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Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems

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Abstract

In the present paper, we deal with 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. Under the assumptions that L(n) is allowed to be sign-changing and satisfies

lim | n | + inf | x | = 1 ( L ( n ) x , x ) =,

W(n,x) is of indefinite sign and superquadratic as |x|+, we establish several existence criteria to guarantee that the above system has infinitely many homoclinic solutions.

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 operator, p,L:Z R N × N and W:Z× R N R, W(n,x) is continuously differentiable in x for every nZ. In general, system (1.1) may be regarded as a discrete analogue of the following second-order Hamiltonian system

[ p ( t ) u ( t ) ] L(t)u(t)+W ( t , u ( t ) ) =0.
(1.2)

Moreover, system (1.1) has applications as is shown in the monographs [1, 2]. In the past 40 years, system (1.2) has been widely investigated, see [39] and references therein. System (1.2) is the special form of the Emden-Fowler equation, appearing in the study of astrophysics, gas dynamics, fluid mechanics, relativistic mechanics, nuclear physics and chemically reacting systems, and many well-known results concerning properties of solutions of (1.2) are collected in [10].

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 the multiplicity of homoclinic solutions of system (1.1) or its special forms have been investigated by many authors. Papers [1113] deal with the periodic case, where p, L and W are N-periodic in n. If the periodicity is lost, the case is quite different from the ones just described, because of lack of compactness of the Sobolev embedding. In this case, either a coercivity condition on L are required to be satisfied, see [1419], or W(n,x) can be dominated by a summable function, see [5, 13]. In the above-mentioned papers, except [16], L was always required to be positive definite. Meanwhile, W was always assumed to be superquadratic as x0 uniformly for nZ, i.e.,

(W0) lim | x | 0 W ( n , x ) | x | 2 =0 uniformly for nZ.

In addition, W(n,x) is subquadratic as |x| in [17, 20], while W(n,x) is superquadratic in [1116, 18, 19, 21]. Moreover, in the superquadratic case, except [5], the well-known global Ambrosetti-Rabinowitz superquadratic condition was always assumed:

(AR) there exists μ>2 such that

0<μW(n,x) ( W ( n , x ) , x ) ,(n,x)Z× R N {0},

where and in the sequel, (,) denotes the standard inner product in R N , and || is the induced norm.

However, in mathematical physics, it is of frequent occurrence in a system like (1.1) that the global positive definiteness of L(n) is not satisfied. This is seen, for example, L(n)=[ l + (n) l (n)] I N , where l + (n)0, l + (n)+ as |n|+, and l (n) is bounded, or L(n)=l(n) I N , l(n) is a polynomial of degree 2m with the property that the coefficient of the leading term is positive.

In this paper, we are interested in the case when L(n) is not global positive definite and satisfies the following assumption.

  1. (L)

    L(n) is an N×N real symmetric matrix for all nZ and the smallest eigenvalue of L(n) as |n|, i.e.,

    lim | n | [ inf | x | = 1 ( L ( n ) x , x ) ] =.

Under assumption (L) above, we will use the symmetric mountain pass theorem to study the existence of infinitely many homoclinic solutions for (1.1) in the case, where W satisfies the following weaker assumptions than (W0) as x0 and (AR) as |x|.

(W1) W(n,x) is continuously differentiable in x for every nZ, W(n,0)0, and there exist constants c 0 >0 and R 0 >0 such that

| W ( n , x ) | c 0 |x|,(n,x)Z× R N ,|x| R 0 ;
(1.3)

(W2) lim | x | | W ( n , x ) | | x | 2 = for all nZ, and

W(n,x)0,(n,x)Z× R N ,|x| R 0 ;

(W3) W(n,x)=W(n,x), (n,x)Z× R N ;

(W4) W ˜ (n,x):= 1 2 (W(n,x),x)W(n,x)g(n), (n,x)Z× R N , where |g| l 1 (Z,R), and there exists c 1 >0 such that

| W ( n , x ) | c 1 | x | 2 W ˜ (n,x),(n,x)Z× R N ,|x| R 0 ;

(W5) there exist μ>2 and ϱ>0 such that

μW(n,x) ( W ( n , x ) , x ) +ϱ | x | 2 ,(n,x)Z× R N ;

(W6) there exists a μ>2 such that

μW(n,x) ( W ( n , x ) , x ) ,(n,x)Z× R N ,|x| R 0 .

