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Existence of homoclinic solutions for a class of difference systems involving p-Laplacian

Abstract

By using the critical point theory, some existence criteria are established which guarantee that the difference p-Laplacian systems of the form Δ( | Δ u ( n 1 ) | p 2 Δu(n1))a(n) | u ( n ) | q p u(n)+W(n,u(n))=0 have at least one or infinitely many homoclinic solutions, where 1<p<(q+2)/2, q>2, nZ, u R N , a:Z(0,+), and W:Z× R N R are not periodic in n.

MSC:34C37, 35A15, 37J45, 47J30.

1 Introduction

Consider homoclinic solutions of the following p-Laplacian system:

Δ ( | Δ u ( n 1 ) | p 2 Δ u ( n 1 ) ) a(n) | u ( n ) | q p u(n)+W ( n , u ( n ) ) =0,nZ,
(1.1)

where 1<p<(q+2)/2, q>2, nZ, u R N , a:Z(0,+), and W:Z× R N R are not periodic in n. Δ is the forward difference operator defined by Δu(n)=u(n+1)u(n), Δ 2 u(n)=Δ(Δu(n)). As usual, we say that a solution u of (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. We may think of (1.1) being a discrete analogue of the following differential system:

d d t ( | u ˙ ( t ) | p 2 u ˙ ( t ) ) a(t) | u ( t ) | q p u(t)+W ( t , u ( t ) ) =0,tR.
(1.2)

When p=2, (1.1) can be regarded as a discrete analogue of the following second-order Hamiltonian system:

u ¨ (t)a(t) | u ( t ) | q 2 u(t)+W ( t , u ( t ) ) =0,tR.
(1.3)

Problem (1.2) has been studied by Shi et al. in [1] and problem (1.3) has been studied in [24]. It is well known that the existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been firstly recognized by Poincaré [5]. If a system has the transversely intersected homoclinic orbits, then it must be chaotic. If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation and its perturbed system probably produces chaotic phenomenon. Therefore, it is of practical importance to investigate the existence of homoclinic orbits of (1.1) emanating from 0.

By applying critical point theory, the authors [622] studied the existence of periodic solutions and subharmonic solutions for difference equations or differential equations, which show that the critical point theory is an effective method to study periodic solutions of difference equations or differential equations. In this direction, several authors [2334] used critical point theory to study the existence of homoclinic orbits for difference equations. Motivated mainly by the ideas of [14, 35], we will consider homoclinic solutions of (1.1) by the mountain pass theorem and the symmetric mountain pass theorem. More precisely, we obtain the following main results, which seem not to have been considered in the literature.

Theorem 1.1 Suppose that a and W satisfy the following conditions:

  1. (A)

    Let 1<p<(q+2)/2 and q>2, a:Z(0,+) is a positive function on such that for all nZ

    a(n)α | n | β ,α>0,β>(q2p+2)/p.

(W1) W(n,x)= W 1 (n,x) W 2 (n,x), W 1 , W 2 are continuously differentiable in x, and there is a bounded set JZ such that

1 a ( n ) | W ( n , x ) | =o ( | x | q p + 1 ) as x0

uniformly in nZJ.

(W2) There is a constant μ>qp+2 such that

0<μ W 1 (n,x) ( W 1 ( n , x ) , x ) ,(n,x)Z× R N {0}.

(W3) W 2 (n,0)=0 and there exists a constant ϱ(qp+2,μ) such that

W 2 (n,x)0, ( W 2 ( n , x ) , x ) ϱ W 2 (n,x),(n,x)Z× R N .

Then problem (1.1) has one nontrivial homoclinic solution.

Theorem 1.2 Suppose that a and W satisfy (A), (W2) and the following conditions:

(W1)′ W(n,x)= W 1 (n,x) W 2 (n,x), W 1 , W 2 are continuously differentiable in x, and

1 a ( n ) | W ( n , x ) | =o ( | x | q p + 1 ) as x0

uniformly in nZ.

(W3)′ W 2 (n,0)=0 and there exists a constant ϱ(qp+2,μ) such that

( W 2 ( n , x ) , x ) ϱ W 2 (n,x),(n,x)Z× R N .

Then problem (1.1) has one nontrivial homoclinic solution.

Theorem 1.3 Suppose that a and W satisfy (A), (W1)-(W3) and

(W4) W(n,x)=W(n,x), (n,x)Z× R N .

Then problem (1.1) has an unbounded sequence of homoclinic solutions.

