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Existence and multiplicity of homoclinic solutions for difference systems involving classical \((\phi_{1},\phi_{2})\)-Laplacian and a parameter

Abstract

In this paper, we investigate the existence and multiplicity of homoclinic solutions for a class of nonlinear difference systems involving classical \((\phi_{1},\phi _{2})\)-Laplacian and a parameter:

$$ \textstyle\begin{cases} \Delta (\rho_{1}(n-1)\phi_{1} (\Delta u_{1}(n-1) ) )-\rho _{3}(n)\phi_{3}(u_{1}(n)) \\ \quad {}+\lambda\nabla_{u_{1}} F (n,u_{1}(n),u_{2}(n) )=f_{1}(n), \\ \Delta (\rho_{2}(n-1)\phi_{2} (\Delta u_{2}(n-1) ) )-\rho _{4}(n)\phi_{4}(u_{2}(n)) \\ \quad {}+\lambda\nabla_{u_{2}} F (n,u_{1}(n),u_{2}(n) )=f_{2}(n). \end{cases} $$

When F is not periodic in n and has \((p,q)\)-sublinear growth or \((p,q)\)-linear growth, by using the least action principle, we obtain that a system with classical \((\phi _{1},\phi_{2})\)-Laplacian has at least one homoclinic solution and, by using Clark’s theorem, we see that a system with \(f_{1}=f_{2}\equiv 0\) has at least m distinct pairs of homoclinic solutions.

Introduction

Let \(\mathbb {R}\) denote the real numbers, \(\mathbb {Z}\) be the integers, and N be a fixed positive integer. \((\cdot,\cdot)\) stands for the usual product in \(\mathbb {R}^{N}\), \(|\cdot|\) is the induced norm, and \(\mathbb {Z}[1,N]=\{ 1,2,\ldots,N\}\). \((\cdot)^{\tau}\) stands for the transpose of a vector. In this paper, we investigate the existence and multiplicity of homoclinic solutions for the following nonlinear difference systems involving classical \((\phi_{1},\phi_{2})\)-Laplacian:

$$ \textstyle\begin{cases} \Delta (\rho_{1}(n-1)\phi_{1} (\Delta u_{1}(n-1) ) )-\rho _{3}(n)\phi_{3}(u_{1}(n)) \\ \quad {}+\lambda\nabla_{u_{1}} F (n,u_{1}(n),u_{2}(n) )=f_{1}(n), \\ \Delta (\rho_{2}(n-1)\phi_{2} (\Delta u_{2}(n-1) ) )-\rho _{4}(n)\phi_{4}(u_{2}(n)) \\ \quad {}+\lambda\nabla_{u_{2}} F (n,u_{1}(n),u_{2}(n) )=f_{2}(n), \end{cases} $$
(1.1)

where \(\lambda>0\), Δ is the forward difference operator, \(n\in \mathbb {Z}\), \(u_{m}(n)\in\mathbb {R}^{N}\), \(f_{m}: \mathbb {Z} \to\mathbb {R}^{N}\) with \(f_{m}=(f_{m1},\ldots,f_{mN})^{\tau}\), \(m=1,2\), and \(\rho_{i}: \mathbb {Z}\rightarrow\mathbb {R}^{+}\) and \(\phi_{i}\), \(i=1,2,3,4\) satisfy the following conditions:

(ρ):

\(0<\inf_{n\in\mathbb {Z}}\rho_{i}\le\sup_{n\in\mathbb {Z}}\rho _{i}<+\infty\), \(i=1,2,3,4\);

(\(\mathcal{A}_{0}\)):

\(\phi_{i}\) is a homeomorphism from \(\mathbb {R}^{N}\) onto \(\mathbb {R}^{N}\) such that \(\phi_{i}(0)=0\) and \(\phi_{i}=\nabla\Phi_{i}\), with \(\Phi_{i}\in C^{1}(\mathbb {R}^{N},[0,+\infty))\) strictly convex and \(\Phi_{i}(0)=0\), \(i=1,2,3,4\).

Remark 1.1

Assumption (\(\mathcal{A}_{0}\)) is given in [1], which is used to characterize the classical homeomorphism. If, furthermore, \(\Phi_{i}:\mathbb {R}^{N}\rightarrow\mathbb {R}\) is coercive (i.e., \(\Phi_{i}(x)\rightarrow+\infty\) as \(|x|\rightarrow \infty\)), then there exists \(\delta_{i}>0\) such that

$$ \Phi_{i}(x)\geq\delta_{i}\bigl( \vert x \vert -1\bigr),\quad x\in\mathbb {R}^{N}, $$
(1.2)

where \(\delta_{i}=\min_{|x|=1}\Phi_{m}(x)\), \(i=1,2,3,4\) (see [1]).

As usual, we say that a solution \(u(n)=(u_{1}(n),u_{2}(n))\) of system (1.1) is homoclinic (to 0) if \(u(n)\rightarrow0\) as \(n\rightarrow\pm \infty\). In addition, if \(u(n)\not\equiv0\), then \(u(n)\) is called a nontrivial homoclinic solution.

It is well known that the existence and multiplicity of homoclinic orbits for difference systems have been extensively studied in many recent papers via critical point theory (for example, see [212]). In [5], by using a linking theorem from [13], the author obtained that a second-order self-adjoint discrete Hamiltonian system has infinitely many nontrivial homoclinic solutions, when potential function W is indefinite sign and subquadratic. In [6], by using a variant of the mountain pass theorem from [14], the authors obtained that a class of p-Laplacian difference systems has at least one nontrivial homoclinic solution when the potential function possesses asymptotically p-linear properties at infinity. In [7], Tang and Lin investigated the following second-order self-adjoint discrete difference system:

$$ \Delta\bigl[p(n)\Delta u(n-1)\bigr]-L(n)u(n)+\nabla W\bigl(n,u(n) \bigr)=0, $$
(1.3)

where \(p(n)\) and \(L(n)\) are \(N\times N\) real symmetric positive definite matrices for all \(n\in\mathbb {Z}\). By using the least action principle, they obtained that system (1.3) has at least one homoclinic solution and, by using the Clark theorem, they obtained that system (1.3) has infinitely many homoclinic solutions. To be precise, they obtained the following theorems.

Theorem A

Assume that \(p(n)\) is an \(N\times N\) real symmetric positive definite matrix for all \(n\in\mathbb {Z}\). Assume L and W satisfy the following conditions:

(L):

\(L(n)\) is an \(N\times N\) real symmetric positive definite matrix for all \(n\in\mathbb {Z}\) and there exists a constant \(\beta>0\) such that

$$\bigl(L(n)x,x\bigr)\ge\beta|x|^{2}, \quad \forall(n,x)\in\mathbb {Z} \times\mathbb {R}^{N}. $$
(W1):

For every \(n\in\mathbb {Z}\), W is continuously differentiable in x and there exist two constants \(1<\gamma_{1}<\gamma_{2}<2\) and two functions \(a_{1},a_{2}\in l^{2/(2-\gamma_{1})}(\mathbb {Z}, [0,+\infty))\) such that

$$\bigl\vert W(n,x) \bigr\vert \le a_{1}(n)|x|^{\gamma_{1}}, \quad \forall(n,x)\in\mathbb {Z}\times \mathbb {R}^{N}, |x|\le1 $$

and

$$\bigl\vert W(n,x) \bigr\vert \le a_{2}(n)|x|^{\gamma_{2}},\quad \forall(n,x)\in\mathbb {Z}\times \mathbb {R}^{N}, |x|\ge1. $$
(W2):

There exist two functions \(b\in l^{2/(2-\gamma_{1})}\) and \(\varphi \in C([0,+\infty),[0,+\infty)) \) such that

$$\bigl\vert \nabla W(n,x) \bigr\vert \le b(n) \varphi\bigl( \vert x \vert \bigr), \quad \forall(n,x)\in\mathbb {Z}\times\mathbb {R}^{N}, $$

where \(\varphi(s)=O(s^{\gamma_{1}-1})\) as \(s \to0^{+}\).

(W3):

There exist \(n_{0}\in\mathbb {Z}\) and two constants \(\eta>0\) and \(\gamma_{3}\in(1,2)\) such that

$$W(n_{0},x)\ge\eta|x|^{\gamma_{3}}, \quad \forall x\in\mathbb {R}^{N}, |x|\le1. $$
Then system (1.3) possesses at least one non-trivial homoclinic solution.

Theorem B

Assume that \(p(n)\) is an \(N\times N\) real symmetric positive definite matrix for all \(n\in\mathbb {Z}\). Assume L and W satisfy (L), (W1), (W2), and the following conditions:

(W3)′:

There exist two constants \(\eta>0\) and \(\gamma_{3}\in(1,2)\) and a set \(J\subset\mathbb {Z}\) with \(m>0\) elements such that

$$W(n,x)\ge\eta|x|^{\gamma_{3}},\quad \forall(n,x)\in J\times\mathbb {R}^{N}, |x|\le1. $$
(W4):

\(W(n,-x)=W(n,x)\), \(\forall(n,x)\in\mathbb {Z}\times\mathbb {R}^{N}\).

Then system (1.3) possesses at least m distinct pairs of non-trivial homoclinic solutions.

Recently, in [1] and [15], Mawhin investigated the following second-order nonlinear difference systems with ϕ-Laplacian:

$$ \Delta\phi\bigl(\Delta u(n-1)\bigr)=\nabla_{u}F \bigl(n,u(n)\bigr)+h(n)\quad (n\in\mathbb {Z}), $$
(1.4)

where ϕ is a homeomorphism from \(X\subset\mathbb {R}^{N}\) onto \(Y\subset\mathbb {R}^{N}\), with three possible cases:

  1. (1)

    classical homeomorphism if \(X=Y=\mathbb {R}^{N}\);

  2. (2)

    bounded homeomorphism if \(X=\mathbb {R}^{N}\), \(Y=B_{a}\) (\(a<+\infty\));

  3. (3)

    singular homeomorphism if \(X=B_{a}\), \(Y=\mathbb {R}^{N}\),

where \(B_{a}\) is a ball with its center at origin and radius a. Inspired by [1, 15], and [10], Zhang and Wang in [8] studied the existence of homoclinic solutions for the following nonlinear difference systems with classical \((\phi_{1},\phi_{2})\)-Laplacian:

$$ \textstyle\begin{cases} \Delta\phi_{1} (\Delta u_{1}(n-1) )+\nabla_{u_{1}} V (n,u_{1}(n),u_{2}(n) )=f_{1}(n), \\ \Delta\phi_{2} (\Delta u_{2}(n-1) )+\nabla_{u_{2}} V (n,u_{1}(n),u_{2}(n) )=f_{2}(n), \end{cases} $$
(1.5)

where \(n\in\mathbb {Z}\), \(u_{m}(n)\in\mathbb {R}^{N}\), \(m=1,2\), and \(\phi_{m}\), \(m=1,2\) satisfy assumption (\(\mathcal{A}_{0}\)) and \(V(n,x_{1},x_{2})=-K(n,x_{1},x_{2})+W(n,x_{1},x_{2})\), where \(K,W:\mathbb {Z}\times\mathbb {R}^{N}\times\mathbb {R}^{N}\rightarrow \mathbb {R}\), \(K(n,x_{1},x_{2})\) and \(W(n,x_{1},x_{2})\) are T-periodic in n, K has p-sublinear growth, W has p-superlinear growth, and \(f_{m}:\mathbb {Z}\rightarrow\mathbb {R}^{N}\), \(m=1,2\) satisfy some reasonable growth conditions. By using a linking theorem due to [16], they obtained some existence results of homoclinic solutions for system (1.5).

