ( t , h ˙ 1 ( t ) ) + 1 p α | h ˙ 2 | p ) d t .
(20)

Let ρ=( ρ 1 , ρ 2 ) R N × R N with | ρ i |=1, i=1,2, and ϵ2πδ/kT. Define the function

h(t)= ϵ k T 2 π ( ρ cos 2 π k T t + J ρ sin 2 π k T t ) .

Obviously, h Y ˜ and J h ˙ (t)= 2 π k T h(t). Meanwhile, from (18), we can see that when z R N with |z|δ, we have

F (t,z) sup x R n ( ( z , x ) F ( t , x ) ) sup x R n ( ( z , x ) δ | x | + β ) β.

So it follows from (20) that

c k X ( h ) 0 k T ( 1 2 ( J h ˙ ( t ) , h ( t ) ) + β α + 1 p α | h ˙ 2 | p ) d t = ϵ 2 T 2 4 π k 2 + ( β α + ϵ p p α ) T k .
(21)

If x k + does not hold when k+. Then there exist a subsequence { k n } and a constant c 1 >0 such that

x k n c 1 .

From (1), we have

d d t Φ p ( x ˙ k n ) c 2 ,

and so x ˙ k n c 3 for some constants c 2 , c 3 >0. This implies that

v k n c 4 , v ˙ k n c 5 .

On the other hand, the inequality

c k n = X ( v k n ) = 0 k n T ( 1 2 ( J v ˙ k n ( t ) , v k n ( t ) ) + H ( t , v ˙ k n ( t ) ) ) d t 0 k n T ( 1 2 v ˙ k n v k n H ( t , 0 ) ) d t ( 1 2 c 4 c 5 + γ α ) T k n
(22)

is incompatible with (21) when n is sufficiently large. Thus x k + when k+.

It remains to prove that the minimal period T k of x k tends to +∞ as k+. If not, there exist R>0 and a subsequence { k n } such that the minimal period T k n of x k n is less than R. Meanwhile, by (18), (19), and Theorem 3.2, we have

0 T k n | x ˙ (t) | p dt α q u 2 L q q p T k n ( β + γ ) p 2 T k n α ,
(23)
0 T k n |x(t)|dt T k n ( β + γ ) δ ( 1 2 T k n α q / p ) .
(24)

We notice that

x k n = x ¯ k n + x ˜ k n | x ¯ k n | + x ˜ k n 1 T k n 0 T k n | x ( t ) | d t + T 1 / q ( 0 T k n | x ˙ ( t ) | p d t ) 1 / p .

Inequalities (23) and (24) imply that { x k n } is bounded, which is a contraction. The proof is complete. □

References

  1. 1.

    Fonda A, Ramos M, Willem M: Subharmonic solutions for second-order differential equations. Topol. Methods Nonlinear Anal. 1993, 1: 49-66.

  2. 2.

    Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, Berlin; 1989.

  3. 3.

    Rabinowitz PH: On subharmonic solutions of Hamiltonian systems. Commun. Pure Appl. Math. 1980, 33: 609-633. 10.1002/cpa.3160330504

  4. 4.

    Tao ZL, Tang CL: Periodic and subharmonic solutions of second order Hamiltonian systems. J. Math. Anal. Appl. 2004, 293: 435-445. 10.1016/j.jmaa.2003.11.007

  5. 5.

    Willem M: Subharmonic oscillations of convex Hamiltonian systems. Nonlinear Anal. 1985, 9: 1303-1311. 10.1016/0362-546X(85)90037-9

  6. 6.

    Clarke FH: Solution périodique des équations hamiltoniennes. C. R. Acad. Sci. Paris 1978, 287: 951-952.

  7. 7.

    Clarke FH: A classical variational principle for periodic Hamiltonian trajectories. Proc. Am. Math. Soc. 1979, 76: 186-188.

  8. 8.