Theorem 1.4 Suppose that a and W satisfy (A), (W1)′, (W2), (W3)′ and (W4). Then problem (1.1) has an unbounded sequence of homoclinic solutions.

The rest of this paper is organized as follows: in Section 2, some preliminaries are presented and we establish an embedding result. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.

2 Preliminaries

Let

S = { { u ( n ) } n Z : u ( n ) R N , n Z } , W = { u S : n Z [ | Δ u ( n 1 ) | p + | u ( n ) | p ] < + } ,

and for uW, let

u= { n Z [ | Δ u ( n 1 ) | p + | u ( n ) | p ] } 1 / p .

Then W is a uniform convex Banach space with this norm. As usual, for 1p<+, let

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

and their norms are given by

u l p = ( n Z | u ( n ) | p ) 1 / p , u l p ( Z , R N ) , u = sup { | u ( n ) | : n Z } , u l ( Z , R N ) ,

respectively.

If σ is a positive function on and 1<s<+, let

l σ s = l σ s ( Z , R N ; σ ) = { u l loc 1 ( Z , R N ) | n Z σ ( n ) | u ( n ) | s < + } .

l σ s equipped with the norm

u s , σ = ( n Z σ ( n ) | u ( n ) | s ) 1 / s

is a reflexive Banach space.

Set E=W l a q p + 2 , where a is the function given in condition (A). Then E with its standard norm is a reflexive Banach space. The functional φ corresponding to (1.1) on E is given by

φ(u)= n Z [ 1 p | Δ u ( n 1 ) | p + a ( n ) q p + 2 | u ( n ) | q p + 2 W ( n , u ( n ) ) ] ,uE.
(2.1)

Clearly, it follows from (W1) or (W1)′ that φ:ER. By Theorem 2.1 of [36], we can deduce that the map

ua(n) | u ( n ) | q p u(n)

is continuous from l a q p + 2 in the dual space l a 1 / ( q p + 1 ) p 1 , where p 1 =(qp+2)/(qp+1). As the embeddings EW l γ for all γp are continuous, if (A) and (W1) or (W1)′ hold, then φ C 1 (E,R) and one can easily check that

φ ( u ) , v = n Z [ | Δ u ( n 1 ) | p 2 ( Δ u ( n 1 ) , Δ v ( n 1 ) ) + a ( n ) | u ( n ) | q p ( u ( n ) , v ( n ) ) ] n Z ( W ( n , u ( n ) ) , v ( n ) ) , u E .
(2.2)

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

Lemma 2.1 [23]

For uE

u u l p 2u.
(2.3)

Lemma 2.2 If a satisfies assumption (A), then

the embedding  l a q p + 2 l p  is continuous.
(2.4)

Moreover, there exists a Sobolev space Z such that

the embeddings  l a q p + 2 Z l p  are continuous,
(2.5)
the embedding WZ l p  is compact.
(2.6)

Proof Let θ=(qp+2)/(q2p+2), θ =(qp+2)/p, we have

u l p p = n Z [ a ( n ) ] 1 / θ [ a ( n ) ] 1 / θ | u ( n ) | p ( n Z [ a ( n ) ] θ / θ ) 1 / θ ( n Z a ( n ) | u ( n ) | p θ ) 1 / θ = a 1 ( n Z a ( n ) | u ( n ) | q p + 2 ) p / q p + 2 = a 1 u q p + 2 , a p ,

where a 1 = ( n Z [ a ( n ) ] p / ( q 2 p + 2 ) ) ( q 2 p + 2 ) / ( q p + 2 ) <+ from (A). Then (2.4) holds.

By (A), there exists a positive function ρ such that ρ(n)+ as |n|+ and

a 2 = ( n Z [ ρ ( n ) ] θ [ a ( n ) ] θ / θ ) 1 / θ <+.

Since

u p , ρ p = n Z ρ ( n ) | u ( n ) | p = n Z ρ ( n ) [ a ( n ) ] 1 / θ [ a ( n ) ] 1 / θ | u ( n ) | p ( n Z [ ρ ( n ) ] θ [ a ( n ) ] θ / θ ) 1 / θ ( n Z a ( n ) | u ( n ) | q p + 2 ) 1 / θ = a 2 u q p + 2 , a p ,

(2.5) holds by taking Z= l ρ p .

Finally, as WZ is the weighted Sobolev space Γ 1 , p (Z,ρ,1), it follows from [36] that (2.6) holds. □

The following two lemmas are the mountain pass theorem and the symmetric mountain pass theorem, which are useful in the proofs of our theorems.