In this paper, motivated by [1, 68, 15], the purpose is to obtain some results like Theorem A and Theorem B for system (1.1). To be precise, by using the least action principle and Clark’s theorem, we obtain some existence and multiplicity results of homoclinic solutions for system (1.1) when \(F(n,x_{1},x_{2})\) is not periodic in n and possesses \((p,q)\)-sublinear growth or \((p,q)\)-linear growth. Our results are different from those in [8]. Moreover, since system (1.1) has a parameter λ and perturbation terms \(f_{m}\) (\(m=1,2\)), some new cases cannot be covered by [7] even if system (1.1) reduces to the second-order difference system. For example, by virtue of perturbation terms \(f_{m}\) (\(m=1,2\)), (I) \(F(n_{0},x_{1},x_{2})\) can be negative in a small interval of \((|x_{1}|,|x_{2}|)\), which is impossible in (W3) (see Theorem 1.1 below), (II) the restriction of \(f_{m}\) (\(m=1,2\)) only aims at two components of \(f_{m}\) (\(m=1,2\)), that is, \(f_{1i_{0}}\) and \(f_{2j_{0}}\), which gives the idea that the other components of \(f_{m}\) (\(m=1,2\)) can be arbitrary even if \(f_{1i_{0}}+f_{2j_{0}}=0\), which is also impossible according to Theorem A (see Theorem 1.2 below), and (III) we consider the case in which F has \((p,q)\)-linear growth, which was not considered in [7] (see Theorem 1.3 below).

Let

$$\underline{\rho_{i}}=\inf_{n\in Z} \rho_{i}(n),\qquad \overline{\rho_{i}}=\sup _{n\in Z}\rho_{i}(n), \quad i=1,2,3,4. $$

Next, we present our main results.

Theorem 1.1

Suppose that (ρ), (\(\mathcal{A}_{0}\)), and the following conditions hold:

(\(\mathcal{A}_{1}\)):

There exist positive constants \(b_{i}\), \(d_{i}\), \(i=1,3\), \(b_{j}\), \(d_{j}\), \(j=2,4\), and \(p>1\), \(q>1\) such that

$$\begin{aligned}& b_{i}|x|^{p}\leq\Phi_{i}(x)\leq d_{i}|x|^{p},\quad i=1,3, \\& b_{j}|y|^{q} \leq\Phi_{j}(y)\leq d_{j}|y|^{q},\quad j=2,4, \forall x,y\in \mathbb {R}^{N}. \end{aligned}$$
(\(\mathcal{A}_{2}\)):

There exist positive constants \(k_{m}\), \(m=1,2\), \(c_{i}\), \(i=1,3\), \(c_{j}\), \(j=2,4\) such that

$$\bigl\vert \phi_{i}(x) \bigr\vert \le k_{m}|x|^{p-1}, \quad m=1,2 $$

and

$$\begin{aligned}& \bigl(\phi_{i}(x)-\phi_{i}(y),x-y\bigr)\geq c_{i}|x-y|^{p},\quad i=1,3, \forall x,y\in \mathbb {R}^{N}, \textit{if } p>2, \\& \bigl(\phi_{j}(x)-\phi_{j}(y),x-y\bigr)\geq c_{j}|x-y|^{q},\quad j=2,4, \forall x,y\in \mathbb {R}^{N}, \textit{if } q>2, \\& \bigl(\phi_{i}(x)-\phi_{i}(y),x-y\bigr)\geq c_{i}|x-y|^{2} \bigl( \vert x \vert + \vert y \vert \bigr)^{p-2}, \quad i=1,3, \forall x,y\in\mathbb {R}^{N}, \textit{if } 1< p \le2, \\& \bigl(\phi_{j}(x)-\phi_{j}(y),x-y\bigr)\geq c_{j}|x-y|^{2} \bigl( \vert x \vert + \vert y \vert \bigr)^{q-2},\quad j=2,4, \forall x,y\in\mathbb {R}^{N}, \textit{if } 1< q \le2. \end{aligned}$$
(\(F_{1}\)):

\(F(n,0,0)=0\) for all \(n\in\mathbb {Z}\) and there exist \(\gamma _{1}\in(1,p)\), \(\gamma_{2}\in(1,q)\), and functions \(a_{1}\in l^{p/(p-\gamma _{1})}(\mathbb {Z},[0,+\infty))\), \(a_{2}\in l^{q/(q-\gamma_{2})}(\mathbb {Z},[0,+\infty))\), \(b_{1}\in l^{\frac{p}{p-1}}(\mathbb {Z},[0,+\infty))\), and \(b_{2}\in l^{\frac{q}{q-1}}(\mathbb {Z},[0,+\infty))\) such that

$$\begin{aligned}& \bigl\vert \nabla_{x_{1}}F(n,x_{1},x_{2}) \bigr\vert \leq a_{1}(n)|x_{1}|^{\gamma_{1}-1}+b_{1}(n), \\& \bigl\vert \nabla_{x_{2}}F(n,x_{1},x_{2}) \bigr\vert \leq a_{2}(n)|x_{2}|^{\gamma_{2}-1}+b_{2}(n), \end{aligned}$$

for all \((n,x_{1},x_{2})\in\mathbb {Z}\times\mathbb {R}^{N}\times\mathbb {R}^{N}\).

(\(F_{2}\)):

There exist \(n_{0}\in\mathbb {Z}\) and constants \(\eta_{j}>0\), \(j=1,2\), \(\delta_{0}\in(0,1)\), and \(\gamma_{3}, \gamma_{4}\in(1,+\infty)\) such that

$$F(n_{0},x_{1},x_{2})\geq-\eta_{1}|x_{1}|^{\gamma_{3}}- \eta_{2}|x_{2}|^{\gamma_{4}},\quad \forall(x_{1},x_{2}) \in\mathbb {R}^{N}\times\mathbb {R}^{N}, |x_{1}|\leq \delta_{0}, |x_{2}|\leq\delta_{0}. $$
(f):

\(f_{1}\in l^{\frac{p}{p-1}}(\mathbb {Z},\mathbb {R}^{N})\), \(f_{2}\in l^{\frac{q}{q-1}}(\mathbb {Z},\mathbb {R}^{N})\), and there exist \(i_{0},j_{0}\in \mathbb {Z}[1,N]\) such that

$$f_{1i_{0}}(n_{0})+f_{2j_{0}}(n_{0})< 0. $$
Then system (1.1) with \(\lambda>0\) possesses at least one nontrivial homoclinic solution.

Remark 1.2

There exist examples satisfying (ρ). For example, let \(\rho_{i}(n)=\frac{1}{n^{2}+1}+1\), \(i=1,2,3,4\). Then \(\overline {\rho_{i}}=2\) and \(\underline{\rho_{i}}=1\), \(i=1,2,3,4\). Moreover, there exist examples satisfying (\(\mathcal{A}_{0}\)), (\(\mathcal{A}_{1}\)), and (\(\mathcal{A}_{2}\)). For example, as in [8]:

  1. (I)

    Assume \(N=1\). Let \(p=3\), \(q=4\),

    $$ \phi_{1}(x_{1})=\phi_{3}(x_{1})= \textstyle\begin{cases} 3|x_{1}|^{2}, & x_{1}>0, \\ 6|x_{1}|^{2}, & x_{1}\le0, \end{cases} $$

    and

    $$ \phi_{2}(x_{2})=\phi_{4}(x_{2})= \textstyle\begin{cases} 4|x_{1}|^{2}, & x_{2}>0, \\ 8|x_{1}|^{2}, & x_{2}\le0. \end{cases} $$
  2. (II)

    Assume \(N\ge1\). Let

    $$\phi_{1}(x_{1})=\phi_{3}(x_{1})=3a_{0}|x_{1}|^{2}, \qquad \phi_{2}(x_{2})=\phi_{4}(x_{2})=4b_{0}|x_{2}|^{3}, $$

    for some \(a_{0}, b_{0}>0\).

Remark 1.3

There exist examples satisfying Theorem 1.1. For example, we take \(N>1\), p, q, \(\rho_{i}\), and \(\phi_{i}\), \(i=1,2,3,4\) as in Remark 1.2. Let

$$\begin{aligned} F(n,x_{1},x_{2}) =&\frac{1}{n^{2}+1}\bigl(|x_{1}|^{\frac{5}{2}}+|x_{2}|^{\frac {7}{2}}+|x_{1}|^{\frac{1}{2}} \ln\bigl(1+|x_{1}|^{2}\bigr) \\ &{}+|x_{2}|^{\frac{5}{2}}\ln\bigl(1+|x_{2}|^{2} \bigr)-\ln\bigl(1+|x_{1}|^{\frac {5}{2}}\bigr)-\ln\bigl(1+|x_{2}|^{\frac{7}{2}} \bigr)\bigr). \end{aligned}$$

Take \(\gamma_{1}=\frac{5}{2}\), \(\gamma_{2}=\frac{7}{2}\), \(a_{1}(n)=a_{2}(n)=\frac{4}{n^{2}+1}\), \(b_{1}(n)=b_{2}(n)=0\), \(\eta_{1}=\eta_{2}=1\), and \(n_{0}=1\). Then it is easy to verify that F satisfies (\(F_{1}\)) and (\(F_{2}\)). Let

$$f_{1}(n)= \biggl(\frac{1}{n^{2}+2},\frac{1}{n^{2}+1},\ldots, \frac {1}{n^{2}+1} \biggr)^{\tau}, \qquad f_{2}(n)= \frac{1}{n^{2}+1}(-1,\ldots,1)^{\tau}. $$

Take \(i_{0}=j_{0}=1\). Then it is easy to see that (f) holds.

Theorem 1.2

Suppose that (ρ), (\(\mathcal{A}_{0}\)), (\(\mathcal{A}_{1}\)), (\(\mathcal{A}_{2}\)), (\(F_{1}\)), and the following conditions hold:

\((F_{2})'\) :

there exist \(n_{0}\in\mathbb {Z}\) and constants \(\eta _{j}>0\), \(j=1,2\), \(\delta_{0}\in(0,1)\), \(\gamma_{3}\in(1,p)\), and \(\gamma_{4}\in (1,q)\) such that

$$F(n_{0},x_{1},x_{2})\geq\eta_{1}|x_{1}|^{\gamma_{3}}+ \eta_{2}|x_{2}|^{\gamma_{4}},\quad \forall(x_{1},x_{2}) \in\mathbb {R}^{N}\times\mathbb {R}^{N}, |x_{1}|\leq \delta_{0}, |x_{2}|\leq\delta_{0}; $$
\((f)'\) :

\(f_{1}\in l^{\frac{p}{p-1}}(\mathbb {Z},\mathbb {R}^{N})\), \(f_{2}\in l^{\frac{q}{q-1}}(\mathbb {Z},\mathbb {R}^{N})\), and there exist \(i_{0},j_{0}\in \mathbb {Z}[1,N]\) such that

$$f_{1i_{0}}(n_{0})+f_{2j_{0}}(n_{0})= 0. $$
Then system (1.1) with \(\lambda>0\) possesses at least one nontrivial homoclinic solution.

