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Homoclinic orbits for second-order discrete Hamiltonian systems with subquadratic potential

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Abstract

Under the assumptions that W(n,x) is indefinite sign and subquadratic as |x|+ and L(n) satisfies

lim inf | n | + [ | n | ν 2 inf | x | = 1 ( L ( n ) x , x ) ] >0

for some constant ν<2, we establish a theorem on the existence of infinitely many homoclinic solutions for the second-order self-adjoint discrete Hamiltonian system

[ p ( n ) u ( n 1 ) ] L(n)u(n)+W ( n , u ( n ) ) =0,

where p(n) and L(n) are N×N real symmetric matrices for all nZ, and p(n) is always positive definite.

MSC:39A11, 58E05, 70H05.

1 Introduction

Consider the second-order self-adjoint discrete Hamiltonian system

[ p ( n ) u ( n 1 ) ] L(n)u(n)+W ( n , u ( n ) ) =0,
(1.1)

where nZ, u R N , u(n)=u(n+1)u(n) is the forward difference, p,L:Z R N × N and W:Z× R N R, W(n,x) is continuously differentiable in x for every nZ.

As usual, we say that a solution u(n) of system (1.1) is homoclinic (to 0) if u(n)0 as n±. In addition, if u(n)0 then u(n) is called a nontrivial homoclinic solution.

The existence and multiplicity of nontrivial homoclinic solutions for problem (1.1) have been extensively investigated in the literature with the aid of critical point theory and variational methods; see, for example, [113]. Most of them treat the case where W(n,x) is superquadratic as |x|.

Compared to the superquadratic case, as far as the author is aware, there are a few papers [10, 12, 13] concerning the case where W(n,x) has subquadratic growth at infinity. Specifically, [12] and [10] dealt with the existence and multiplicity of homoclinic solutions for (1.1) under the following assumptions on L:

( L ) L(n) is an N×N real symmetric positive definite matrix for all nZ and there exists a constant β>0 such that

( L ( n ) x , x ) β | x | 2 ,(n,x)Z× R N ;

( L ν ) L(n) is an N×N real symmetric positive definite matrix for all nZ and there exists a constant ν<2 such that

lim inf | n | + [ | n | ν 2 inf | x | = 1 ( L ( n ) x , x ) ] >0,

respectively. In the above two cases, since L(n) is positive definite, the variational functional associated with system (1.1) is bounded from below, techniques based on the genus properties have been well applied. In particular, Clark’s theorem is an efficacious tool to prove the existence and multiplicity of homoclinic solutions for system (1.1). However, if L(n) is not global positive definite on , the problem is far more difficult as 0 is a saddle point rather than a local minimum of the variational functional, which is strongly indefinite and it is not easy to obtain the boundedness of the Palais-Smale sequence. In a recent paper [13], based on a new direct sum decomposition of the ‘work space’, Tang and Lin proved the following theorem by using a linking theorem which was developed in [14].

Theorem 1.1 [13]

Assume that p(n) is an N×N real symmetric positive definite matrix for all nZ, L and W satisfy the following assumptions:

( L ν ) L(n) is an N×N real symmetric matrix for all nZ and there exists a constant ν<2 such that

lim inf | n | + [ | n | ν 2 inf | x | = 1 ( L ( n ) x , x ) ] >0;

(W1) there exist constants max{1,2/(3ν)}< γ 1 < γ 2 <2 and a 1 , a 2 0 such that

| W ( n , x ) | a 1 | x | γ 1 + a 2 | x | γ 2 ,(n,x)Z× R N ;

(W2) there exists a function φC([0,+),[0,+)) such that

| W ( n , x ) | φ ( | x | ) ,(n,x)Z× R N ,

where φ(s)=O( s γ 3 1 ) as s 0 + , max{1,2/(3ν)}< γ 3 <2;

(W3) there exist constants b 1 >0, b 2 , b 3 0 and max{1,2/(3ν)}< γ 6 < γ 5 < γ 4 <2 such that

2W(n,x)W(n,x) b 1 | x | γ 4 b 2 | x | γ 5 b 3 | x | γ 6 ,(n,x)Z× R N ;

(W4) there exist constants b 4 >0, b 5 , b 6 0 and max{1,2/(3ν)}< γ 7 < γ 8 < γ 9 <2 such that

W(n,x) b 4 | x | γ 7 b 5 | x | γ 8 b 6 | x | γ 9 ,(n,x)Z× R N ;

(W5) W(n,x)=W(n,x), (n,x)Z× R N .

Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.

We remark that the condition ‘positive definite’ is removed in ( L ν ), i.e., L(n) is not required to be global positive definite on . The main goal of this paper is to weaken conditions (W1), (W2), (W3) and (W4) of Theorem 1.1 under assumption ( L ν ).

To state our result, we first introduce the following assumptions:

(W1) there exist constants σ i [0,2ν), a i 0 and max{1,2(1+ σ i )/(3ν)}< γ i <2 with i=1,2 such that

| W ( n , x ) | i = 1 2 a i ( 1 + | n | σ i ) | x | γ i ,(n,x)Z× R N ;

(W2) there exist two constants max{1,2(1+ σ i )/(3ν)}< γ i + 2 <2, i=1,2 and two functions φ 1 , φ 2 C([0,+),[0,+)) such that

| W ( n , x ) | i = 1 2 ( 1 + | n | σ i ) φ i ( | x | ) ,(n,x)Z× R N ,

where φ i (s)=O( s γ i + 2 1 ) as s 0 + , i=1,2;

(W3) there exist constants b 1 >0, b 2 0 and 1< γ 6 < γ 5 <2 such that

2W(n,x)W(n,x) b 1 | x | γ 5 b 2 | x | γ 6 ,(n,x)Z× R N ;

(W4) there exist constants b 3 >0, b 4 0 and 1< γ 7 < γ 8 <2 such that

W(n,x) b 3 | x | γ 7 b 4 | x | γ 8 ,(n,x)Z× R N .

We are now in a position to state the main result of this paper.

Theorem 1.2 Assume that p(n) is an N×N real symmetric positive definite matrix for all nZ, L and W satisfy ( L ν ), (W1), (W2), (W3), (W4) and (W5). Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.

2 Preliminaries

In what follows, we always assume that p(n) is a real symmetric positive definite matrix for all nZ. As done in [13], we define

l(n)= inf x R N , | x | = 1 ( L ( n ) x , x )
(2.1)

and

Z 1 = { n Z : l ( n ) 0 } , Z 2 = { n Z : l ( n ) > 0 } .
(2.2)

Then by ( L ν ), l(n) is bounded from below and so Z 1 is a finite set and

l :=min { l ( n ) : n Z 2 } >0.
(2.3)

Define

L ˜ (n)={ l I N , n Z 1 , L ( n ) , n Z 2 ; l ˜ (n)={ l , n Z 1 , l ( n ) , n Z 2 .
(2.4)

Then, it follows from (2.1), (2.2), (2.3) and (2.4) that

( L ˜ ( n ) x , x ) l ˜ (n) | x | 2 l | x | 2 ,(n,x)Z× R N .
(2.5)

Let

S = { { u ( n ) } n Z : u ( n ) R N , n Z } , E = { u S : n Z [ ( p ( n + 1 ) u ( n ) , u ( n ) ) + ( L ˜ ( n ) u ( n ) , u ( n ) ) ] < + } ,

and for u,vE, let

(u,v)= n Z [ ( p ( n + 1 ) u ( n ) , v ( n ) ) + ( L ˜ ( n ) u ( n ) , v ( n ) ) ] .

Then E is a Hilbert space with the above inner product, and the corresponding norm is

u= { n Z [ ( p ( n + 1 ) u ( n ) , u ( n ) ) + ( L ˜ ( n ) u ( n ) , u ( n ) ) ] } 1 / 2 ,uE.

As usual, for 1q<+, set

l q ( Z , R N ) = { { u ( n ) } n Z : u ( n ) R N , n Z , n Z | u ( n ) | q < + }

and

l ( Z , R N ) = { { u ( n ) } n Z : u ( n ) R N , n Z , sup n Z | u ( n ) | < + } ,

and their norms are defined by

u q = ( n Z | u ( n ) | q ) 1 / q , u l q ( Z , R N ) ; u = sup n Z | u ( n ) | , u l ( Z , R N ) ,

respectively.