Lemma 2.3 [37]

Let E be a real Banach space and I C 1 (E,R) satisfying the (PS)-condition. Suppose I(0)=0 and

  1. (i)

    There exist constants ρ,α>0 such that I B ρ ( 0 ) α.

  2. (ii)

    There exists an eE B ¯ ρ (0) such that I(e)0.

Then I possesses a critical value cα which can be characterized as

c= inf h Φ max s [ 0 , 1 ] I ( h ( s ) ) ,

where Φ={hC([0,1],E)|h(0)=0,h(1)=e}, and B ρ (0) is an open ball in E of radius ρ centered at 0.

Lemma 2.4 [37]

Let E be a real Banach space and I C 1 (E,R) with I even. Assume that I(0)=0 and I satisfies (PS)-condition, (i) of Lemma  2.3 and the following condition:

  1. (iii)

    For each finite dimensional subspace E E, there is r=r( E )>0 such that I(u)0 for u E B r (0), B r (0) is an open ball in E of radius r centered at 0.

Then I possesses an unbounded sequence of critical values.

Lemma 2.5 Assume that (W2) and (W3) or (W3)′ hold. Then for every (n,x)Z× R N ,

  1. (i)

    s μ W 1 (n,sx) is nondecreasing on (0,+);

  2. (ii)

    s ϱ W 2 (n,sx) is nonincreasing on (0,+).

The proof of Lemma 2.5 is routine and we omit it. In the following, C i (i=1,2,) denote different positive constants.

3 Proofs of theorems

Proof of Theorem 1.1 Firstly, we prove that the functional φ satisfies the (PS)-condition. Let { u k }E satisfying φ( u k ) is bounded and φ ( u k )0 as k. Hence, there exists a constant C 1 >0 such that

| φ ( u k ) | C 1 , φ ( u k ) E μ C 1 .
(3.1)

From (2.1), (2.2), (3.1), (W2), and (W3), we have

p C 1 + p C 1 u k p φ ( u k ) p μ φ ( u k ) , u k = μ p μ Δ u k l p p + p n Z [ W 2 ( n , u k ( n ) ) 1 μ ( W 2 ( n , u k ( n ) ) , u k ( n ) ) ] p n Z [ W 1 ( n , u k ( n ) ) 1 μ ( W 1 ( n , u k ( n ) ) , u k ( n ) ) ] + ( p q p + 2 p μ ) n Z a ( n ) | u k ( n ) | q p + 2 μ p μ Δ u k l p p + ( p q p + 2 p μ ) u k q p + 2 , a q p + 2 .
(3.2)

It follows from Lemma 2.2, p<(q+2)/2, μ>qp+2, and (3.2) that there exists a constant C 2 >0 such that

u k C 2 ,kN.
(3.3)

Now we prove that u k u in E. Passing to a subsequence if necessary, it can be assumed that u k u in E. For any given ε>0, by (W1), we can choose δ(0,1) such that

| W ( n , x ) | εa(n) | x | q p + 1 for nZJ and |x|δ.
(3.4)

Since uE, we can also choose a positive integer K>max{|k|:kJ} such that

| u ( n ) | δfor |n|K.

Hence,

| W ( n , u ( n ) ) | εa(n) | u ( n ) | q p + 1 for nZJ and  | u ( n ) | δ.
(3.5)

Furthermore,

| W ( n , u k ( n ) ) | εa(n) | u k ( n ) | q p + 1 for nZJ and  | u k ( n ) | δ.
(3.6)

Hence, from (3.5) and (3.6), we have

| W ( n , u k ( n ) ) W ( n , u ( n ) ) | p [ ε a ( n ) ( | u k ( n ) | q p + 1 + | u ( n ) | q p + 1 ) ] p [ ε 2 q p + 1 a ( n ) | u k ( n ) u ( n ) | q p + 1 + ε ( 1 + 2 q p + 1 ) a ( n ) | u ( n ) | q p + 1 ] p 2 p ( q p + 2 ) ε p [ a ( n ) ] p | u k ( n ) u ( n ) | p ( q p + 1 ) + 2 p ε p ( 1 + 2 q p + 1 ) p [ a ( n ) ] p | u ( n ) | p ( q p + 1 ) : = g k ( n ) ,
(3.7)

where p =p/(p1). Moreover, since a(n) is a positive function on , p<qp+2, and u k (n)u(n) for almost every nZ, we have

lim k g k (n)= 2 p ε p ( 1 + 2 q p + 1 ) p [ a ( n ) ] p | u ( n ) | p ( q p + 1 ) :=g(n),for a.e. nZ,
(3.8)