Remark 1.4

There exist examples satisfying Theorem 1.2. For example, we take \(N>1\), p, q, \(\rho_{i}\), and \(\phi_{i}\), \(i=1,2,3,4\) as in Remark 1.2. Let

$$F(n,x_{1},x_{2})=\frac{1}{n^{2}+1} \bigl(|x_{1}|^{\frac{5}{2}}+|x_{2}|^{\frac {7}{2}}+ \ln\bigl(1+|x_{1}|^{\frac{5}{2}}\bigr)+\ln\bigl(1+|x_{2}|^{\frac{7}{2}} \bigr) \bigr). $$

Take \(\gamma_{1}=\gamma_{3}=\frac{5}{2}\), \(\gamma_{2}=\gamma_{4}=\frac{7}{2}\), \(a_{1}(n)=a_{2}(n)=\frac{4}{n^{2}+1}\), \(b_{1}(n)=b_{2}(n)=0\), \(\eta_{1}=\eta_{2}=1\), and \(n_{0}=1\). Then it is easy to verify that F satisfies (\(F_{1}\)) and \((F_{2})'\). Let

$$f_{1}(n)=\frac{1}{n^{2}+1} (1,\ldots, 1 )^{\tau},\qquad f_{2}(n)=\frac {1}{n^{2}+1}(-1,\ldots,1)^{\tau}. $$

Take \(i_{0}=j_{0}=1\). Then it is easy to see that \((f)'\) holds.

Theorem 1.3

Suppose that (ρ), (\(\mathcal{A}_{0}\)), (\(\mathcal{A}_{1}\)), (\(\mathcal{A}_{2}\)), (f), (\(F_{2}\)), and the following condition hold:

\((F_{1})'\) :

\(F(n,0,0)=0\) and there exist functions \(a_{1}, a_{2}\in l^{\infty }(\mathbb {Z},[0,+\infty))\) with \(a_{i}(n)\to0\) as \(n\to\infty\), \(i=1,2\), \(b_{1}\in l^{\frac{p}{p-1}}(\mathbb {Z},[0,+\infty))\), and \(b_{2}\in l^{\frac {q}{q-1}}(\mathbb {Z},[0,+\infty))\) such that

$$\begin{aligned}& \bigl\vert \nabla_{x_{1}}F(n,x_{1},x_{2}) \bigr\vert \leq a_{1}(n)|x_{1}|^{p-1}+b_{1}(n), \\& \bigl\vert \nabla_{x_{2}}F(n,x_{1},x_{2}) \bigr\vert \leq a_{2}(n)|x_{2}|^{q-1}+b_{2}(n), \end{aligned}$$

for all \((n,x_{1},x_{2})\in\mathbb {Z}\times\mathbb {R}^{N}\times \mathbb {R}^{N}\).

Then system (1.1) with \(\lambda\in (0,\min \{\frac{p\min \{\underline{\rho_{1}}b_{1},\underline{\rho_{3}}b_{3}\}}{\|a_{1}\|_{\infty}},\frac {q\min\{\underline{\rho_{2}}b_{2},\underline{\rho_{4}}b_{4}\}}{\|a_{2}\|_{\infty }} \} )\) possesses at least one nontrivial homoclinic solution.

Remark 1.5

There exist examples satisfying Theorem 1.3. For example, we take \(N>1\), p, q, \(\rho_{i}\), and \(\phi_{i}\), \(i=1,2,3,4\), as in Remark 1.2. Let

$$F(n,x_{1},x_{2})=\frac{1}{n^{2}+1} \bigl(|x_{1}|^{3}+|x_{2}|^{4}+|x_{2}|^{2} \ln \bigl(1+|x_{2}|^{2}\bigr)-\ln\bigl(1+|x_{1}|^{3} \bigr)-\ln\bigl(1+|x_{2}|^{4}\bigr) \bigr). $$

Take \(a_{1}(n)=a_{2}(n)=\frac{4}{n^{2}+1}\), \(b_{1}(n)=b_{2}(n)=0\), \(\eta_{1}=\eta _{2}=1\), \(\gamma_{3}=3\), \(\gamma_{4}=4\), and \(n_{0}=1\). Then it is easy to verify that F satisfies \((F_{1})'\) and (\(F_{2}\)). Let

$$f_{1}(n)= \biggl(\frac{1}{n^{2}+2},\frac{1}{n^{2}+1},\ldots, \frac {1}{n^{2}+1} \biggr)^{\tau}, \qquad f_{2}(n)= \frac{1}{n^{2}+1}(-1,\ldots,1)^{\tau}. $$

Take \(i_{0}=j_{0}=1\). Then it is easy to see that (f) holds.

Theorem 1.4

Suppose that (ρ), (\(\mathcal{A}_{0}\)), (\(\mathcal{A}_{1}\)), (\(\mathcal{A}_{2}\)), (\(F_{1}\)), and the following conditions hold:

\((F_{2})'''\) :

there exist constants \(\delta_{0}\in(0,1)\), \(\eta_{j}>0\), \(j=1,2\), \(\gamma_{3},\gamma_{4}\in(1,\min\{p,q\})\), and a set \(J\subset Z\) with \(m\in\mathbb {Z}[1,N]\) elements such that

$$F(n,x_{1},x_{2})\geq\eta_{1}|x_{1}|^{\gamma_{3}}+ \eta_{2}|x_{2}|^{\gamma_{4}},\quad \forall(n,x_{1},x_{2}) \in J\times\mathbb {R}^{N}\times\mathbb {R}^{N}, |x_{1}|\leq\delta_{0}, |x_{2}|\leq \delta_{0}; $$
(\(F_{3}\)):

\(F(n,-x_{1},-x_{2})=F(n,x_{1},x_{2})\), \(\forall(n,x_{1},x_{2})\in\mathbb {Z}\times\mathbb {R}^{N}\times\mathbb {R}^{N}\);

\((f)''\) :

\(f_{1}=f_{2}\equiv0\).

Then, for every \(\lambda>0\), system (1.1) possesses at least m distinct pairs of nontrivial homoclinic solutions.

Remark 1.6

There exist examples satisfying Theorem 1.4. For example, we take \(N>4\), p, q, \(\rho_{i}\), and \(\phi_{i}\), \(i=1,2,3,4\), as in Remark 1.2. Let

$$F(n,x_{1},x_{2})=\frac{1}{n^{2}+1} \bigl(|x_{1}|^{\frac{5}{2}}+|x_{2}|^{\frac {7}{2}}+ \ln\bigl(1+|x_{1}|^{\frac{5}{2}}\bigr)+\ln\bigl(1+|x_{2}|^{\frac{7}{2}} \bigr) \bigr). $$

Take \(\gamma_{1}=\gamma_{3}=\frac{5}{2}\), \(\gamma_{2}=\gamma_{4}=\frac{7}{2}\), \(a_{1}(n)=a_{2}(n)=\frac{4}{n^{2}+1}\), \(b_{1}(n)=b_{2}(n)=0\), \(\eta_{1}=\eta_{2}=\frac {1}{18}\), and \(J=\{1,2,3,4\}\). Then it is easy to verify that F satisfies (\(F_{1}\)) and \((F_{2})'''\). Hence, Theorem 1.4 implies that system (1.1) possesses at least four distinct pairs of nontrivial homoclinic solutions for every \(\lambda>0\).

Preliminaries

Define

$$ \begin{aligned} &S= \bigl\{ \bigl\{ u(n)\bigr\} _{n\in\mathbb {Z}}:u(n)\in\mathbb {R}^{N},n\in\mathbb {Z} \bigr\} , \\ &E_{\kappa}= \biggl\{ u\in S:\sum_{n\in\mathbb {Z}}\bigl[ \bigl\vert \Delta u(n) \bigr\vert ^{\kappa}+ \bigl\vert u(n) \bigr\vert ^{\kappa}\bigr]< +\infty \biggr\} , \end{aligned} $$
(2.1)

where \(1<\kappa<+\infty\) and for \(v\in E_{\kappa}\) we define

$$ \|v\|_{\kappa}= \biggl\{ \sum_{n\in\mathbb {Z}} \bigl[ \bigl\vert \Delta v(n) \bigr\vert ^{\kappa }+ \bigl\vert v(n) \bigr\vert ^{\kappa}\bigr] \biggr\} ^{1/\kappa}. $$
(2.2)

Let \(E=E_{p}\times E_{q}\). For \(u=(u_{1},u_{2})\in E\), we define

$$ \|u\|=\|u_{1}\|_{p}+\|u_{2} \|_{q}. $$
(2.3)

Then E is a uniformly convex Banach space with this norm. As in [7], for \(1<\kappa<+\infty\), set

$$ \begin{aligned} &l^{\kappa}:=l^{\kappa}\bigl(\mathbb {Z}, \mathbb {R}^{N}\bigr)= \biggl\{ u\in S: \sum_{n\in\mathbb {Z}} \bigl\vert u(n) \bigr\vert ^{\kappa}< +\infty \biggr\} , \\ &l^{\infty}:=l^{\infty}\bigl(\mathbb {Z},\mathbb {R}^{N} \bigr)= \Bigl\{ u\in S: \sup_{n\in\mathbb {Z}} \bigl\vert u(n) \bigr\vert < +\infty \Bigr\} , \end{aligned} $$
(2.4)

with the norms

$$ \begin{aligned} &\|u\|_{l^{\kappa}}= \biggl(\sum _{n\in\mathbb {Z}} \bigl\vert u(n) \bigr\vert ^{\kappa}\biggr)^{1/\kappa}, \quad \forall u\in l^{\kappa}\bigl(\mathbb {Z},\mathbb {R}^{N}\bigr), \\ &\|u\|_{\infty}=\sup\bigl\{ \bigl\vert u(n) \bigr\vert :n\in\mathbb {Z} \bigr\} , \quad \forall u\in l^{\infty}\bigl(\mathbb {Z},\mathbb {R}^{N}\bigr), \end{aligned} $$
(2.5)

respectively. For \(u\in E_{\kappa}\), it is easy to obtain

$$ \|u\|_{\infty}\leq\|u\|_{l^{\kappa}}\leq \|u \|_{\kappa}. $$
(2.6)

Lemma 2.1

Assume that (ρ), (\(\mathcal{A}_{0}\)), (\(\mathcal{A}_{1}\)), and (\(F_{1}\)) hold. Then, for all \(\lambda>0\), \(f_{1}\in l^{\frac{p}{p-1}}(\mathbb {Z},\mathbb {R}^{N})\), and \(f_{2}\in l^{\frac{q}{q-1}}(\mathbb {Z},\mathbb {R}^{N})\), the functional \(\mathcal{J}:E\rightarrow\mathbb {R}\) defined by

$$\begin{aligned} \mathcal{J}(u) =&\sum_{n\in\mathbb {Z}}\bigl[ \rho_{1}(n)\Phi_{1}\bigl(\Delta u_{1}(n)\bigr)+ \rho_{2}(n)\Phi_{2}\bigl(\Delta u_{2}(n)\bigr)+ \rho_{3}(n)\Phi _{3}\bigl(u_{1}(n)\bigr) \\ &{}+\rho_{4}(n)\Phi_{4}\bigl(u_{2}(n)\bigr)- \lambda F\bigl(n,u_{1}(n),u_{2}(n)\bigr) \\ &{}+\bigl(f_{1}(n),u_{1}(n)\bigr)+\bigl(f_{2}(n),u_{2}(n) \bigr)\bigr], \quad \forall u\in E, \end{aligned}$$
(2.7)

is well defined and of class \(C^{1}(E,\mathbb {R})\) and

$$\begin{aligned} \bigl\langle \mathcal{J}'(u),v \bigr\rangle =& \bigl\langle \mathcal {J}'(u_{1},u_{2}),(v_{1},v_{2}) \bigr\rangle \\ = &\sum_{n\in\mathbb {Z}} \bigl[\rho_{1}(n) \bigl( \phi_{1}\bigl(\Delta u_{1}(n)\bigr),\Delta v_{1}(n)\bigr) \\ &{} +\rho_{2}(n) \bigl(\phi_{2}\bigl(\Delta u_{2}(n)\bigr),\Delta v_{2}(n)\bigr) \\ &{} +\rho_{3}(n) \bigl(\phi_{3}\bigl(u_{1}(n) \bigr),v_{1}(n)\bigr)+\rho_{4}(n) \bigl(\phi _{4}\bigl(u_{2}(n)\bigr),v_{2}(n)\bigr) \\ & {} -\lambda\bigl(\nabla_{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr),v_{1}(n)\bigr) \\ & {} -\lambda\bigl(\nabla_{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr),v_{2}(n)\bigr) \\ & {} +\bigl(f_{1}(n),v_{1}(n)\bigr)+\bigl(f_{2}(n),v_{2}(n) \bigr) \bigr], \quad \forall u,v\in E. \end{aligned}$$
(2.8)

Furthermore, the critical points of \(\mathcal{J}\) in E are solutions of (1.1) with \(u(\pm\infty) =0\).