Lemma 2.1 [[9], Lemma 2.2]

For uE, one has

u 1 ( l + 4 α ) l 4 u,
(2.6)

where α=inf{(p(n)x,x):nZ,x R N ,|x|=1}.

Set

b(u,v)= n Z [ ( p ( n + 1 ) u ( n ) , v ( n ) ) + ( L ( n ) u ( n ) , v ( n ) ) ] ,u,vE.
(2.7)

Lemma 2.2 [[13], Lemma 2.3]

Suppose that L satisfies ( L ν ). Then

  1. (i)

    b(u,v) is a bilinear function on E, and there exists a constant C 0 >0 such that

    | b ( u , v ) | C 0 uv,u,vE;
    (2.8)
  2. (ii)
    b(u,u)= u 2 n Z 1 ( ( L ˜ ( n ) L ( n ) ) u ( n ) , u ( n ) ) ,uE.
    (2.9)

By ( L ν ), there exist an integer N 0 >max{|n|:n Z 1 } and M 0 >0 such that

| n | ν 2 inf | x | = 1 ( L ( n ) x , x ) M 0 ,|n| N 0 ,

which implies

| n | ν 2 ( L ( n ) x , x ) M 0 | x | 2 ,|n| N 0 ,x R N .
(2.10)

Lemma 2.3 Suppose that L satisfies ( L ν ). Then, for σ[0,2ν) and 1q(2(1+σ)/(3ν),2), E is compactly embedded in l q (Z, R N ); moreover,

| n | > N ( 1 + | n | σ ) | u ( n ) | q K ( σ , q ) N κ u q ,uE,N N 0
(2.11)

and

n Z ( 1 + | n | σ ) | u ( n ) | q [ ( | n | N ( 1 + | n | σ ) 2 / ( 2 q ) [ l ˜ ( n ) ] q / ( 2 q ) ) 1 q 2 + K ( σ , q ) N κ ] u q , u E , N N 0 ,
(2.12)

where

κ= ( 3 ν ) q 2 ( 1 + σ ) 2 >0,K(σ,q)=2 [ 2 ( 2 q ) ( 3 ν ) q 2 ( 1 + σ ) ] 1 q 2 M 0 q / 2 .
(2.13)

Proof Let r=[(3ν)q2(1+σ)]/(2q). Then r>0. For uE and N N 0 , it follows from (2.10), (2.13) and the Hölder inequality that

| n | > N ( 1 + | n | σ ) | u ( n ) | q 2 ( | n | > N | n | [ ( 2 ν ) q 2 σ ] / ( 2 q ) ) 1 q 2 ( | n | > N | n | 2 ν | u ( n ) | 2 ) q 2 = 2 ( | n | > N | n | ( r + 1 ) ) 1 q 2 ( | n | > N | n | 2 ν | u ( n ) | 2 ) q 2 2 ( 2 r N r ) 1 q 2 [ 1 M 0 | n | > N ( L ( n ) u ( n ) , u ( n ) ) ] q 2 2 1 + ( 2 q ) / 2 M 0 q / 2 r ( 2 q ) / 2 N κ u q = K ( σ , q ) N κ u q .

This shows that (2.11) holds. Hence, from (2.5), (2.11) and the Hölder inequality, one has

n Z ( 1 + | n | σ ) | u ( n ) | q = | n | N ( 1 + | n | σ ) | u ( n ) | q + | n | > N ( 1 + | n | σ ) | u ( n ) | q ( | n | N ( 1 + | n | σ ) 2 / ( 2 q ) [ l ˜ ( n ) ] q / ( 2 q ) ) 1 q 2 ( | n | N l ˜ ( n ) | u ( n ) | 2 ) q 2 + K ( σ , q ) N κ u q ( | n | N ( 1 + | n | σ ) 2 / ( 2 q ) [ l ˜ ( n ) ] q / ( 2 q ) ) 1 q 2 u q + K ( σ , q ) N κ u q .

This shows that (2.12) holds.