Now, we are ready to state the main results of this paper.

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

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

It is easy to check that (W1) and (W6) imply (W5). Thus, we have the following corollary.

Corollary 1.3 Assume that p(n) is an N×N real symmetric positive definite matrix for all nZ, L and W satisfy (L), (W1), (W2), (W3) and (W6). Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.

Remark 1.4 In our theorems, L(n) is allowed to be sign-changing, for example,

L(n)= ( n 2 10 ) I N .
(1.4)

Moreover, W(n,x) is also allowed to be sign-changing. Even if W(n,x)0, assumptions (W2), (W4), (W5) and (W6) are weaker than the superquadratic conditions, obtained in the aforementioned references. It is easy to check that the following functions W satisfy (W1), (W2), (W3) and (W4) or (W6):

W(n,x)= ( 1 + sin 2 n ) | x | 2 ln ( 1 2 + | x | ) ,
(1.5)
W(n,x)= e ( 1 + n 2 ) | x | 2 / ( 2 + n 2 ) 1,
(1.6)
W(n,x)= 1 1 + n 2 [ 4 | x | 5 + 2 | x | 3 sin | x | 4 | x | 2 cos | x | ]
(1.7)

and

W(n,x)=a(n) i = 1 m b i | x | β i ,
(1.8)

where b 1 >0, b i R, i=2,3,,m, β 1 > β 2 >> β m 2, and 0< inf Z a sup Z a<. One can see that they do not satisfy (W0) or (AR).

2 Preliminaries

Throughout this section, we always assume that p(n) is real symmetric positive definite matrix for all nZ. Set

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

and make the following assumption on L(n) instead of (L):

(L′) L(n) is an N×N real symmetric matrix for all nZ, lim | n | l(n)=, and

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

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 inner product above, 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 S : n Z | u ( n ) | q < + }

and

l ( Z , R N ) = { u S : 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. Evidently, E is continuously embedded into l q (Z, R N ) for 2q+, i.e., there exists γ q >0 such that

u q γ q u,uE.
(2.2)

Lemma 2.1 (Lin and Tang [15])

For uE, one has

u 1 1 + 4 α 4 u,
(2.3)

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

Lemma 2.2 (Tang and Lin [17])

Suppose that L satisfies (L′). Then E is compactly embedded in l q (Z, R N ) for 2q<, and

u q q [ 1 + 4 α ] ( 2 q ) / 4 u q ,uE.
(2.4)

Now, we define a functional Φ on E by

Φ(u)= 1 2 n Z [ ( p ( n + 1 ) u ( n ) , u ( n ) ) + ( L ( n ) u ( n ) , u ( n ) ) ] n Z W ( n , u ( n ) ) .
(2.5)

For any uE, there exists an NN such that |u(n)| R 0 for |n|N. Hence, by (W1), one has

| W ( n , u ( n ) ) | c 0 2 | u ( n ) | 2 ,|n|N.
(2.6)

Consequently, under assumptions (L′) and (W1), the functional Φ is of class C 1 (E,R). Moreover,

Φ(u)= 1 2 u 2 n Z W ( n , u ( n ) ) ,uE,
(2.7)
Φ ( u ) , v =u,v n Z ( W ( n , u ( n ) ) , v ( n ) ) ,u,vE.
(2.8)

Furthermore, the critical points of Φ in E are solutions of system (1.1) with u(±)=0, see [14, 16].

Lemma 2.3 Under assumptions (L′), (W1), (W2) and (W4), any sequence { u k }E satisfying

Φ( u k )c>0, Φ ( u k ) , u k 0,
(2.9)

is bounded in E.

Proof To prove the boundedness of { u k }, arguing by contradiction, suppose that u k . Let v k = u k / u k . Then v k =1 and v k q γ q v k = γ q for 2q. Observe that for k large

c+1Φ( u k ) 1 2 Φ ( u k ) , u k = n Z W ˜ ( n , u k ( n ) ) .
(2.10)

It follows from (2.7) and (2.9) that

1 2 lim sup k n Z | W ( n , u k ( n ) ) | u k 2 .
(2.11)

For 0a<b, let

Ω k (a,b)= { n Z : a | u k ( n ) | < b } .
(2.12)

Passing to a subsequence, we may assume that v k v in E, then by Lemma 2.2, v k v in l q (Z, R N ), 2q<, and v k (n)v(n) for all nZ.