Proof

Firstly, we show that \(\mathcal{J}:E\rightarrow\mathbb {R}\) is well defined. In fact,

$$\begin{aligned} F(n,x_{1},x_{2}) = & \int_{0}^{1}\bigl(\nabla_{x_{1}}F(n,sx_{1},x_{2}),x_{1} \bigr)\,ds+F(n,0,x_{2}) \\ = & \int_{0}^{1}\bigl(\nabla_{x_{1}}F(n,sx_{1},x_{2}),x_{1} \bigr)\,ds \\ &{}+ \int_{0}^{1}\bigl(\nabla _{x_{2}}F(n,0,tx_{2}),x_{2} \bigr)\,dt \\ &{}+F(n,0,0). \end{aligned}$$
(2.9)

Then, by (\(F_{1}\)), we have

$$\begin{aligned} \begin{aligned}[b] \bigl\vert F(n,x_{1},x_{2}) \bigr\vert \leq{}& \int_{0}^{1} \bigl\vert \nabla_{x_{1}}F(n,sx_{1},x_{2}) \bigr\vert \vert x_{1} \vert \,ds+ \int _{0}^{1} \bigl\vert \nabla_{x_{2}}F(n,0,tx_{2}) \bigr\vert \vert x_{2} \vert \,dt \\ \leq{}& \int_{0}^{1}\bigl( \bigl\vert a_{1}(n) \bigr\vert \vert sx_{1} \vert ^{\gamma_{1}-1}+b_{1}(n) \bigr) \vert x_{1} \vert \,ds + \int_{0}^{1}\bigl( \bigl\vert a_{2}(n) \bigr\vert \vert tx_{2} \vert ^{\gamma_{2}-1} \\ &{}+b_{2}(n)\bigr) \vert x_{2} \vert \,dt \\ = {}&\frac{ \vert a_{1}(n) \vert }{\gamma_{1}} \vert x_{1} \vert ^{\gamma_{1}}+ \frac{ \vert a_{2}(n) \vert }{\gamma _{2}} \vert x_{2} \vert ^{\gamma_{2}}+b_{1}(n) \vert x_{1} \vert +b_{2}(n) \vert x_{2} \vert . \end{aligned} \end{aligned}$$
(2.10)

So, for \(u=(u_{1},u_{2})^{\tau}\in E\), by (2.10), the Hölder inequality, and (2.6), we have

$$\begin{aligned} \biggl\vert \sum_{n\in\mathbb {Z}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr) \biggr\vert \leq&\sum_{n\in\mathbb {Z}} \bigl\vert F \bigl(n,u_{1}(n),u_{2}(n)\bigr) \bigr\vert \\ \leq& \sum_{n\in\mathbb {Z}} \biggl(\frac{ \vert a_{1}(n) \vert }{\gamma _{1}} \bigl\vert u_{1}(n) \bigr\vert ^{\gamma_{1}} +\frac{ \vert a_{2}(n) \vert }{\gamma_{2}} \bigl\vert u_{2}(n) \bigr\vert ^{\gamma_{2}} \biggr) \\ &{} +\sum_{n\in\mathbb {Z}} \bigl( \bigl\vert b_{1}(n) \bigr\vert \bigl\vert u_{1}(n) \bigr\vert + \bigl\vert b_{2}(n) \bigr\vert \bigl\vert u_{2}(n) \bigr\vert \bigr) \\ \leq&\frac{1}{\gamma_{1}} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert a_{1}(n) \bigr\vert ^{\frac {p}{p-\gamma_{1}}} \biggr)^{\frac{p-\gamma_{1}}{p}} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert u_{1}(n) \bigr\vert ^{p} \biggr)^{\frac{\gamma _{1}}{p}} \\ &{} +\frac{1}{\gamma_{2}} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert a_{2}(n) \bigr\vert ^{\frac {q}{q-\gamma_{2}}} \biggr)^{\frac{q-\gamma_{2}}{q}} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert u_{2}(n) \bigr\vert ^{q} \biggr)^{\frac{\gamma _{2}}{q}} \\ &{}+ \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert b_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \biggr)^{\frac{p-1}{p}} \biggl( \sum_{n\in\mathbb {Z}} \bigl\vert u_{1}(n) \bigr\vert ^{p} \biggr)^{\frac{1}{p}} \\ &{}+ \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert b_{2}(n) \bigr\vert ^{\frac{q}{q-1}} \biggr)^{\frac{q-1}{q}} \biggl( \sum_{n\in\mathbb {Z}} \bigl\vert u_{2}(n) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\ = &\frac{1}{\gamma_{1}}\|a_{1}\|_{l^{p/(p-\gamma_{1})}}\|u_{1}\| _{l^{p}}^{\gamma_{1}} +\frac{1}{\gamma_{2}}\|a_{2} \|_{l^{q/(q-\gamma_{2})}}\|u_{2}\|_{l^{q}}^{\gamma _{2}} \\ &{}+\|b_{1}\|_{l^{p/(p-1)}}\|u_{1}\|_{l^{p}} + \|b_{2}\|_{l^{q/(q-1)}}\|u_{2}\|_{l^{q}} \\ \leq&\frac{1}{\gamma_{1}}\|a_{1}\|_{l^{p/(p-\gamma_{1})}}\|u_{1}\| _{p}^{\gamma_{1}} +\frac{1}{\gamma_{2}}\|a_{2} \|_{l^{q/(q-\gamma_{2})}}\|u_{2}\|_{q}^{\gamma _{2}} \\ &{}+\|b_{1}\|_{l^{p/(p-1)}}\|u_{1}\|_{p} + \|b_{2}\|_{l^{q/(q-1)}}\|u_{2}\|_{q}. \end{aligned}$$
(2.11)

It follows from (ρ), (\(\mathcal{A}_{1}\)), (2.7), and (2.11) that

$$\begin{aligned} \mathcal{J}(u) \le& \sum_{n\in\mathbb {Z}}\bigl[\overline{ \rho_{1}}d_{1} \bigl\vert \Delta u_{1}(n) \bigr\vert ^{p}+\overline{\rho_{2}}d_{2} \bigl\vert \Delta u_{2}(n) \bigr\vert ^{q}+\overline{\rho _{3}}d_{3} \bigl\vert u_{1}(n) \bigr\vert ^{p}+\overline{\rho_{4}}d_{4} \bigl\vert u_{2}(n) \bigr\vert ^{q}\bigr] \\ &{} +\frac{\lambda}{\gamma_{1}}\|a_{1}\|_{l^{p/(p-\gamma_{1})}}\|u_{1}\| _{p}^{\gamma_{1}} +\frac{\lambda}{\gamma_{2}}\|a_{2} \|_{l^{q/(q-\gamma_{2})}}\|u_{2}\| _{q}^{\gamma_{2}} \\ &{} +\lambda\|b_{1}\|_{l^{p/(p-1)}}\|u_{1} \|_{p} +\lambda\|b_{2}\| _{l^{q/(q-1)}}\|u_{2} \|_{q} \\ &{} +\|f_{1}\|_{l^{\frac{p}{p-1}}} \biggl(\sum _{n\in\mathbb {Z}} \bigl\vert u_{1}(n) \bigr\vert ^{p} \biggr)^{1/p}+\|f_{2}\|_{l^{\frac{q}{q-1}}} \biggl( \sum_{n\in \mathbb {Z}} \bigl\vert u_{2}(n) \bigr\vert ^{q} \biggr)^{1/q} \\ \le& \max\{\overline{\rho_{1}}d_{1}, \overline{ \rho_{3}}d_{3}\}\|u_{1}\| _{p}^{p}+ \max\{\overline{\rho_{2}}d_{2}, \overline{ \rho_{4}}d_{4}\}\|u_{2}\|_{q}^{q} \\ &{} +\frac{\lambda}{\gamma_{1}}\|a_{1}\|_{l^{p/(p-\gamma_{1})}}\|u_{1}\| _{p}^{\gamma_{1}} +\frac{\lambda}{\gamma_{2}}\|a_{2} \|_{l^{q/(q-\gamma_{2})}}\|u_{2}\| _{q}^{\gamma_{2}} \\ &{} +\lambda\|b_{1}\|_{l^{p/(p-1)}}\|u_{1} \|_{p} +\lambda\|b_{2}\| _{l^{q/(q-1)}}\|u_{2} \|_{q} \\ &{} +\|f_{1}\|_{l^{\frac{p}{p-1}}} \|u_{1}\|_{p}+ \|f_{2}\|_{l^{\frac {q}{q-1}}}\|u_{2}\|_{q}, \end{aligned}$$

which shows that J is well defined.

Next, we prove that \(\mathcal{J}\in C^{1}(E,\mathbb {R})\). We denote \(\mathcal{J}\) as follows:

$$ \mathcal{J}(u)=\mathcal{J}_{1}(u)-\lambda \mathcal{J}_{2}(u)+\mathcal{J}_{3}(u), $$
(2.12)

where

$$\begin{aligned} \begin{aligned} &\mathcal{J}_{1}(u):=\sum _{n\in\mathbb {Z}}\bigl[\rho_{1}(n)\Phi_{1}\bigl( \Delta u_{1}(n)\bigr)+\rho_{2}(n)\Phi_{2}\bigl( \Delta u_{2}(n)\bigr) \\ &\hphantom{\mathcal{J}_{1}(u):={}}{}+\rho_{3}(n)\Phi_{3}\bigl(u_{1}(n) \bigr)+\rho_{4}(n)\Phi _{4}\bigl(u_{2}(n)\bigr) \bigr], \\ &\mathcal{J}_{2}(u):=\sum_{n\in\mathbb {Z}}F \bigl(n,u_{1}(n),u_{2}(n)\bigr), \\ &\mathcal{J}_{3}(u):=\sum_{n\in\mathbb {Z}}\bigl[ \bigl(f_{1}(n),u_{1}(n)\bigr)+\bigl(f_{2}(n),u_{2}(n) \bigr)\bigr]. \end{aligned} \end{aligned}$$
(2.13)

First, by (\(\mathcal{A}_{0}\)), it is easy to prove that \(\mathcal{J}_{1}\in C^{1}(E,\mathbb {R})\) and

$$\begin{aligned} \bigl\langle \mathcal{J}_{1}'(u),v \bigr\rangle = &\sum _{n\in\mathbb {Z}}\bigl[\rho_{1}(n) \bigl(\phi_{1} \bigl(\Delta u_{1}(n)\bigr),\Delta v_{1}(n)\bigr) \\ &{}+\rho_{2}(n) \bigl(\phi_{2}\bigl(\Delta u_{2}(n)\bigr),\Delta v_{2}(n)\bigr)+\rho _{3}(n) \bigl(\phi_{3}\bigl(u_{1}(n)\bigr),v_{1}(n) \bigr) \\ &{}+\rho_{4}(n) \bigl(\phi_{4}\bigl(u_{2}(n) \bigr),v_{2}(n)\bigr)\bigr],\quad \forall u,v\in E. \end{aligned}$$
(2.14)