    Clarke FH: Periodic solutions to Hamiltonian inclusions. J. Differ. Equ. 1981, 40: 1-6. 10.1016/0022-0396(81)90007-3

  9. 9.

    Clarke FH, Ekeland I: Nonlinear oscillations and boundary value problems for Hamiltonian systems. Arch. Ration. Mech. Anal. 1982, 78: 315-333. 10.1007/BF00249584

  10. 10.

    Ekeland I: Periodic solutions of Hamiltonian equation and a theorem of P. Rabinowitz. J. Differ. Equ. 1979, 34: 523-534. 10.1016/0022-0396(79)90034-2

  11. 11.

    Ekeland I: Convexity Methods in Hamiltonian Mechanics. Springer, Berlin; 1990.

  12. 12.

    Rockafellar RT: Convex Analysis. Princeton University Press, Princeton; 1970.

  13. 13.

    Mawhin J, Willem M: Critical points of convex pertubations of some indefinite quadratic forms and semi-linear boundary value problems at resonance. Ann. Inst. Henri Poincaré 1986, 3: 431-453.

  14. 14.

    Willem M: Jean Mawhin’s contributions to critical point theory. Bound. Value Probl. 2013., 2013: 10.1186/1687-2770-2013-115

  15. 15.

    Tian Y, Ge WG: Periodic solutions of non-autonomous second order system with a p -Laplacian. Nonlinear Anal. 2007, 66: 192-203. 10.1016/j.na.2005.11.020

  16. 16.

    Tian Y, Graef JR, Kong LJ, Wang M: Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Discrete Contin. Dyn. Syst. 2013, 2013: 759-769. suppl.

  17. 17.

    Lin XY, Tang XH: Existence of infinitely many solutions for p -Laplacian equations in R N . Nonlinear Anal. 2013, 92: 72-81. 10.1016/j.na.2013.06.011

  18. 18.

    Lv X, Lu SP, Yan P: Periodic solutions of non-autonomous ordinary p -Laplacian system. J. Appl. Math. Comput. 2011, 35: 11-18. 10.1007/s12190-009-0336-4

  19. 19.

    Ma SW, Zhang YX: Existence of infinitely many periodic solutions for ordinary p -Laplacian systems. J. Math. Anal. Appl. 2009, 351: 469-479. 10.1016/j.jmaa.2008.10.027

  20. 20.

    Tang XH, Zhang XY: Periodic solutions for second-order Hamiltonian systems with a p -Laplacian. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 2010, LXIV: 93-113.

  21. 21.

    Wang ZY, Zhang JH: Periodic solutions of non-autonomous second order systems with p -Laplacian. Electron. J. Differ. Equ. 2009., 2009:

  22. 22.

    Xu B, Tang CL: Some existence results on periodic solutions of ordinary p -Laplacian systems. J. Math. Anal. Appl. 2007, 333: 1228-1236. 10.1016/j.jmaa.2006.11.051

  23. 23.

    Zhang L, Chen Y:Existence of periodic solutions of p(t)-Laplacian systems. Bull. Malays. Math. Soc. 2012, 35(1):25-38.

  24. 24.

    Zhang XY, Tang XH: Periodic solutions for second order Hamiltonian system with a p -Laplacian. Bull. Belg. Math. Soc. Simon Stevin 2011, 18: 301-309.

  25. 25.

    Manasevich R, Mawhin J: The spectrum of p -Laplacian systems under Dirichlet, Neumann and periodic boundary conditions. In Morse Theory, Minimax Theory and Their Applications to Nonlinear Differential Equations. International Press, Somerville; 2003:201-216.

Download references

Acknowledgements

This research is supported by the National Natural Science Foundation of China (No. 11101385) and by the Beijing Higher Education Young Elite Teacher Project.

Author information

Correspondence to Hairong Lian.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Keywords

  • p-Laplacian
  • periodic solution
  • subharmonics
  • dual least action principle