Finally, we prove that E is compactly embedded in l q (Z, R N ). Let { u k }E be a bounded sequence. Then by (2.6), there exists a constant Λ>0 such that

u k 1 ( l + 4 α ) l 4 u k Λ,kN.
(2.14)

Since E is reflexive, { u k } possesses a weakly convergent subsequence in E. Passing to a subsequence if necessary, it can be assumed that u k u 0 in E. It is easy to verify that

lim k u k (n)= u 0 (n),nZ.
(2.15)

For any given number ε>0, we can choose N ε >0 such that

2 q 1 K ( σ , q ) N ε κ { [ ( l + 4 α ) l 4 Λ ] q + u 0 q } <ε.
(2.16)

It follows from (2.15) that there exists k 0 N such that

| n | N ε | u k ( n ) u 0 ( n ) | q <εfor k k 0 .
(2.17)

On the other hand, it follows from (2.11), (2.14) and (2.16) that

| n | > N ε | u k ( n ) u 0 ( n ) | q 2 q 1 | n | > N ε ( | u k ( n ) | q + | u 0 ( n ) | q ) 2 q 1 K ( σ , q ) N ε κ ( u k q + u 0 q ) 2 q 1 K ( σ , q ) N ε κ { [ ( l + 4 α ) l 4 Λ ] q + u 0 q } ε , k N .
(2.18)

Since ε is arbitrary, combining (2.17) with (2.18), we get

u k u 0 q q = n Z | u k ( n ) u 0 ( n ) | q 0as k.

This shows that { u k } possesses a convergent subsequence in l q (Z, R N ). Therefore, E is compactly embedded in l q (Z, R N ) for 1q(2(1+σ)/(3ν),2). □

Lemma 2.4 Suppose that L and W satisfy ( L ν ) and (W1). Then, for uE,

n Z | W ( n , u ( n ) ) | ϕ 1 (N) u γ 1 + ϕ 2 (N) u γ 2 ,N N 0 ,
(2.19)

where

κ 1 = ( 3 ν ) γ 1 2 ( 1 + σ 1 ) 2 , κ 2 = ( 3 ν ) γ 2 2 ( 1 + σ 2 ) 2 ;
(2.20)
ϕ 1 (N)= a 1 [ ( | n | N ( 1 + | n | σ 1 ) 2 / ( 2 γ 1 ) [ l ˜ ( n ) ] γ 1 / ( 2 γ 1 ) ) 1 γ 1 2 + K ( σ 1 , γ 1 ) N κ 1 ] ,
(2.21)
ϕ 2 (N)= a 2 [ ( | n | N ( 1 + | n | σ 2 ) 2 / ( 2 γ 2 ) [ l ˜ ( n ) ] γ 2 / ( 2 γ 2 ) ) 1 γ 2 2 + K ( σ 2 , γ 2 ) N κ 2 ] .
(2.22)

Proof For N N 0 , it follows from (W1), (2.12), (2.20), (2.21) and (2.22) that

n Z | W ( n , u ( n ) ) | i = 1 2 a i n Z ( 1 + | n | σ i ) | u ( n ) | γ i i = 1 2 a i [ ( | n | N ( 1 + | n | σ i ) 2 / ( 2 γ i ) [ l ˜ ( n ) ] γ i / ( 2 γ i ) ) 1 γ i 2 + K ( σ i , γ i ) N κ i ] u γ i = ϕ 1 ( N ) u γ 1 + ϕ 2 ( N ) u γ 2 .

This shows that (2.19) holds. □

Lemma 2.5 Assume that L and W satisfy ( L ν ), (W1) and (W2). Then the functional f:ER defined by

f(u)= 1 2 b(u,u) n Z W ( n , u ( n ) ) ,uE
(2.23)

is well defined and of class C 1 (E,R) and

f ( u ) , v =b(u,v) n Z ( W ( n , u ( n ) ) , v ( n ) ) ,u,vE.
(2.24)

Furthermore, the critical points of f in E are the solutions of system (1.1) with u(±)=0.