If v=0, then v k 0 in l q (Z, R N ), 2q<, v k (n)0 for all nZ. Hence, it follows from (W1) that

n Ω k ( 0 , R 0 ) | W ( n , u k ( n ) ) | | u k ( n ) | 2 | v k ( n ) | 2 c 0 2 n Ω k ( 0 , R 0 ) | v k ( n ) | 2 c 0 2 v k 2 2 0.
(2.13)

By virtue of (W4) and (2.10), one can get that

n Ω k ( R 0 , ) | W ( n , u k ( n ) ) | | u k ( n ) | 2 | v k ( n ) | 2 v k 2 n Ω k ( R 0 , ) | W ( n , u k ( n ) ) | | u k ( n ) | 2 c 1 [ n Ω k ( R 0 , ) W ˜ ( n , u k ( n ) ) ] n Z | v k ( n ) | 2 c 1 [ c + 1 n Ω k ( 0 , R 0 ) W ˜ ( n , u k ( n ) ) ] v k 2 2 c 1 [ c + 1 n Ω k ( 0 , R 0 ) g ( n ) ] v k 2 2 c 1 [ c + 1 + n Z | g ( n ) | ] v k 2 2 0 .
(2.14)

Combining (2.13) with (2.14), we have

n Z | W ( n , u k ( n ) ) | u k 2 = n Ω k ( 0 , R 0 ) | W ( n , u k ( n ) ) | | u k ( n ) | 2 | v k ( n ) | 2 + n Ω k ( R 0 , ) | W ( n , u k ( n ) ) | | u k ( n ) | 2 | v k ( n ) | 2 0 ,

which contradicts (2.11).

Set A:={nZ:|v(n)|0}. If v0, then A. For any nA, we have lim k + | u k (n)|=. Hence A Ω k ( R 0 ,) for large kN, and it follows from (2.7), (W1), (W2) and Fatou’s lemma that

0 = lim k c + o ( 1 ) u k 2 = lim k Φ ( u k ) u k 2 = lim k [ 1 2 n Z W ( n , u k ( n ) ) | u k ( n ) | 2 | v k ( n ) | 2 ] = lim k [ 1 2 n Ω k ( 0 , R 0 ) W ( n , u k ( n ) ) | u k ( n ) | 2 | v k ( n ) | 2 n Ω k ( R 0 , ) W ( n , u k ( n ) ) | u k ( n ) | 2 | v k ( n ) | 2 ] lim sup k [ 1 2 + c 0 2 n Z | v k ( n ) | 2 n Ω k ( R 0 , ) W ( n , u k ( n ) ) | u k ( n ) | 2 | v k ( n ) | 2 ] 1 2 + c 0 γ 2 2 2 lim inf k n Ω k ( R 0 , ) W ( n , u k ( n ) ) | u k ( n ) | | v k ( n ) | 2 = 1 2 + c 0 γ 2 2 2 lim inf k n Z | W ( n , u k ( n ) ) | | u k ( n ) | 2 [ χ Ω k ( R 0 , ) ( n ) ] | v k ( n ) | 2 1 2 + c 0 γ 2 2 2 n Z lim inf k | W ( n , u k ( n ) ) | | u k ( n ) | 2 [ χ Ω k ( R 0 , ) ( n ) ] | v k ( n ) | 2 = ,
(2.15)

which is a contradiction. Thus { u k } is bounded in E. □

Lemma 2.4 Under assumptions (L′), (W1), (W2) and (W4), any sequence { u k }E satisfying (2.9) has a convergent subsequence in E.