Next, we prove that \(\mathcal{J}_{2}\in C^{1}(E,\mathbb {R})\) and

$$\begin{aligned} \bigl\langle \mathcal{J}_{2}'(u),v \bigr\rangle =&\sum _{n\in\mathbb {Z}}\bigl[\bigl(\nabla _{u_{1}}F \bigl(n,u_{1}(n),u_{2}(n)\bigr),v_{1}(n)\bigr) \\ &{}+\bigl(\nabla_{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr),v_{2}(n)\bigr)\bigr]. \end{aligned}$$
(2.15)

For any given \(u=(u_{1},u_{2}), v=(v_{1},v_{2})\in E\) and for any sequence \(\{ \theta_{n}\}_{n\in\mathbb {Z}}\subset\mathbb {R}\) with \(|\theta_{n}|<1\) for \(n\in\mathbb {Z}\) and any number \(h\in(0,1)\), by (\(F_{1}\)) and the Hölder inequality, we have

$$\begin{aligned}& \sum_{n\in\mathbb {Z}}\max_{h\in[0,1]} \bigl\vert \bigl(\nabla _{u_{1}}F\bigl(n,u_{1}(n)+ \theta_{n} hv_{1}(n),u_{2}(n)+hv_{2}(n) \bigr),v_{1}(n)\bigr) \bigr\vert \\& \qquad {}+\sum_{n\in\mathbb {Z}}\max_{h\in[0,1]} \bigl\vert \bigl(\nabla _{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n)+ \theta_{n} hv_{2}(n)\bigr),v_{2}(n)\bigr) \bigr\vert \\& \quad \leq \sum_{n\in\mathbb {Z}}\max_{h\in[0,1]} \bigl\vert \nabla _{u_{1}}F\bigl(n,u_{1}(n)+ \theta_{n} hv_{1}(n),u_{2}(n)+hv_{2}(n) \bigr) \bigr\vert \bigl\vert v_{1}(n) \bigr\vert \\& \qquad {}+\sum_{n\in\mathbb {Z}}\max_{h\in[0,1]} \bigl\vert \nabla _{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n)+ \theta_{n} hv_{2}(n)\bigr) \bigr\vert \bigl\vert v_{2}(n) \bigr\vert \\& \quad \leq \sum_{n\in\mathbb {Z}}\max_{h\in[0,1]} \bigl( \bigl\vert a_{1}(n) \bigr\vert \bigl\vert u_{1}(n)+\theta _{n} hv_{1}(n) \bigr\vert ^{\gamma_{1}-1}+b_{1}(n)\bigr) \bigl\vert v_{1}(n) \bigr\vert \\& \qquad {}+\sum_{n\in\mathbb {Z}}\max_{h\in[0,1]} \bigl( \bigl\vert a_{2}(n) \bigr\vert \bigl\vert u_{2}(n)+\theta_{n} hv_{2}(n) \bigr\vert ^{\gamma_{2}-1}+b_{2}(n)\bigr) \bigl\vert v_{2}(n) \bigr\vert \\& \quad \leq 2^{\gamma_{1}-1}\sum_{n\in\mathbb {Z}} \bigl\vert a_{1}(n) \bigr\vert \bigl( \bigl\vert u_{1}(n) \bigr\vert ^{\gamma _{1}-1}+ \bigl\vert v_{1}(n) \bigr\vert ^{\gamma_{1}-1}\bigr) \bigl\vert v_{1}(n) \bigr\vert \\& \qquad {}+2^{\gamma_{2}-1}\sum_{n\in\mathbb {Z}} \bigl\vert a_{2}(n) \bigr\vert \bigl( \bigl\vert u_{2}(n) \bigr\vert ^{\gamma _{2}-1}+ \bigl\vert v_{2}(n) \bigr\vert ^{\gamma_{2}-1}\bigr) \bigl\vert v_{2}(n) \bigr\vert \\& \qquad {}+\sum_{n\in\mathbb {Z}} \bigl\vert b_{1}(n) \bigr\vert \bigl\vert v_{1}(n) \bigr\vert + \sum_{n\in\mathbb {Z}} \bigl\vert b_{2}(n) \bigr\vert \bigl\vert v_{2}(n) \bigr\vert \\& \quad \leq 2^{\gamma_{1}-1} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert a_{1}(n) \bigr\vert ^{\frac {p}{p- \gamma_{1}}} \biggr)^{\frac{p-\gamma_{1}}{p}} \biggl( \sum_{n\in\mathbb {Z}} \bigl\vert u_{1}(n) \bigr\vert ^{p} \biggr)^{\frac{\gamma_{1}-1}{p}} \biggl( \sum_{n\in\mathbb {Z}} \bigl\vert v_{1}(n) \bigr\vert ^{p} \biggr)^{\frac{1}{p}} \\& \qquad {}+2^{\gamma_{1}-1} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert a_{1}(n) \bigr\vert ^{\frac {p}{p- \gamma_{1}}} \biggr)^{\frac{p-\gamma_{1}}{p}} \biggl( \sum_{n\in\mathbb {Z}} \bigl\vert v_{1}(n) \bigr\vert ^{p} \biggr)^{\frac{\gamma_{1}}{p}} \\& \qquad {}+ \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert b_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \biggr)^{\frac {p-1}{p}} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert v_{1}(n) \bigr\vert ^{p} \biggr)^{\frac{1}{p}} \\& \qquad {}+2^{\gamma_{2}-1} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert a_{2}(n) \bigr\vert ^{\frac {q}{q- \gamma_{2}}} \biggr)^{\frac{q-\gamma_{2}}{q}} \biggl( \sum_{n\in\mathbb {Z}} \bigl\vert u_{2}(n) \bigr\vert ^{q} \biggr)^{\frac{\gamma_{2}-1}{q}} \biggl( \sum_{n\in\mathbb {Z}} \bigl\vert v_{2}(n) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\& \qquad {}+2^{\gamma_{2}-1} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert a_{2}(n) \bigr\vert ^{\frac {q}{q- \gamma_{2}}} \biggr)^{\frac{q-\gamma_{2}}{q}} \biggl( \sum_{n\in\mathbb {Z}} \bigl\vert v_{2}(n) \bigr\vert ^{q} \biggr)^{\frac{\gamma_{2}}{q}} \\& \qquad {}+ \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert b_{2}(n) \bigr\vert ^{\frac{q}{q-1}} \biggr)^{\frac {q-1}{q}} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert v_{2}(n) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\& \quad \leq 2^{\gamma_{1}-1}\|a_{1}\|_{l^{p/(p-\gamma_{1})}}\bigl( \|u_{1}\|_{p}^{\gamma _{1}-1}+\|v_{1} \|_{p}^{\gamma_{1}-1}\bigr)\|v_{1}\|_{p} \\& \qquad {}+2^{\gamma_{2}-1}\|a_{2}\|_{l^{q/(q-\gamma_{2})}}\bigl( \|u_{2}\|_{q}^{\gamma _{2}-1}+\|v_{2} \|_{q}^{\gamma_{2}-1}\bigr)\|v_{2}\|_{q} \\& \qquad {}+\|b_{1}\|_{l^{p/(p-1)}}\|v_{1}\|_{p} + \|b_{2}\|_{l^{q/(q-1)}}\|v_{2}\| _{q} \\& \quad < +\infty. \end{aligned}$$
(2.16)

Then it follows from (2.13) and (2.16) that

$$\begin{aligned} \begin{aligned} \bigl\langle \mathcal{J}_{2}'(u),v \bigr\rangle ={}&\lim_{h\rightarrow{0^{+}}} \frac {\mathcal{J}_{2}(u+hv)-\mathcal{J}_{2}(u)}{h} \\ = {}&\lim_{h\rightarrow{0^{+}}}\frac{1}{h}\sum _{n\in\mathbb {Z}}\bigl[F\bigl(n,u_{1}(n)+hv_{1}(n),u_{2}(n)+hv_{2}(n) \bigr) -F\bigl(n,u_{1}(n),u_{2}(n)\bigr)\bigr] \\ = {}&\lim_{h\rightarrow{0^{+}}}\sum_{n\in\mathbb {Z}}\bigl[ \bigl(\nabla _{u_{1}}F\bigl(n,u_{1}(n)+\theta_{n} hv_{1}(n),u_{2}(n)+hv_{2}(n)\bigr),v_{1}(n) \bigr) \\ & {} +\bigl(\nabla_{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n)+ \theta_{n} hv_{2}(n)\bigr),v_{2}(n)\bigr)\bigr] \\ ={} &\sum_{n\in\mathbb {Z}}\bigl[\bigl(\nabla _{u_{1}}F \bigl(n,u_{1}(n),u_{2}(n)\bigr),v_{1}(n)\bigr) \\ &{} +\bigl(\nabla_{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr),v_{2}(n)\bigr)\bigr], \end{aligned} \end{aligned}$$

which implies that (2.15) holds. Next, we prove \(\mathcal{J}_{2}\in C^{1}(E,\mathbb {R})\). For any sequence \(\{u_{k}\}=\{(u_{1}^{k},u_{2}^{k})\}\) and any given \(v\in E\), by the Hölder inequality and (2.6), we obtain

$$\begin{aligned}& \bigl\vert \bigl\langle \mathcal{J}_{2}'(u_{k})- \mathcal{J}_{2}'(u),v\bigr\rangle \bigr\vert \\ & \quad \leq \biggl\vert \sum_{n\in\mathbb {Z}}\bigl(\nabla _{u_{1}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr),v_{1}(n)\bigr) \biggr\vert \\ & \qquad {} + \biggl\vert \sum_{n\in\mathbb {Z}}\bigl(\nabla _{u_{2}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr),v_{2}(n)\bigr) \biggr\vert \\ & \quad \leq \sum_{n\in\mathbb {Z}} \bigl\vert \nabla _{u_{1}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr) \bigr\vert \bigl\vert v_{1}(n) \bigr\vert \\ & \qquad {} +\sum_{n\in\mathbb {Z}} \bigl\vert \nabla _{u_{2}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr) \bigr\vert \bigl\vert v_{2}(n) \bigr\vert \\ & \quad \leq \Vert v_{1} \Vert _{l^{p}} \biggl(\sum _{n\in\mathbb {Z}} \bigl\vert \nabla _{u_{1}}F \bigl(n,u_{1}^{k}(n),u_{2}^{k}(n)\bigr) -\nabla _{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n)\bigr) \bigr\vert ^{\frac{p}{p-1}} \biggr)^{\frac {p-1}{p}} \\ & \qquad {} + \Vert v_{2} \Vert _{l^{q}} \biggl(\sum _{n\in\mathbb {Z}} \bigl\vert \nabla _{u_{2}}F \bigl(n,u_{1}^{k}(n),u_{2}^{k}(n)\bigr) -\nabla _{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n)\bigr) \bigr\vert ^{\frac{q}{q-1}} \biggr)^{\frac {q-1}{q}} \\ & \quad \leq \Vert v_{1} \Vert _{p} \biggl(\sum _{n\in\mathbb {Z}} \bigl\vert \nabla _{u_{1}}F \bigl(n,u_{1}^{k}(n),u_{2}^{k}(n)\bigr) -\nabla _{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n)\bigr) \bigr\vert ^{\frac{p}{p-1}} \biggr)^{\frac {p-1}{p}} \\ & \qquad {}+ \Vert v_{2} \Vert _{q} \biggl(\sum _{n\in\mathbb {Z}} \bigl\vert \nabla _{u_{2}}F \bigl(n,u_{1}^{k}(n),u_{2}^{k}(n)\bigr) -\nabla _{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n)\bigr) \bigr\vert ^{\frac{q}{q-1}} \biggr)^{\frac {q-1}{q}}. \end{aligned}$$
(2.17)