Proof Lemmas 2.2 and 2.4 imply that f defined by (2.23) is well defined on E. Next, we prove that (2.24) holds. By (W2), there exist M 1 , M 2 >0 such that

φ i ( | x | ) M i | x | γ 2 + i 1 ,x R N ,|x|1,i=1,2.
(2.25)

For any u,vE, there exists an integer N 1 > N 0 such that |u(n)|+|v(n)|<1 for |n|> N 1 . Then, for any sequence { θ n } n Z R with | θ n |<1 for nZ and any number h(0,1), by (W2), (2.11) and (2.25), we have

n Z max h [ 0 , 1 ] | ( W ( n , u ( n ) + θ n h v ( n ) ) , v ( n ) ) | | n | N 1 max h [ 0 , 1 ] | W ( n , u ( n ) + θ n h v ( n ) ) | | v ( n ) | + | n | > N 1 max h [ 0 , 1 ] | W ( n , u ( n ) + θ n h v ( n ) ) | | v ( n ) | | n | N 1 max | x | u + v | W ( n , x ) | | v ( n ) | + i = 1 2 M i | n | > N 1 ( 1 + | n | σ i ) ( | u ( n ) | + | v ( n ) | ) γ 2 + i 1 | v ( n ) | | n | N 1 max | x | u + v | W ( n , x ) | | v ( n ) | + i = 1 2 M i | n | > N 1 ( 1 + | n | σ i ) | v ( n ) | γ 2 + i + i = 1 2 M i ( | n | > N 1 ( 1 + | n | σ i ) | u ( n ) | γ 2 + i ) 1 1 γ 2 + i × ( | n | > N 1 ( 1 + | n | σ i ) | v ( n ) | γ 2 + i ) 1 γ 2 + i | n | N 1 max | x | u + v | W ( n , x ) | | v ( n ) | + i = 1 2 M i K ( σ i , γ 2 + i ) N 1 κ 2 + i ( u γ 2 + i 1 + v γ 2 + i 1 ) v < + ,
(2.26)

where κ 2 + i =[ γ 2 + i (3ν)2(1+ σ i )]/2>0, i=1,2. Then by (2.23), (2.26) and Lebesgue’s dominated convergence theorem, we have

f ( u ) , v = lim h 0 + f ( u + h v ) f ( u ) h = lim h 0 + [ b ( u , v ) + h b ( v , v ) 2 n Z ( W ( n , u ( n ) + θ n h v ( n ) ) , v ( n ) ) ] = b ( u , v ) n Z ( W ( n , u ( n ) ) , v ( n ) ) .

This shows that (2.24) holds. In view of the proof of [[13], Lemma 2.6], the critical points of f in E are the solutions of system (1.1) with u(±)=0. □

Let us prove now that f is continuous. Let u k u in E. Then there exists a constant δ>0 such that

u ( l + 4 α ) l 4 δ, u k ( l + 4 α ) l 4 δ,k=1,2,.
(2.27)

It follows from (2.6) that

u δ, u k δ,k=1,2,.
(2.28)

By (W2), there exist M 3 , M 4 >0 such that

φ i ( | x | ) M 2 + i | x | γ 2 + i 1 ,x R N ,|x|δ,i=1,2.
(2.29)

From (2.11), (2.24), (2.27), (2.28), (2.29), (W2) and the Hölder inequality, we have

| f ( u k ) f ( u ) , v | | b ( u k u , v ) | + n Z | ( W ( n , u k ( n ) ) W ( n , u ( n ) ) , v ( n ) ) | C 0 u k u v + | n | N | W ( n , u k ( n ) ) W ( n , u ( n ) ) | | v ( n ) | + | n | > N ( | W ( n , u k ( n ) ) | + | W ( n , u ( n ) ) | ) | v ( n ) | o ( 1 ) + i = 1 2 M 2 + i | n | > N ( 1 + | n | σ i ) ( | u k ( n ) | γ 2 + i 1 + | u ( n ) | γ 2 + i 1 ) | v ( n ) | o ( 1 ) + i = 1 2 M 2 + i ( | n | > N ( 1 + | n | σ i ) | u k ( n ) | γ 2 + i ) 1 1 γ 2 + i ( | n | > N ( 1 + | n | σ i ) | v ( n ) | γ 2 + i ) 1 γ 2 + i + i = 1 2 M 2 + i ( | n | > N ( 1 + | n | σ i ) | u ( n ) | γ 2 + i ) 1 1 γ 2 + i ( | n | > N ( 1 + | n | σ i ) | v ( n ) | γ 2 + i ) 1 γ 2 + i o ( 1 ) + i = 1 2 M 2 + i K ( σ i , γ 2 + i ) N κ 2 + i ( u k γ 2 + i 1 + u γ 2 + i 1 ) v = o ( 1 ) , k + , N + , v E ,

which implies the continuity of f . The proof is complete.  □

Lemma 2.6 [14]