and

lim k n Z g k ( n ) = lim k n Z [ 2 p ( q p + 2 ) ε p [ a ( n ) ] p | u k ( n ) u ( n ) | p ( q p + 1 ) + 2 p ε p ( 1 + 2 q p + 1 ) p [ a ( n ) ] p | u ( n ) | p ( q p + 1 ) ] = 2 p ( q p + 2 ) ε p lim k n Z [ a ( n ) ] p | u k ( n ) u ( n ) | p ( q p + 1 ) + 2 p ε p ( 1 + 2 q p + 1 ) p n Z [ a ( n ) ] p | u ( n ) | p ( q p + 1 ) = 2 p ε p ( 1 + 2 q p + 1 ) p n Z [ a ( n ) ] p | u ( n ) | p ( q p + 1 ) = n Z g ( n ) < + .
(3.9)

It follows from (3.7), (3.8), (3.9), and the Lebesgue dominated convergence theorem that

lim k n Z | W ( n , u k ( n ) ) W ( n , u ( n ) ) | p =0.

This shows that

W(n, u k )W(n,u)in  l p ( Z , R N ) .
(3.10)

From (2.2), we have

φ ( u k ) φ ( u ) , u k u ) = n Z ( | Δ u k ( n 1 ) | p 2 Δ u k ( n 1 ) | Δ u ( n 1 ) | p 2 Δ u ( n 1 ) , Δ u k ( n 1 ) Δ u ( n 1 ) ) + n Z a ( n ) ( | u k ( n ) | q p u k ( n ) | u ( n ) | q p u ( n ) ) ( u k ( n ) u ( n ) ) n Z ( W ( n , u k ( n ) ) W ( n , u ( n ) ) , u k ( n ) u ( n ) ) Δ u k l p p + Δ u l p p Δ u l p Δ u k l p p 1 Δ u k l p Δ u l p p 1 + n Z a ( n ) ( | u k ( n ) | q p u k ( n ) | u ( n ) | q p u ( n ) ) ( u k ( n ) u ( n ) ) n Z ( W ( n , u k ( n ) ) W ( n , u ( n ) ) , u k ( n ) u ( n ) ) = ( Δ u k l p p 1 Δ u l p p 1 ) ( Δ u k l p Δ u l p ) + n Z a ( n ) ( | u k ( n ) | q p u k ( n ) | u ( n ) | q p u ( n ) ) ( u k ( n ) u ( n ) ) n Z ( W ( n , u k ( n ) ) W ( n , u ( n ) ) , u k ( n ) u ( n ) ) .
(3.11)

It is easy to see that for any α>1 there exists a constant C 3 >0 such that

( | x | α 1 x | y | α 1 y ) (xy) C 3 | x y | α + 1 ,x,yR.
(3.12)

Hence, we have

( Δ u k l p p 1 Δ u l p p 1 ) ( Δ u k l p Δ u l p ) C 4 | Δ u k l p Δ u l p | p
(3.13)

and

n Z a ( n ) ( | u k ( n ) | q p u k ( n ) | u ( n ) | q p u ( n ) ) ( u k ( n ) u ( n ) ) C 5 n Z a ( n ) | u k ( n ) u ( n ) | q p + 2 .
(3.14)

Since φ ( u k )0 as k+, u k u in E and the embeddings EW l γ for all γp are continuous, it follows from Lemma 2.2, (3.10), (3.11), (3.13), and (3.14) that

Δ u k l p Δ u l p as k
(3.15)

and

n Z a(n) | u k ( n ) | q p + 2 n Z a(n) | u ( n ) | q p + 2 as k.
(3.16)

Hence, we have u k u in E by (3.15) and (3.16). This shows that φ satisfies the (PS)-condition.

Secondly, we prove that there exist ρ,α>0 such that φ B ρ ( 0 ) α. From (W1), there exists δ 1 (0,1) such that

| W ( n , x ) | 1 p a(n) | x | q p + 1 for |n|ZJ and |x| δ 1 .
(3.17)

From (3.17), we have

| W ( n , x ) | 1 p ( q p + 2 ) a(n) | x | q p + 2 for |n|ZJ and |x| δ 1 .
(3.18)

Let

C 6 =sup { W 1 ( n , x ) a ( n ) | n J , x R N , | x | = 1 } .
(3.19)