Finally, we claim that

$$ \sum_{n\in\mathbb {Z}} \bigl\vert \nabla _{u_{1}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr) \bigr\vert ^{\frac{p}{p-1}}\rightarrow0,\quad \mbox{as } k\rightarrow \infty, $$
(2.18)

and

$$ \sum_{n\in\mathbb {Z}} \bigl\vert \nabla _{u_{2}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr) \bigr\vert ^{\frac{q}{q-1}}\rightarrow0, \quad \mbox{as } k\rightarrow \infty, $$
(2.19)

if \(u_{k}\rightarrow u\) in E. In fact, since \(u_{k}\rightarrow u\), \(\|u_{1}^{k}-u_{1}\|_{p}^{p}\to0\) and \(\|u_{2}^{k}-u_{2}\|_{q}^{q}\to0\). Furthermore, by (2.6), we have \(u_{1}^{k}\rightarrow u_{1}\) in \(l^{p}\) and \(u_{2}^{k}\rightarrow u_{2}\) in \(l^{q}\) and

$$ \lim_{k\to\infty} u_{i}^{k}(n)=u_{i}(n), \quad \forall n\in\mathbb {Z}, i=1,2. $$
(2.20)

Therefore, there exists a constant \(C_{0}>0\) such that

$$\bigl\Vert u_{1}^{k} \bigr\Vert _{l^{p}}+ \Vert u_{1} \Vert _{l^{p}}+ \bigl\Vert u_{2}^{k} \bigr\Vert _{l^{q}}+ \Vert u_{2} \Vert _{l^{q}}\le C_{0}. $$

By (\(F_{1}\)), we have

$$\begin{aligned} \begin{aligned}[b] &\bigl\vert \nabla_{u_{1}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr) \bigr\vert ^{\frac{p}{p-1}} \\ &\quad \leq \bigl[ \bigl\vert a_{1}(n) \bigr\vert \bigl( \bigl\vert u_{1}^{k}(n) \bigr\vert ^{\gamma _{1}-1}+ \bigl\vert u_{1}(n) \bigr\vert ^{\gamma_{1}-1} \bigr)+2b_{1}(n) \bigr]^{\frac{p}{p-1}} \\ &\quad \leq 2^{\frac{1}{p-1}} \bigl\vert a_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \bigl( \bigl\vert u_{1}^{k}(n) \bigr\vert ^{\gamma_{1}-1}+ \bigl\vert u_{1}(n) \bigr\vert ^{\gamma_{1}-1} \bigr)^{\frac {p}{p-1}}+ 2^{\frac{p+1}{p-1}}\bigl(b_{1}(n) \bigr)^{\frac{p}{p-1}} \\ &\quad \leq 2^{\frac{2}{p-1}} \bigl\vert a_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \bigl\vert u_{1}^{k}(n) \bigr\vert ^{\frac {p(\gamma_{1}-1)}{p-1}}+ 2^{\frac{2}{p-1}} \bigl\vert a_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \bigl\vert u_{1}(n) \bigr\vert ^{\frac{p(\gamma _{1}-1)}{p-1}} \\ &\qquad {}+ 2^{\frac{p+1}{p-1}} \bigl\vert b_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \\ &\quad :=g(n), \quad \forall k\in\mathbb {N}, n\in\mathbb {Z}. \end{aligned} \end{aligned}$$
(2.21)

By (2.21) and the Hölder inequality, we obtain

$$\begin{aligned} \sum_{n\in\mathbb {Z}}g(n) = &2^{\frac{2}{p-1}}\sum _{n\in\mathbb {Z}} \bigl[ \bigl\vert a_{1}(n) \bigr\vert ^{\frac {p}{p-1}} \bigl\vert u_{1}^{k}(n) \bigr\vert ^{\frac{p(\gamma_{1}-1)}{p-1}} + \bigl\vert a_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \bigl\vert u_{1}(n) \bigr\vert ^{\frac{p(\gamma_{1}-1)}{p-1}} \bigr] \\ &{}+2^{\frac{p+1}{p-1}}\sum_{n\in\mathbb {Z}} \bigl\vert b_{1}(n) \bigr\vert ^{\frac {p}{p-1}} \\ \leq& 2^{\frac{2}{p-1}}\|a_{1}\|_{l^{\frac{p}{p-\gamma_{1}}}}^{\frac {p}{p-1}} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert u_{1}^{k}(n) \bigr\vert ^{p} \biggr)^{\frac{\gamma _{1}-1}{p-1}} \\ &{} +2^{\frac{2}{p-1}}\|a_{1}\|_{l^{\frac{p}{p-\gamma_{1}}}}^{\frac{p}{p-1}} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert u_{1}(n) \bigr\vert ^{p} \biggr)^{\frac{\gamma _{1}-1}{p-1}} \\ &{}+ 2^{\frac{p+1}{p-1}}\sum_{n\in\mathbb {Z}} \bigl\vert b_{1}(n) \bigr\vert ^{\frac {p}{p-1}} \\ \leq& 2^{\frac{2}{p-1}}\|a_{1}\|_{l^{p/(p-\gamma_{1})}}^{\frac{p}{p-1}} \bigl\Vert u_{1}^{k} \bigr\Vert _{l^{p}}^{\frac{p(\gamma_{1}-1)}{p-1}}+2^{\frac{2}{p-1}}\|a_{1}\| _{l^{p/(p-\gamma_{1})}}^{\frac{p}{p-1}} \|u_{1}\|_{l^{p}}^{\frac{p(\gamma_{1}-1)}{p-1}} \\ &{}+ 2^{\frac{p+1}{p-1}}\|b_{1}\|_{l^{\frac{p}{p-1}}}^{{\frac {p}{p-1}}} \\ \leq& 2^{\frac{2}{p-1}}\|a_{1}\|_{l^{p/(p-\gamma_{1})}}^{\frac {p}{p-1}}C_{0}^{\frac{p(\gamma_{1}-1)}{p-1}}+2^{\frac{2}{p-1}}\|a_{1}\| _{l^{p/(p-\gamma_{1})}}^{\frac{p}{p-1}} C_{0}^{\frac{p(\gamma_{1}-1)}{p-1}}+ 2^{\frac{p+1}{p-1}}\|b_{1}\|_{l^{\frac {p}{p-1}}}^{{\frac{p}{p-1}}} \\ < &+\infty. \end{aligned}$$
(2.22)

Since F is continuously differentiable in \((x_{1},x_{2})\in\mathbb {R}^{N}\times\mathbb {R}^{N}\), (2.20) implies that, for all \(n\in\mathbb {Z}\),

$$ \bigl\vert \nabla_{u_{1}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr) \bigr\vert \to0,\quad \mbox{as }k\to\infty. $$
(2.23)

Then it follows from (2.22) and (2.23) that

$$ \sum_{n\in\mathbb {Z}} \bigl\vert \nabla_{u_{1}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr) \bigr\vert ^{\frac{p}{p-1}}\rightarrow0, \quad \mbox{as } k \rightarrow\infty. $$
(2.24)

Hence, (2.18) holds. Similarly, we can obtain (2.19). Combining (2.18) and (2.19) with (2.17), we conclude that \(\mathcal{J}_{2}\in C^{1}(E,\mathbb {R})\).

Finally, it is easy to check that \(\mathcal{J}_{3}\in C^{1}(E,\mathbb {R})\) and

$$ \bigl\langle \mathcal{J}_{3}'(u),v \bigr\rangle :=\sum _{n\in\mathbb {Z}}\bigl[\bigl(f_{1}(n),v_{1}(n)\bigr)+ \bigl(f_{2}(n),v_{2}(n)\bigr)\bigr]. $$
(2.25)

Combining (2.14) and (2.15) with (2.25), we deduce that (2.8) holds. By (\(\mathcal{A}_{2}\)) and the Hölder inequality, we obtain, for any given \(u=(u_{1},u_{2}),v=(v_{1},v_{2})\in E\),

$$\begin{aligned}& \sum_{n\in\mathbb {Z}}\Delta\bigl(\rho_{1}(n-1) \phi_{1}\bigl(\Delta u_{1}(n-1)\bigr),v_{1}(n) \bigr) \\& \quad \le \sum_{n\in\mathbb {Z}}\bigl[ \bigl\vert \rho_{1}(n) \bigr\vert \bigl\vert \phi_{1}\bigl(\Delta u_{1}(n)\bigr) \bigr\vert \bigl\vert v_{1}(n+1) \bigr\vert + \bigl\vert \rho_{1}(n-1) \bigr\vert \bigl\vert \phi_{1}\bigl(\Delta u_{1}(n-1)\bigr) \bigr\vert \bigl\vert v_{1}(n) \bigr\vert \bigr] \\& \quad \le \overline{\rho_{1}}\sum_{n\in\mathbb {Z}}k_{1} \bigl\vert \Delta u_{1}(n) \bigr\vert ^{p-1} \bigl\vert v_{1}(n+1) \bigr\vert +\overline{\rho_{1}}\sum _{n\in\mathbb {Z}}k_{1} \bigl\vert \Delta u_{1}(n-1) \bigr\vert ^{p-1} \bigl\vert v_{1}(n) \bigr\vert \\& \quad \le \overline{\rho_{1}}k_{1} \biggl(\sum _{n\in\mathbb {Z}} \bigl\vert \Delta u_{1}(n) \bigr\vert ^{p} \biggr)^{\frac{p-1}{p}} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert v_{1}(n+1) \bigr\vert ^{p} \biggr)^{1/p} \\& \qquad {} +\overline{\rho_{1}}k_{1} \biggl(\sum _{n\in\mathbb {Z}} \bigl\vert \Delta u_{1}(n-1) \bigr\vert ^{p} \biggr)^{\frac{p-1}{p}} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert v_{1}(n) \bigr\vert ^{p} \biggr)^{1/p}, \end{aligned}$$

which, together with the definition of E, implies that the series \(\sum_{n\in\mathbb {Z}}\Delta(\rho_{1}(n-1)\phi_{1}(\Delta u_{1}(n-1)),v_{1}(n))\) is absolutely convergent and then it is easy to see that

$$\sum_{n\in\mathbb {Z}}\Delta\bigl(\rho_{1}(n-1) \phi_{1}\bigl(\Delta u_{1}(n-1)\bigr),v_{1}(n) \bigr)=0. $$

Similarly, we have

$$\sum_{n\in\mathbb {Z}}\Delta\bigl(\rho_{2}(n-1) \phi_{2}\bigl(\Delta u_{2}(n-1)\bigr),v_{2}(n) \bigr)=0. $$