Let X be an infinite dimensional Banach space and let f C 1 (X,R) be even, satisfy the (PS)-condition, and f(0)=0. If X= X 1 X 2 (direct sum), where X 1 is finite dimensional, and f satisfies

  1. (i)

    f is bounded from below on X 2 ;

  2. (ii)

    for each finite dimensional subspace X ˜ X, there are positive constants ρ=ρ( X ˜ ) and σ=σ( X ˜ ) such that f | B ρ X ˜ 0 and f | B ρ X ˜ σ, where B ρ ={xX:x=ρ}.

Then f possesses infinitely many nontrivial critical points.

3 Proof of the theorem

Proof of Theorem 1.2 For uE, we define two functions as follows:

u (n)={ u ( n ) , n Z 1 , 0 , n Z 2 ; u + (n)={ 0 , n Z 1 , u ( n ) , n Z 2 .
(3.1)

Set

X 1 = { u : u E } , X 2 = { u + : u E } .
(3.2)

Then X:=E= X 1 X 2 (direct sum) and dim( X 1 )<+. Obviously, (W1) and (W5) imply f(0)=0 and f is even. In view of Lemma 2.5, f C 1 (E,R). In what follows, we first prove that f satisfies the (PS)-condition. Assume that { u k } k N E is a (PS)-sequence: { f ( u k ) } k N is bounded and f ( u k )0 as k+. From (2.23), (2.24) and (W3), we have

f ( u k ) , u k 2 f ( u k ) = n Z [ 2 W ( n , u k ( n ) ) ( W ( n , u k ( n ) ) , u k ( n ) ) ] b 1 n Z | u k ( n ) | γ 5 b 2 n Z | u k ( n ) | γ 6 = b 1 u k γ 5 γ 5 b 2 u k γ 6 γ 6 .

It follows that there exists a constant C 1 >0 such that

b 1 u k γ 5 γ 5 b 2 u k γ 6 γ 6 C 1 ( 1 + u k ) .
(3.3)

Since dim( X 1 )<+, it follows that there exists a constant C 2 >0 such that

u k 2 2 = ( u k , u k ) l 2 u k γ 5 u k γ 5 C 2 u k 2 u k γ 5 ,
(3.4)

where γ 5 = γ 5 /( γ 5 1). Combining (3.3) with (3.4), one has

n Z 1 ( ( L ˜ ( n ) L ( n ) ) u k ( n ) , u k ( n ) ) = n Z 1 ( ( L ˜ ( n ) L ( n ) ) u k ( n ) , u k ( n ) ) C 3 u k 2 2 C 4 ( 1 + u k 2 / γ 5 + u k 2 γ 6 / γ 5 ) .
(3.5)

From (2.19), (2.23) and (3.5), we obtain

u k 2 = n Z [ ( p ( n + 1 ) u k ( n ) , u k ( n ) ) + ( L ˜ ( n ) u k ( n ) , u k ( n ) ) ] = b ( u k , u k ) + n Z 1 ( ( L ˜ ( n ) L ( n ) ) u k ( n ) , u k ( n ) ) = n Z 1 ( ( L ˜ ( n ) L ( n ) ) u k ( n ) , u k ( n ) ) + 2 f ( u k ) + 2 n Z W ( n , u k ( n ) ) C 5 ( 1 + u k 2 / γ 5 + u k 2 γ 6 / γ 5 ) + 2 ϕ 1 ( N 0 ) u k γ 1 + 2 ϕ 2 ( N 0 ) u k γ 2 C 6 ( 1 + u k γ 1 + u k γ 2 + u k 2 / γ 5 + u k 2 γ 6 / γ 5 ) .
(3.6)