Set σ=min{1/ ( p ( q p + 2 ) C 6 + 1 ) 1 / ( μ q + p 2 ) , δ 1 } and u=σ/2:=ρ, it follows from (2.3) that

u 2uσ,

which shows that |u(n)|σ δ 1 <1. From Lemma 2.5(i) and (3.19), we have

n J W 1 ( n , u ( n ) ) { n J : u ( n ) 0 } W 1 ( n , u ( n ) | u ( n ) | ) | u ( n ) | μ C 6 n J a ( n ) | u ( n ) | μ C 6 σ μ q + p 2 n J a ( n ) | u ( n ) | q p + 2 1 p ( q p + 2 ) n J a ( n ) | u ( n ) | q p + 2 .
(3.20)

It follows from (W3), (3.18), and (3.20) that

φ ( u ) = 1 p n Z | Δ u ( n 1 ) | p + n Z a ( n ) q p + 2 | u ( n ) | q p + 2 n Z W ( n , u ( n ) ) = 1 p Δ u l p p + 1 q p + 2 u q p + 2 , a q p + 2 Z J W ( n , u ( n ) ) n J W ( n , u ( n ) ) 1 p Δ u l p p + 1 q p + 2 u q p + 2 , a q p + 2 n J W 1 ( n , u ( n ) ) Z J 1 p ( q p + 2 ) a ( n ) | u ( n ) | q p + 2 1 p Δ u l p l p + 1 q p + 2 u q p + 2 , a q p + 2 1 p ( q p + 2 ) n J a ( n ) | u ( n ) | q p + 2 Z J 1 p ( q p + 2 ) a ( n ) | u ( n ) | q p + 2 = 1 p Δ u l p p + p 1 p ( q p + 2 ) u q p + 2 , a q p + 2 .

Therefore, we can choose a constant α>0 depending on ρ such that φ(u)α for any uE with u=ρ.

Thirdly, we prove that assumption (ii) of Lemma 2.3 holds. From Lemma 2.5(ii) and (2.3), we have for any uE

n [ 3 , 3 ] W 2 ( n , u ( n ) ) = { n [ 3 , 3 ] : | u ( n ) | > 1 } W 2 ( n , u ( n ) ) + { n [ 3 , 3 ] : | u ( n ) | 1 } W 2 ( n , u ( n ) ) { n [ 3 , 3 ] : | u ( n ) | > 1 } W 2 ( n , u ( n ) | u ( n ) | ) | u ( n ) | ϱ + n [ 3 , 3 ] max | x | 1 W 2 ( n , x ) u ϱ n [ 3 , 3 ] max | x | = 1 W 2 ( n , x ) + n [ 3 , 3 ] max | x | 1 W 2 ( n , x ) 2 ϱ u ϱ n [ 3 , 3 ] max | x | = 1 W 2 ( n , x ) + n [ 3 , 3 ] max | x | 1 W 2 ( n , x ) = C 7 u ϱ + C 8 ,
(3.21)

where C 7 = 2 ϱ n [ 3 , 3 ] max | x | = 1 W 2 (n,x), C 8 = n [ 3 , 3 ] max | x | 1 W 2 (n,x). Take ωE such that

| ω ( n ) | = { 1 , for  | n | 1 , 0 , for  | n | 3 ,
(3.22)

and |ω(n)|1 for |n|(1,3]. For s>1, from Lemma 2.5(i) and (3.22), we get

n [ 1 , 1 ] W 1 ( n , s ω ( n ) ) s μ n [ 1 , 1 ] W 1 ( n , ω ( n ) ) = C 9 s μ ,
(3.23)

where C 9 = n [ 1 , 1 ] W 1 (n,ω(n))>0. From (W3), (2.1), (3.21), (3.22), (3.23), we have for s>1

φ ( s ω ) = s p p Δ ω l p p + s q p + 2 q p + 2 ω q p + 2 , a q p + 2 + n Z [ W 2 ( n , s ω ( n ) ) W 1 ( n , s ω ( n ) ) ] s p p Δ ω l p p + s q p + 2 q p + 2 ω q p + 2 , a q p + 2 + n [ 3 , 3 ] W 2 ( n , s ω ( n ) ) n [ 1 , 1 ] W 1 ( n , s ω ( n ) ) s p p Δ ω l p p + s q p + 2 q p + 2 ω q p + 2 , a q p + 2 + C 7 s ϱ ω ϱ + C 8 C 9 s μ .
(3.24)

Since μ>ϱ>qp+2 and C 9 >0, it follows from (3.24) that there exists s 1 >1 such that s 1 ω>ρ and φ( s 1 ω)<0. Let e= s 1 ω(n), then eE, e= s 1 ω>ρ, and φ(e)=φ( s 1 ω)<0.