Thus, for \(u,v\in E\),

$$\begin{aligned}& \sum_{n\in\mathbb {Z}} \bigl[\rho_{1}(n) \bigl(\phi_{1}\bigl(\Delta u_{1}(n)\bigr),\Delta v_{1}(n)\bigr)+\rho_{2}(n) \bigl(\phi_{2}\bigl( \Delta u_{2}(n)\bigr),\Delta v_{2}(n)\bigr) \\& \qquad {}+\rho_{3}(n) \bigl(\phi_{3}\bigl(u_{1}(n) \bigr),v_{1}(n)\bigr)+\rho_{4}(n) \bigl(\phi _{4}\bigl(u_{2}(n)\bigr),v_{2}(n)\bigr) \\& \qquad {}-\bigl(\nabla_{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr),v_{1}(n)\bigr)-\bigl(\nabla _{u_{2}}F \bigl(n,u_{1}(n),u_{2}(n)\bigr),v_{2}(n)\bigr) \\& \qquad {}+\bigl(f_{1}(n),v_{1}(n)\bigr)+ \bigl(f_{2}(n),v_{2}(n)\bigr) \bigr] \\& \quad = \sum_{n\in\mathbb {Z}} \bigl[\Delta\bigl( \rho_{1}(n-1)\phi_{1}\bigl(\Delta u_{1}(n-1) \bigr),v_{1}(n)\bigr) \\& \qquad {}-\bigl(\Delta\bigl(\rho_{1}(n-1)\phi_{1}\bigl( \Delta u_{1}(n-1)\bigr)\bigr),v_{1}(n)\bigr) \\& \qquad {}+\Delta\bigl(\rho_{2}(n-1)\phi_{2}\bigl(\Delta u_{2}(n-1)\bigr),v_{2}(n)\bigr) \\& \qquad {}-\bigl(\Delta\bigl(\rho_{2}(n-1)\phi_{2}\bigl( \Delta u_{2}(n-1)\bigr)\bigr),v_{2}(n)\bigr) \\& \qquad {}+\rho_{3}(n) \bigl(\phi_{3}\bigl(u_{1}(n) \bigr),v_{1}(n)\bigr)+\rho_{4}(n) \bigl(\phi _{4}\bigl(u_{2}(n)\bigr),v_{2}(n)\bigr) \\& \qquad {}-\bigl(\nabla_{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr),v_{1}(n)\bigr)-\bigl(\nabla _{u_{2}}F \bigl(n,u_{1}(n),u_{2}(n)\bigr),v_{2}(n)\bigr) \\& \qquad {}+\bigl(f_{1}(n),v_{1}(n)\bigr)+ \bigl(f_{2}(n),v_{2}(n)\bigr) \bigr] \\& \quad = \sum_{n\in\mathbb {Z}} \bigl[\bigl(-\Delta\bigl( \rho_{1}(n-1)\phi_{1}\bigl(\Delta u_{1}(n-1)\bigr) \bigr)+\rho_{3}(n)\phi_{3}\bigl(u_{1}(n)\bigr) \\& \qquad {}-\nabla_{u_{1}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr),v_{1}(n)\bigr) \bigr] \\& \qquad {}+\sum_{n\in\mathbb {Z}} \bigl[\bigl(-\Delta\bigl( \rho_{2}(n-1)\phi_{2}\bigl(\Delta u_{2}(n-1)\bigr) \bigr)+\rho_{4}(n)\phi_{4}\bigl(u_{2}(n)\bigr) \\& \qquad {}-\nabla_{u_{2}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr),v_{2}(n)\bigr) \bigr] \\& \qquad {}+\sum_{n\in\mathbb {Z}} \bigl[\bigl(f_{1}(n),v_{1}(n) \bigr)+\bigl(f_{2}(n),v_{2}(n)\bigr) \bigr]. \end{aligned}$$

Using the above equation, it is easy to show that the critical points of \(\mathcal{J}\) in E are weak solutions of (1.1) with \(u(\pm \infty)=0\). The proof is complete. □

Lemma 2.2

Assume that (ρ), (\(\mathcal{A}_{0}\)), (\(\mathcal{A}_{1}\)), and \((F_{1})'\) hold. Then, for all \(\lambda>0\), \(f_{1}\in l^{\frac{p}{p-1}}(\mathbb {Z},\mathbb {R}^{N})\), and \(f_{2}\in l^{\frac{q}{q-1}}(\mathbb {Z},\mathbb {R}^{N})\), the functional \(\mathcal{J}:E\rightarrow\mathbb {R}\) defined by (2.7) is well defined and of class \(C^{1}(E,\mathbb {R})\) and (2.8) holds. Furthermore, the critical points of \(\mathcal{J}\) in E are weak solutions of (1.1) with \(u(\pm\infty) =0\).

Proof

The proof is similar to Lemma 2.1. In the proof of Lemma 2.1, we only need to replace \(\gamma_{1}\), \(\gamma_{2}\), \(\|a_{1}\| _{l^{p/(p-\gamma_{1})}}\), and \(\|a_{2}\|_{l^{q/(q-\gamma_{2})}}\) with p, q, \(\|a_{1}\|_{l^{\infty}}\), and \(\|a_{2}\|_{l^{\infty}}\), respectively. We omit the details. □

Next, we introduce two lemmas which will be used to prove our main results.

Assume that E is a real Banach space. For \(\varphi\in C^{1}(E,\mathbb {R})\), we say that φ satisfies the Palais-Smale (PS) condition if any sequence \(\{u_{m}\}\subset E\) for which \(\varphi(u_{m})\) is bounded and \(\varphi'(u_{m})\rightarrow0\) as \(m\rightarrow\infty\) has a convergent subsequence.

Lemma 2.3

(see [17])

Assume that E is a real Banach space and let \(\varphi\in C^{1}(E,\mathbb {R})\) satisfy the PS condition. If φ is bounded from below, then \(c=\inf_{E}\varphi\) is a critical value of φ.

Lemma 2.4

(see [18])

Assume that E is a real Banach space and \(\varphi\in C^{1}(E,\mathbb {R})\) with φ even, bounded from below, and satisfying the PS condition. Suppose \(\varphi (0)=0\). Then there exists a set \(K\subset E\) such that K is homeomorphic to \(S^{j-1}\) (\(j-1\) dimension unit sphere) by an odd map and \(\sup_{K}\varphi<0\). Then φ has at least j distinct pairs of critical points.

Proofs

Proof of Theorem 1.1

By Lemma 2.1, we have \(\mathcal{J}\in C^{1}(E,\mathbb {R})\). It follows from (ρ), (\(\mathcal{A}_{1}\)), and (2.11) that

$$\begin{aligned} \mathcal{J}(u) = & \sum_{n\in\mathbb {Z}} \rho_{1}(n)\Phi_{1}\bigl(\Delta u_{1}(n)\bigr)+\sum _{n\in \mathbb {Z}}\rho_{2}(n)\Phi_{2}\bigl( \Delta u_{2}(n)\bigr) \\ &{} +\sum_{n\in\mathbb {Z}}\rho_{3}(n) \Phi_{3}\bigl(u_{1}(n)\bigr)+\sum _{n\in\mathbb {Z}}\rho_{4}(n)\Phi_{4} \bigl(u_{2}(n)\bigr) \\ &{}-\lambda\sum_{n\in\mathbb {Z}}F\bigl(n,u_{1}(n),u_{2}(n) \bigr) \\ &{} +\sum_{n\in\mathbb {Z}}\bigl(f_{1}(n),u_{1}(n) \bigr)+\sum_{n\in\mathbb {Z}}\bigl(f_{2}(n),u_{2}(n) \bigr) \\ \geq& \underline{\rho_{1}}\sum_{n\in\mathbb {Z}}b_{1} \bigl\vert \Delta u_{1}(n) \bigr\vert ^{p}+\underline{ \rho_{2}}\sum_{n\in\mathbb {Z}}b_{2} \bigl\vert \Delta u_{2}(n) \bigr\vert ^{q} \\ &{} +\underline{\rho_{3}}\sum_{n\in\mathbb {Z}}b_{3} \bigl\vert u_{1}(n) \bigr\vert ^{p} +\underline{ \rho_{4}}\sum_{n\in\mathbb {Z}}b_{4} \bigl\vert u_{2}(n) \bigr\vert ^{q} \\ &{}-\lambda\sum _{n\in\mathbb {Z}}F\bigl(n,u_{1}(n),u_{2}(n)\bigr) \\ &{} -\|f_{1}\|_{l^{\frac{p}{p-1}}} \biggl(\sum _{n\in\mathbb {Z}} \bigl\vert u_{1}(n) \bigr\vert ^{p} \biggr)^{1/p}-\|f_{2}\|_{l^{\frac{q}{q-1}}} \biggl( \sum_{n\in \mathbb {Z}} \bigl\vert u_{2}(n) \bigr\vert ^{q} \biggr)^{1/q} \\ \geq& \min\{\underline{\rho_{1}}b_{1},\underline{ \rho_{3}}b_{3}\}\|u_{1}\| _{p}^{p}+ \min\{\underline{\rho_{2}}b_{2},\underline{ \rho_{4}}b_{4}\}\|u_{2}\| _{q}^{q} \\ &{} -\frac{\lambda}{\gamma_{1}}\|a_{1}\|_{l^{p/(p-\gamma_{1})}}\|u_{1}\| _{p}^{\gamma_{1}} -\frac{\lambda}{\gamma_{2}}\|a_{2} \|_{l^{q/(q-\gamma_{2})}}\|u_{2}\| _{q}^{\gamma_{2}} \\ &{}-\lambda\|b_{1}\|_{l^{p/(p-1)}}\|u_{1} \|_{p} -\lambda\|b_{2}\| _{l^{q/(q-1)}}\|u_{2} \|_{q} \\ &{}- \|f_{1}\|_{l^{p/(p-1)}}\|u_{1}\|_{p} - \|f_{2}\|_{l^{q/(q-1)}}\|u_{2}\|_{q}. \end{aligned}$$
(3.1)

Note that \(1<\gamma_{1}<p\), \(1< \gamma_{2}<q\). Then (3.1) and (ρ) show that \(\mathcal{J}(u)\rightarrow+\infty\) as \(\|u\|\rightarrow+\infty \), which implies that \(\mathcal{J}\) is bounded from below.