Since 1< γ 1 < γ 2 <2, 1< γ 6 < γ 5 <2, it follows from (3.6) that { u k } is bounded. Let A>0 such that

u k 1 ( l + 4 α ) l 4 u k A,kN.
(3.7)

So, passing to a subsequence if necessary, it can be assumed that u k u 0 in E. It is easy to verify that

lim k u k (n)= u 0 (n),nZ.
(3.8)

By (W2), there exist M 5 , M 6 >0 such that

φ i ( | x | ) M 4 + i | x | γ 2 + i 1 ,x R N ,|x|A,i=1,2.
(3.9)

For any given number ε>0, we can choose an integer N 3 > N 0 such that

M 4 + i K ( σ i , γ 2 + i ) N 3 κ 2 + i { [ ( l + 4 α ) l 4 A ] γ 2 + i + u 0 γ 2 + i } <ε,i=1,2.
(3.10)

It follows from (3.8) and the continuity of W(n,x) on x that there exists k 0 N such that

n = N 2 N 2 | W ( n , u k ( n ) ) W ( n , u 0 ( n ) ) | | u k ( n ) u 0 ( n ) | <εfor k k 0 .
(3.11)

On the other hand, it follows from (2.11), (3.7), (3.9), (3.10) and (W2) that

| n | > N 2 | W ( n , u k ( n ) ) W ( n , u 0 ( n ) ) | | u k ( n ) u 0 ( n ) | | n | > N 2 [ | W ( n , u k ( n ) ) | + | W ( n , u 0 ( n ) ) | ] ( | u k ( n ) | + | u 0 ( n ) | ) i = 1 2 | n | > N 2 ( 1 + | n | σ i ) [ φ i ( | u k ( n ) | ) + φ i ( | u 0 ( n ) | ) ] ( | u k ( n ) | + | u 0 ( n ) | ) i = 1 2 M 4 + i | n | > N 2 ( 1 + | n | σ i ) ( | u k ( n ) | γ 2 + i 1 + | u 0 ( n ) | γ 2 + i 1 ) ( | u k ( n ) | + | u 0 ( n ) | ) 2 i = 1 2 M 4 + i | n | > N 2 ( 1 + | n | σ i ) ( | u k ( n ) | γ 2 + i + | u 0 ( n ) | γ 2 + i ) i = 1 2 2 M 4 + i K ( σ i , γ 2 + i ) N 2 κ 2 + i ( u k γ 2 + i + u 0 γ 2 + i ) i = 1 2 2 M 4 + i K ( σ i , γ 2 + i ) N 2 κ 2 + i { [ ( l + 4 α ) l 4 A ] γ 2 + i + u 0 γ 2 + i } 4 ε , k N .
(3.12)

Since ε is arbitrary, combining (3.11) with (3.12), we get

n Z ( W ( n , u k ( n ) ) W ( n , u 0 ( n ) ) , u k ( n ) u 0 ( n ) ) 0as k.
(3.13)

It follows from (2.24) that

f ( u k ) f ( u 0 ) , u k u 0 = b ( u k u 0 , u k u 0 ) n Z ( W ( n , u k ( n ) ) W ( n , u 0 ( n ) ) , u k ( n ) u 0 ( n ) ) = u k u 0 2 n Z 1 ( ( L ˜ ( n ) L ( n ) ) ( u k u 0 ) , u k u 0 ) n Z ( W ( n , u k ( n ) ) W ( n , u 0 ( n ) ) , u k ( n ) u 0 ( n ) ) .
(3.14)

Since f ( u k ) f ( u 0 ), u k u 0 0, it follows from (3.8), (3.13) and (3.14) that u k u 0 in E. Hence, f satisfies the (PS)-condition.

Next, for u X 2 , it follows from (2.9), (2.19) and (2.23) that

f ( u ) = 1 2 b ( u , u ) n Z W ( n , u ( n ) ) = 1 2 u 2 n Z W ( n , u ( n ) ) 1 2 u 2 ϕ 1 ( N 0 ) u γ 1 ϕ 2 ( N 0 ) u γ 2 +
(3.15)

as u+ and u X 2 , since 1< γ 1 < γ 2 <2.