By Lemma 2.3, φ has a critical value c>α given by

c= inf g Φ max s [ 0 , 1 ] φ ( g ( s ) ) ,
(3.25)

where

Φ= { g C ( [ 0 , 1 ] , E ) : g ( 0 ) = 0 , g ( 1 ) = e } .

Hence, there exists u E such that

φ ( u ) =c, φ ( u ) =0.

The function u is a desired solution of problem (1.1). Since c>0, u is a nontrivial homoclinic solution. The proof is complete. □

Proof of Theorem 1.2 In the proof of Theorem 1.1, the condition W 2 (t,x)0 in (W3) is only used in the proofs of (3.3) and assumption (i) of Lemma 2.3. Therefore, we only need to prove that (3.3) and assumption (i) of Lemma 2.3 still hold if we use (W1)′ and (W3)′ instead of (W1) and (W3), respectively. We first prove that (3.3) holds. From (W2), (W3)′, (2.1), (2.2), and (3.1), we have

p ( q p + 2 ) C 1 + p ( q p + 2 ) C 1 μ ϱ u k p ( q p + 2 ) φ ( u k ) p ( q p + 2 ) ϱ φ ( u k ) , u k = ( ϱ p ) ( q p + 2 ) ϱ Δ u k l p p + p ( q p + 2 ) n Z [ W 2 ( n , u k ( n ) ) 1 ϱ ( W 2 ( n , u k ( n ) ) , u k ( n ) ) ] p ( q p + 2 ) n Z [ W 1 ( n , u k ( n ) ) 1 ϱ ( W 1 ( n , u k ( n ) ) , u k ( n ) ) ] + p ( 1 q p + 2 ϱ ) n Z a ( n ) | u k ( n ) | q p + 2 ( ϱ p ) ( q p + 2 ) ϱ Δ u k l p p + p ( 1 q p + 2 ϱ ) u n q p + 2 , a q p + 2 ,

which implies that there exists a constant C 2 >0 such that (3.3) holds. Next, we prove that assumption (i) of Lemma 2.3 still holds. From (W1)′, there exists δ 2 (0,1) such that

| W ( n , x ) | 1 p a(n) | x | q p + 1 for nZ and |x| δ 2 .
(3.26)

By (3.26), we have

| W ( n , x ) | 1 p ( q p + 2 ) a(n) | x | q p + 2 for nZ and |x| δ 2 .
(3.27)

Let 0<σ δ 2 and u=σ/2:=ρ, it follows from (2.3) that

u 2uσ,

which shows that |u(n)|σ δ 2 <1. It follows from (2.1) and (3.27) that

φ ( u ) = 1 p n Z | Δ u ( n 1 ) | p + n Z a ( n ) q p + 2 | u ( n ) | q p + 2 n Z W ( n , u ( n ) ) 1 p Δ u l p p + 1 q p + 2 u q p + 2 , a q p + 2 n Z 1 p ( q p + 2 ) a ( n ) | u ( n ) | q p + 2 = 1 p Δ u l p p + p 1 p ( q p + 2 ) u q p + 2 , a q p + 2 .

Therefore, we can choose a constant α>0 depending on ρ such that φ(u)α for any uE with u=ρ. The proof of Theorem 1.2 is complete. □

Proof of Theorem 1.3 Condition (W4) shows that φ is even. In view of the proof of Theorem 1.1, we know that φ C 1 (E,R) and satisfies (PS)-condition and assumption (i) of Lemma 2.3. Now, we prove that assumption (iii) of Lemma 2.4 holds. Let E be a finite dimensional subspace of E. Since all norms of a finite dimensional space are equivalent, there exists C 10 >0 such that

u C 10 u .
(3.28)

Assume that dim E =m and { u 1 , u 2 ,, u m } is a base of E such that

u i = C 10 ,i=1,2,,m.
(3.29)

For any u E , there exists λ i R, i=1,2,,m such that

u(n)= i = 1 m λ i u i (n)for nZ.
(3.30)

Let

u = i = 1 m | λ i | u i .
(3.31)

It is easy to see that is a norm of E . Hence, there exists a constant C 11 >0 such that C 11 u u. Since u i E, by Lemma 2.2, we can choose K 1 >K such that

| u i ( n ) | < C 11 δ 1 1 + C 11 ,|n|> K 1 ,i=1,2,,m,
(3.32)

where δ 1 is given in (3.17). Let

Θ= { i = 1 m λ i u i ( n ) : λ i R , i = 1 , 2 , , m ; i = 1 m | λ i | = 1 } = { u E : u = C 10 } .
(3.33)