Proof Lemma 2.3 implies that { u k } is bounded in E. Going if necessary to a subsequence, we can assume that u k u in E. By Lemma 2.2, u k u in l q (Z, R N ) for 2q<, and u k (n)u(n) for all nZ. By (L′), there exists an integer NN such that

| u k ( n ) | 2 1 min | s | | n | l ( s ) | s | | n | l(s) | u k ( s ) | 2 1 min | s | | n | l ( s ) u k 2 < R 0 2 ,|n|N.
(2.16)

It is easy to see that

| n | N | W ( n , u k ( n ) ) W ( n , u ( n ) ) | | u k ( n ) u ( n ) | 0,k+.
(2.17)

Next, we prove that

| n | > N | W ( n , u k ( n ) ) W ( n , u ( n ) ) | | u k ( n ) u ( n ) | 0,k+.
(2.18)

If (2.18) is not true, then there exist a constant ε 0 >0 and a subsequence { u k i } such that

| n | > N | W ( n , u k i ( n ) ) W ( n , u ( n ) ) | | u k i ( n ) u ( n ) | ε 0 ,iN.
(2.19)

Since u k u in l 2 (Z, R N ), passing to a subsequence if necessary, it can be assumed that i = 1 u k i u 2 2 <+. Set

w(n)= [ i = 1 | u k i ( n ) u ( n ) | 2 ] 1 / 2 ,nZ.
(2.20)

Then w l 2 (Z,R). From (2.16), (2.20) and (W1), one has

| W ( n , u k i ( n ) ) W ( n , u ( n ) ) | | u k i ( n ) u ( n ) | ( | W ( n , u k i ( n ) ) | + | W ( n , u ( n ) ) | ) ( | u k i ( n ) | + | u ( n ) | ) c 0 ( | u k i ( n ) | + | u ( n ) | ) 2 2 c 0 ( | u k i ( n ) | 2 + | u ( n ) | 2 ) 8 c 0 ( | u k i ( n ) u ( n ) | 2 + | u ( n ) | 2 ) 8 c 0 ( [ w ( n ) ] 2 + | u ( n ) | 2 ) : = h ( n ) , i N , | n | N
(2.21)

and

| n | > N h ( n ) = 8 c 0 | n | > N ( [ w ( n ) ] 2 + | u ( n ) | 2 ) 8 c 0 ( w 2 2 + u 2 2 ) < + .
(2.22)

Since u k i (n)u(n) for all nZ, then by (2.21), (2.22) and Lebesgue’s dominated convergence theorem, we have

lim i | n | > N | W ( n , u k i ( n ) ) W ( n , u ( n ) ) | | u k i ( n ) u ( n ) | =0,

which contradicts (2.19). Hence (2.18) holds. Combining (2.17) with (2.18), one has

n Z | W ( n , u k ( n ) ) W ( n , u ( n ) ) | | u k ( n ) u ( n ) | 0,n.
(2.23)

Observe that

u k u 2 = Φ ( u k ) Φ ( u ) , u k u + n Z ( W ( n , u k ( n ) ) W ( n , u ( n ) ) , u k ( n ) u ( n ) ) .
(2.24)

It is clear that

Φ ( u k ) Φ ( u ) , u k u 0,n.
(2.25)

From (2.23), (2.24) and (2.25), we have u k u0, n. □

Lemma 2.5 Under assumptions (L′), (W1) and (W5), any sequence { u k }E satisfying (2.9) has a convergent subsequence in E.

Proof First, we prove that { u k } is bounded in E. To this end, arguing by contradiction, suppose that u k . Let v k = u k / u k . Then v k =1 and v k q γ q v k = γ q for 2q. By (2.7), (2.8), (2.9) and (W5), one has

c + 1 Φ ( u k ) 1 μ Φ ( u k ) , u k = μ 2 2 μ u k 2 + n Z [ 1 μ ( W ( n , u k ( n ) ) , u k ( n ) ) W ( n , u k ( n ) ) ] μ 2 2 μ u k 2 ϱ μ u k 2 2 for large  k N ,
(2.26)

which implies

1 2 ϱ μ 2 lim sup k v k 2 2 .
(2.27)

Passing to a subsequence, we may assume that v k v in E, then by Lemma 2.1, v k v in l q (Z, R N ), 2q<, and v k (n)v(n) for all nZ. Hence, it follows from (2.27) that v0. Analogous to the proof of (2.15), we can deduce a contradiction. Thus, { u k } is bounded in E. The rest of the proof is the same as the one in Lemma 2.4. □

Lemma 2.6 Under assumptions (L′), (W1) and (W2), for any finite-dimensional subspace E ˜ E, there holds

Φ(u),u,u E ˜ .
(2.28)