Finally, we prove that assumption (ii) in Lemma 2.6 holds. Let X ˜ X be any finite dimensional subspace. Then there exist constants c 0 =c( X ˜ )>0 and c =c( X ˜ )>0 such that

c 0 u u γ i c u,i=7,8,u X ˜ .
(3.16)

From (2.9), (2.23), (3.16) and (W4), one has

f ( u ) = 1 2 b ( u , u ) n Z W ( n , u ( n ) ) 1 2 u 2 b 3 n Z | u ( n ) | γ 7 + b 4 n Z | u ( n ) | γ 8 = 1 2 u 2 b 3 u γ 7 γ 7 + b 4 u γ 8 γ 8 1 2 u 2 b 3 c 0 γ 7 u γ 7 + b 4 c γ 8 u γ 8 , u X ˜ .

Since 1< γ 7 < γ 8 <2, the above estimation implies that there exist ρ=ρ( b 3 , b 4 , c 0 , c )=ρ( X ˜ )>0 and σ=σ( b 3 , b 4 , c 0 , c )=σ( X ˜ )>0 such that

f(u)0,u B ρ X ˜ ;f(u)σ,u B ρ X ˜ .

This shows that assumption (ii) in Lemma 2.6 holds. By Lemma 2.6, f has infinitely many critical points which are homoclinic solutions for system (1.1). □

4 Example

In this section, we give an example to illustrate our result.

Example 4.1 In system (1.1), let p(n) be an N×N real symmetric positive definite matrix for all nZ, L(n)=(1+ sin 2 n)( | n | 4 / 5 6) I N , and let

W(n,x)= ( 1 + sin 2 n ) [ ( 1 + | n | 1 / 9 ) | x | 5 / 4 3 | x | 3 / 2 + ( 1 + | n | 1 / 2 ) | x | 7 / 4 ] .
(4.1)

Then L satisfies ( L ν ) with ν=6/5, and

W ( n , x ) = ( 1 + sin 2 n ) [ 5 4 ( 1 + | n | 1 / 9 ) | x | 3 / 4 x 9 2 | x | 1 / 2 x + 7 4 ( 1 + | n | 1 / 2 ) | x | 1 / 4 x ] , | W ( n , x ) | 5 ( 1 + | n | 1 / 9 ) | x | 5 / 4 + 5 ( 1 + | n | 1 / 2 ) | x | 7 / 4 , ( n , x ) Z × R N , | W ( n , x ) | 7 ( 1 + | n | 1 / 9 ) | x | 1 / 4 + 8 ( 1 + | n | 1 / 2 ) | x | 3 / 4 , ( n , x ) Z × R N , 2 W ( n , x ) W ( n , x ) 1 4 | x | 7 / 4 3 | x | 3 / 2 , ( n , x ) Z × R N

and

W(n,x) | x | 5 / 4 6 | x | 3 / 2 ,(n,x)Z× R N .

Thus all the conditions of Theorem 1.2 are satisfied with

5 4 = γ 1 = γ 3 = γ 7 < γ 6 = γ 8 = 3 2 < γ 5 = γ 4 = γ 2 = 7 4 ; a 1 = a 2 = 5 ; b 1 = 1 4 , b 2 = 3 , b 3 = 1 , b 4 = 6 ; σ 1 = 1 9 , σ 2 = 1 2 ; φ 1 ( s ) = 7 s 1 / 4 , φ 2 ( s ) = 8 s 3 / 4 .

Hence, by Theorem 1.2, system (1.1) has infinitely many nontrivial homoclinic solutions. However, one can see that W(n,x) defined by (4.1) does not satisfy (W1) and (W2).

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Acknowledgements

The author would like to express their thanks to the referees for their helpful suggestions. This work is partially supported by the NNSF (No. 11171351) of China and supported by the Scientific Research Fund of Hunan Provincial Education Department (08A053) and supported by the Hunan Provincial Natural Science Foundation of China (No. 11JJ2005).

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Keywords

  • homoclinic solution
  • discrete Hamiltonian system
  • subquadratic
  • critical point