Hence, for uΘ, let n 0 = n 0 (u)Z such that

| u ( n 0 ) | = u .
(3.34)

It follows from (3.28)-(3.31), (3.33), and (3.34) that

C 10 C 11 = C 10 C 11 i = 1 m | λ i | = C 11 i = 1 m | λ i | u i = C 11 u u C 10 u = C 10 | u ( n 0 ) | C 10 i = 1 m | λ i | | u i ( n 0 ) | , u Θ .
(3.35)

This shows that |u( n 0 )| C 11 and there exists i 0 {1,2,,m} such that | u i 0 ( n 0 )| C 11 , which together with (3.32), implies that | n 0 | K 1 . Let

γ=min { W 1 ( n , x ) : K 1 n K 1 , | x | C 11 } .
(3.36)

Since W 1 (n,x)>0 for all nZ and x R N {0}, and W 1 (n,x) is continuous in x, it follows that γ>0. For any uE, from Lemma 2.5(ii) and (2.3), we have

n = K 1 K 1 W 2 ( n , u ( n ) ) = { n [ K 1 , K 1 ] : | u ( n ) | > 1 } W 2 ( n , u ( n ) ) + { n [ K 1 , K 1 ] : | u ( n ) | 1 } W 2 ( n , u ( n ) ) { n [ K 1 , K 1 ] : | u ( n ) | > 1 } W 2 ( n , u ( n ) | u ( n ) | ) | u ( n ) | ϱ + n = K 1 K 1 max | x | 1 W 2 ( n , x ) u ϱ n = K 1 K 1 max | x | = 1 W 2 ( n , x ) + n = K 1 K 1 max | x | 1 W 2 ( n , x ) 2 ϱ u ϱ n = K 1 K 1 max | x | = 1 W 2 ( n , x ) + n = K 1 K 1 max | x | 1 W 2 ( n , x ) = C 12 u ϱ + C 13 ,
(3.37)

where C 12 = 2 ϱ n = K 1 K 1 max | x | = 1 W 2 (n,x), C 13 = n = K 1 K 1 max | x | 1 W 2 (n,x). It follows from Lemma 2.5(i) and (3.36) that

n = K 1 K 1 W 1 ( n , u ( n ) ) W 1 ( n 0 , u ( n 0 ) ) W 1 ( n 0 , C 11 u ( n 0 ) | u ( n 0 ) | ) ( | u ( n 0 ) | C 11 ) μ min | x | 1 { W 1 ( n 0 , x ) } γ for  u Θ .
(3.38)

By (3.18), (3.37), (3.38), and Lemma 2.5, we have for uΘ and r>1

φ ( r u ) = r p p Δ u l p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + n Z [ W 2 ( n , r u ( n ) ) W 1 ( n , r u ( n ) ) ] r p p Δ u l p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + r ϱ n Z W 2 ( n , u ( n ) ) r μ n Z W 1 ( n , u ( n ) ) = r p p Δ u l p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + r ϱ | n | K 1 W 2 ( n , u ( n ) ) r μ | n | K 1 W 1 ( n , u ( n ) ) + r ϱ n = K 1 K 1 W 2 ( n , u ( n ) ) r μ n = K 1 K 1 W 1 ( n , u ( n ) ) r p p Δ u l p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 r ϱ | n | K 1 W ( n , u ( n ) ) r μ n = K 1 K 1 W 1 ( n , u ( n ) ) + r ϱ n = K 1 K 1 W 2 ( n , u ( n ) ) r p p Δ u l p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + r ϱ p ( q p + 2 ) | n | K 1 a ( n ) | u ( n ) | q p + 2 + r ϱ ( C 12 u ϱ + C 13 ) γ r μ r p p Δ u l p p + ( r q p + 2 q p + 2 + r ϱ p ( q p + 2 ) ) u q p + 2 , a q p + 2 + r ϱ ( C 12 u ϱ + C 13 ) γ r μ r p p C 10 p + ( r q p + 2 q p + 2 + r ϱ p ( q p + 2 ) ) C 10 q p + 2 + C 12 ( r C 10 ) ϱ + C 13 r ϱ γ r μ .
(3.39)

Since μ>ϱ>qp+2>p, we deduce that there exists r 0 = r 0 ( C 10 , C 11 , C 12 , C 13 ,K, K 1 ,ε,γ)= r 0 ( E )>1 such that

φ(ru)<0for uΘ and r r 0 .