Proof Arguing indirectly, assume that for some sequence { u k } E ˜ with u k , and there is M 2 >0 such that Φ( u k ) M 2 for all kN. Set v k = u k / u k , then v k =1. Passing to a subsequence, we may assume that v k v in E. Since E ˜ is finite-dimensional, then v k v E ˜ in E, v k (n)v(n) for all nZ, and so v=1. Hence, we can deduce a contradiction in the same way as (2.15). □

Corollary 2.7 Under assumptions (L′), (W1) and (W2), for any finite-dimensional subspace E ˜ E, there is R=R( E ˜ )>0 such that

Φ(u)0,u E ˜ ,uR.
(2.29)

Let { e j } is an orthonomormal basis of E and define X j =R e j ,

Y k = j = 1 k X j , Z k = j = k + 1 X j ,kN.
(2.30)

Lemma 2.8 Under assumption (L′), for 2q<,

β k (q):= sup u Z k , u = 1 u q 0,k.
(2.31)

Proof Since the embedding from E into l q (Z, R N ) is compact for 2q<, then Lemma 2.8 can be proved in a similar way as [[22], Lemma 3.8]. □

Applying Lemma 2.8, we can choose an integer m1 such that

u 2 2 1 2 c 0 u 2 ,u Z m .
(2.32)

Lemma 2.9 Under assumptions (L′) and (W1), there exist constants ρ,α>0 such that Φ | B ρ Z m α.

Proof If u= R 0 , then u R 0 . Hence, it follows from (W1) that

| W ( n , u ( n ) ) | c 0 2 | u ( n ) | 2 ,uE,u= R 0 .
(2.33)

By (2.7), (2.32) and (2.33), we have

Φ ( u ) = 1 2 u 2 n Z W ( n , u ( n ) ) 1 2 u 2 c 0 2 n Z | u ( n ) | 2 = 1 2 u 2 c 0 2 u 2 2 1 4 u 2 = 1 4 R 0 2 : = α , u Z m , u = R 0 : = ρ .

 □

We say that I C 1 (X,R) satisfies ( C ) c -condition if any sequence { u k } such that

I( u k )c, I ( u k ) ( 1 + u k ) 0
(2.34)

has a convergent subsequence.

Lemma 2.10 (Bartolo, Benci and Fortunato [23])

Let X be an infinite-dimensional Banach space, X=YZ, where Y is finite-dimensional. If I C 1 (X,R) satisfies ( C ) c -condition for all c>0, and

(I1) I(0)=0, I(u)=I(u) for all uX;

(I2) there exist constants ρ,α>0 such that Φ | B ρ Z α;

(I3) for any finite-dimensional subspace X ˜ X, there is R=R( X ˜ )>0 such that I(u)0 on X ˜ B R ;

then I possesses an unbounded sequence of critical values.

3 Proofs of the main results

By (L), there exists a constant a 0 >0 such that

( ( L ( n ) + 2 a 0 I N ) x , x ) | x | 2 ,(n,x)Z× R N .
(3.1)

Let L ¯ (n):=L(n)+2 a 0 I N and W ¯ (n,x)=W(n,x)+ a 0 | x | 2 . Then it is easy to verify the following lemma.

Lemma 3.1 uS is a solution of system (1.1) if and only if it is a solution of the following system

[ p ( n ) u ( n 1 ) ] L ¯ (n)u(n)+ W ¯ ( n , u ( n ) ) =0.
(3.2)

Proof of Theorem 1.1 Let X=E, Y= Y m , and let Z= Z m . Obviously, W ¯ satisfies (W1), (W2), (W3) and (W4). By Lemmas 2.3, 2.4, 2.9 and Corollary 2.7, all conditions of Lemma 2.10 are satisfied. Thus, system (3.2) possesses infinitely many nontrivial solutions. By Lemma 3.1, system (1.1) also possesses infinitely many nontrivial solutions. □

Proof of Theorem 1.2 Let X=E, Y= Y m , and let Z= Z m . Obviously, W ¯ satisfies (W1), (W2), (W3) and (W5). The rest of the proof is the same as that of Theorem 1.1 by using Lemma 2.5 instead of Lemmas 2.3 and 2.4. □

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Acknowledgements

The authors 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 Hunan Provincial Natural Science Foundation of China (No. 11JJ2005).

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Correspondence to Xianhua Tang.

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The authors declare that they have no competing interests.

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All authors jointly worked on the results, and they read and approved the final manuscript.

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Keywords

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