It follows that

φ(u)<0for u E  and u C 10 r 0 ,

which shows that assumption (iii) of Lemma 2.4 holds. By Lemma 2.4, φ possesses an unbounded sequence { c k } k = 1 of critical values with c k =φ( u k ), where u k is such that φ ( u k )=0 for k=1,2, . If { u k } is bounded, then there exists C 14 >0 such that

u k C 14 for kN.
(3.40)

In a similar fashion to the proof of (3.5) and (3.6), for the given δ 1 in (3.18), there exists K 2 >max{|k|:kJ} such that

| u k ( n ) | δ 1 for |n| K 2  and kN.
(3.41)

Hence, by (2.1), (2.3), (3.18), (3.40), and (3.41), we have

1 p Δ u k l p p + 1 q p + 2 u k q p + 2 , a q p + 2 = c k + n Z W ( n , u k ( n ) ) = c k + | n | K 2 W ( n , u k ( n ) ) + n = K 2 K 2 W ( n , u k ( n ) ) c k 1 p ( q p + 2 ) | n | K 2 a ( n ) | u k ( n ) | q p + 2 n = K 2 K 2 | W ( n , u k ( n ) ) | c k 1 p ( q p + 2 ) u k q p + 2 , a q p + 2 n = K 2 K 2 max | x | 2 C 14 | W ( n , x ) | .

It follows that

c k 1 p Δ u k l p p + p + 1 q p + 2 u k q p + 2 , a q p + 2 + n = K 2 K 2 max | x | 2 C 14 | W ( n , x ) | <+.

This contradicts the fact that { c k } k = 1 is unbounded, and so { u k } is unbounded. The proof is complete. □

Proof of Theorem 1.4 In view of the proofs of Theorem 1.2 and Theorem 1.3, the conclusion of Theorem 1.4 holds. The proof is complete. □

4 Examples

Example 4.1 Consider the following system:

Δ ( | Δ u ( n 1 ) | 2 Δ u ( n 1 ) ) a(n) | u ( n ) | 3 u(n)+W ( n , u ( n ) ) =0,a.e. nZ,
(4.1)

where p=4, q=7, nZ, u R N , a:Z(0,), and a satisfies (A). Let

W(n,x)=a(n) ( i = 1 m 1 a i | x | μ i j = 1 m 2 b j | x | ϱ j ) ,

where μ 1 > μ 2 >> μ m 1 > ϱ 1 > ϱ 2 >> ϱ m 2 >5, a i , b j >0, i=1,, m 1 , j=1,, m 2 . Let

W 1 (n,x)=a(n) i = 1 m 1 a i | x | μ i , W 2 (n,x)=a(n) j = 1 m 2 b j | x | ϱ j .

Then it is easy to check that all the conditions of Theorem 1.3 are satisfied with μ= μ m 1 and ϱ= ϱ 1 . Hence, problem (4.1) has an unbounded sequence of homoclinic solutions.

Example 4.2 Consider the following system:

Δ ( | Δ u ( n 1 ) | 1 / 2 Δ u ( n 1 ) ) a(n) | u ( n ) | 3 / 2 u(n)+W ( n , u ( n ) ) =0,a.e. nZ,
(4.2)

where p=3/2, q=3, nZ, u R N , a:Z(0,) and a satisfies (A). Let

W(n,x)=a(n) [ a 1 | x | μ 1 + a 2 | x | μ 2 b 1 ( sin n ) | x | ϱ 1 b 2 | x | ϱ 2 ] ,

where μ 1 > μ 2 > ϱ 1 > ϱ 2 >7/2, a 1 , a 2 >0, b 1 , b 2 >0. Let

W 1 (n,x)=a(n) ( a 1 | x | μ 1 + a 2 | x | μ 2 ) , W 2 (n,x)=a(n) [ b 1 ( sin n ) | x | ϱ 1 + b 2 | x | ϱ 2 ] .

Then it is easy to check that all the conditions of Theorem 1.4 are satisfied with μ= μ 2 and ϱ= ϱ 1 . Hence, by Theorem 1.4, problem (4.2) has an unbounded sequence of homoclinic solutions.

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Acknowledgements

This work was supported by the NNSF of China (No. 11301108), Guangxi Natural Science Foundation (No. 2013GXNSFBA019004) and the Scientific Research Foundation of Guangxi Education Office (No. 201203YB093).

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Zhang, Q. Existence of homoclinic solutions for a class of difference systems involving p-Laplacian. Adv Differ Equ 2014, 291 (2014). https://doi.org/10.1186/1687-1847-2014-291

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Keywords

  • homoclinic solutions
  • variational methods
  • difference p-Laplacian systems