Next, we show that \(\mathcal{J}\) satisfies the PS condition. Suppose that \(\{u_{k}=(u_{1}^{k},u_{2}^{k})\}_{k\in\mathbb {N}}\subset E\) is a sequence such that \(\{\mathcal{J}(u_{k})\}_{k\in\mathbb {N}}\) is bounded and \(\mathcal {J}'(u_{k})\rightarrow0\) as \(k\rightarrow+\infty\). Then, by (3.1), there exists a constant \(M_{0}>0\) such that

$$ \Vert u_{k} \Vert = \bigl\Vert u_{1}^{k} \bigr\Vert _{p}+ \bigl\Vert u_{2}^{k} \bigr\Vert _{q}\leq M_{0},\quad k\in\mathbb {N}. $$

By (2.6), we have

$$ \bigl\Vert u_{1}^{k} \bigr\Vert _{\infty}\leq \bigl\Vert u_{1}^{k} \bigr\Vert _{p}\leq M_{0},\qquad \bigl\Vert u_{2}^{k} \bigr\Vert _{\infty}\leq \bigl\Vert u_{2}^{k} \bigr\Vert _{q}\leq M_{0}. $$
(3.2)

Hence, there exists a subsequence, still denoted by \(\{u_{k}\}\), such that \(u_{k}\rightharpoonup u_{0}\) for some \(u_{0}=(u_{1}^{0},u_{2}^{0})\) in E. Like the argument of Proposition 1.2 in [17], it is easy to verify that

$$ \lim_{k\rightarrow+\infty}u_{k}(n)=u_{0}(n), \quad \forall n\in\mathbb {Z}. $$
(3.3)

Hence, by (3.2), (3.3), and the lower semi-continuity of norm, we have

$$ \bigl\Vert u_{1}^{0} \bigr\Vert _{\infty}\leq M_{0}, \qquad \bigl\Vert u_{2}^{0} \bigr\Vert _{\infty}\leq M_{0}. $$
(3.4)

Note that \(a_{1}\in l^{p/(p-\gamma_{1})}(\mathbb {Z},[0,+\infty))\) and \(b_{1}\in l^{\frac{p}{p-1}}(\mathbb {Z},[0,+\infty))\). Then, for any given \(\varepsilon>0\), there exists an integer \(M_{1}>0\) such that

$$ \biggl(\sum_{|n|>M_{1}} \bigl\vert a_{1}(n) \bigr\vert ^{\frac{p}{p-\gamma_{1}}} \biggr)^{\frac{p-\gamma _{1}}{p}}< \varepsilon, \qquad \biggl( \sum_{|n|>M_{1}} \bigl\vert b_{1}(n) \bigr\vert ^{\frac {p}{p-1}} \biggr)^{\frac{p-1}{p}}< \varepsilon. $$
(3.5)

It follows from (3.2)-(3.4) and (\(F_{1}\)) that

$$\begin{aligned}& \sum_{n=-M_{1}}^{M_{1}} \bigl\vert \nabla_{u_{1}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr) \\& \quad {}-\nabla _{u_{1}}F\bigl(n,u_{1}^{0}(n),u_{2}^{0}(n) \bigr) \bigr\vert \bigl\vert u_{1}^{k}(n)-u_{1}^{0}(n) \bigr\vert \to0,\quad \mbox{as } k\to\infty. \end{aligned}$$
(3.6)

On the other hand, it follows from (3.2), (3.4), (3.5), (\(F_{1}\)), and Young’s inequality that

$$\begin{aligned}& \sum_{ \vert n \vert >M_{1}} \bigl\vert \nabla_{u_{1}}F \bigl(n,u_{1}^{k}(n),u_{2}^{k}(n)\bigr)- \nabla _{u_{1}}F\bigl(n,u_{1}^{0}(n),u_{2}^{0}(n) \bigr) \bigr\vert \bigl\vert u_{1}^{k}(n)-u_{1}^{0}(n) \bigr\vert \\& \quad \leq\sum_{ \vert n \vert >M_{1}}\bigl[ \bigl\vert a_{1}(n) \bigr\vert \bigl( \bigl\vert u_{1}^{k}(n) \bigr\vert ^{\gamma _{1}-1}+ \bigl\vert u_{1}^{0}(n) \bigr\vert ^{\gamma _{1}-1}\bigr)+2b_{1}(n)\bigr]\bigl( \bigl\vert u_{1}^{k}(n) \bigr\vert + \bigl\vert u_{1}^{0}(n) \bigr\vert \bigr) \\& \quad \leq3\sum_{ \vert n \vert >M_{1}} \bigl\vert a_{1}(n) \bigr\vert \bigl( \bigl\vert u_{1}^{k}(n) \bigr\vert ^{\gamma _{1}}+ \bigl\vert u_{1}^{0}(n) \bigr\vert ^{\gamma_{1}}\bigr) \\& \qquad {}+2\sum_{ \vert n \vert >M_{1}}b_{1}(n) \bigl( \bigl\vert u_{1}^{k}(n) \bigr\vert + \bigl\vert u_{1}^{0}(n) \bigr\vert \bigr) \\& \quad \leq3 \biggl(\sum_{|n|>M_{1}} \bigl\vert a_{1}(n) \bigr\vert ^{\frac{p}{p-\gamma_{1}}} \biggr)^{\frac {p-\gamma_{1}}{p}} \bigl( \bigl\Vert u_{1}^{k} \bigr\Vert _{l^{p}}^{\gamma_{1}}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{l^{p}}^{\gamma_{1}} \bigr) \\& \qquad {}+2 \biggl(\sum_{|n|>M_{1}} \bigl\vert b_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \biggr)^{\frac{p-1}{p}} \bigl( \bigl\Vert u_{1}^{k} \bigr\Vert _{l^{p}}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{l^{p}}\bigr) \\& \quad \leq3 \biggl(\sum_{|n|>M_{1}} \bigl\vert a_{1}(n) \bigr\vert ^{\frac{p}{p-\gamma_{1}}} \biggr)^{\frac {p-\gamma_{1}}{p}} \bigl( \bigl\Vert u_{1}^{k} \bigr\Vert _{p}^{\gamma_{1}}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{p}^{\gamma_{1}} \bigr) \\& \qquad {}+2 \biggl(\sum_{|n|>M_{1}} \bigl\vert b_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \biggr)^{\frac {p-1}{p}}\bigl( \bigl\Vert u_{1}^{k} \bigr\Vert _{p}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{p}\bigr) \\& \quad \leq3\varepsilon\bigl(M_{0}^{\gamma_{1}}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{p}^{\gamma _{1}} \bigr)+2\varepsilon\bigl(M_{0}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{p}\bigr), \quad k\in N. \end{aligned}$$
(3.7)

Then the arbitrariness of ε, together with (3.6), implies that

$$\begin{aligned}& \sum_{n\in\mathbb {Z}}\bigl(\nabla _{u_{1}}F \bigl(n,u_{1}^{k}(n),u_{2}^{k}(n)\bigr) \\& \quad {}-\nabla _{u_{1}}F\bigl(n,u_{1}^{0}(n),u_{2}^{0}(n) \bigr),u_{1}^{k}(n)-u_{1}^{0}(n)\bigr) \rightarrow0,\quad \mbox{as } k\rightarrow+\infty. \end{aligned}$$
(3.8)

Similarly, we have

$$\begin{aligned}& \sum_{n\in\mathbb {Z}}\bigl(\nabla _{u_{2}}F \bigl(n,u_{1}^{k}(n),u_{2}^{k}(n)\bigr) \\& \quad {}-\nabla _{u_{2}}F\bigl(n,u_{1}^{0}(n),u_{2}^{0}(n) \bigr),u_{2}^{k}(n)-u_{2}^{0}(n)\bigr) \rightarrow0, \quad \mbox{as } k\rightarrow+\infty. \end{aligned}$$
(3.9)

By (\(\mathcal{A}_{2}\)), we have

$$\bigl(\phi_{i}(x)-\phi_{i}(y), x-y\bigr)\ge0,\quad \forall x, y\in R^{N}, i=1,2,3,4. $$

Then

$$\begin{aligned}& \bigl\langle \mathcal{J}'(u_{k})-\mathcal{J}'(u_{0}),u_{k}-u_{0} \bigr\rangle \\& \quad = \bigl\langle \mathcal{J}'\bigl(u_{1}^{k},u_{2}^{k} \bigr)-\mathcal {J}'\bigl(u_{1}^{0},u_{2}^{0} \bigr),\bigl(u_{1}^{k}-u_{1}^{0},u_{2}^{k}-u_{2}^{0} \bigr) \bigr\rangle \\& \quad \ge \underline{\rho_{1}}\sum_{n\in\mathbb {Z}} \bigl(\phi_{1}\bigl(\Delta u_{1}^{k}(n)\bigr)- \phi_{1}\bigl(\Delta u_{1}^{0}(n)\bigr),\Delta u_{1}^{k}(n)-\Delta u_{1}^{0}(n)\bigr) \\& \qquad {} +\underline{\rho_{2}}\sum_{n\in\mathbb {Z}} \bigl(\phi_{2}\bigl(\Delta u_{2}^{k}(n)\bigr)- \phi_{2}\bigl(\Delta u_{2}^{0}(n)\bigr),\Delta u_{2}^{k}(n)-\Delta u_{2}^{0}(n)\bigr) \\& \qquad {} +\underline{\rho_{3}}\sum_{n\in\mathbb {Z}} \bigl(\phi_{3}\bigl(u_{1}^{k}(n)\bigr)-\phi _{3}\bigl(u_{1}^{0}(n)\bigr),u_{1}^{k}(n)-u_{1}^{0}(n) \bigr) \\& \qquad {} +\underline{\rho_{4}}\sum_{n\in\mathbb {Z}} \bigl(\phi_{4}\bigl(u_{2}^{k}(n)\bigr)-\phi _{4}\bigl(u_{2}^{0}(n)\bigr),u_{2}^{k}(n)-u_{2}^{0}(n) \bigr) \\& \qquad {} -\lambda\sum_{n\in\mathbb {Z}} \bigl[\bigl(\nabla _{u_{1}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{1}}F\bigl(n,u_{1}^{0}(n),u_{2}^{0}(n) \bigr),u_{1}^{k}(n)-u_{1}^{0}(n)\bigr) \\& \qquad {} +\bigl(\nabla_{u_{2}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{2}}F\bigl(n,u_{1}^{0}(n),u_{2}^{0}(n) \bigr),u_{2}^{k}(n)-u_{2}^{0}(n)\bigr) \bigr]. \end{aligned}$$
(3.10)

Moreover, since \(\mathcal{J}'(u_{k})\to0\) and \(u_{k}\rightharpoonup u_{0}\) as \(k\to\infty\), we have

$$ \bigl\langle \mathcal{J}'(u_{k})- \mathcal{J}'(u_{0}),u_{k}-u_{0} \bigr\rangle \to0,\quad \mbox{as } k\to\infty. $$
(3.11)

Since \((\phi_{i}(x)-\phi_{i}(y),x-y)\geq0\) for all \(x,y\in R^{N}\), \(\lambda >0\), (3.10) and (3.11), together with (3.8) and (3.9), imply that

$$\begin{aligned}& \sum_{n\in\mathbb {Z}}\bigl(\phi_{1} \bigl(\Delta u_{1}^{k}(n)\bigr)-\phi _{1}\bigl( \Delta u_{1}^{0}(n)\bigr),\Delta u_{1}^{k}(n)- \Delta u_{1}^{0}(n)\bigr)\to0,\quad \mbox{as } k\rightarrow+ \infty, \end{aligned}$$
(3.12)
$$\begin{aligned}& \sum_{n\in\mathbb {Z}}\bigl(\phi_{2} \bigl(\Delta u_{2}^{k}(n)\bigr)-\phi _{2}\bigl( \Delta u_{2}^{0}(n)\bigr),\Delta u_{2}^{k}(n)- \Delta u_{2}^{0}(n)\bigr)\to0,\quad \mbox{as } k\rightarrow+ \infty, \end{aligned}$$
(3.13)
$$\begin{aligned}& \sum_{n\in\mathbb {Z}}\bigl(\phi_{3} \bigl(u_{1}^{k}(n)\bigr)-\phi _{3} \bigl(u_{1}^{0}(n)\bigr),u_{1}^{k}(n)-u_{1}^{0}(n) \bigr)\to0, \quad \mbox{as } k\rightarrow +\infty, \end{aligned}$$
(3.14)
$$\begin{aligned}& \sum_{n\in\mathbb {Z}}\bigl(\phi_{4} \bigl(u_{2}^{k}(n)\bigr)-\phi _{4} \bigl(u_{2}^{0}(n)\bigr),u_{2}^{k}(n)-u_{2}^{0}(n) \bigr)\to0, \quad \mbox{as } k\rightarrow +\infty. \end{aligned}$$
(3.15)

If \(1< p\le2\), then it follows from (\(\mathcal{A}_{2}\)) and the Hölder inequality that

$$\begin{aligned}& \sum_{n\in\mathbb {Z}} \bigl\vert \Delta u_{1}^{k}(n)- \Delta u_{1}^{0}(n) \bigr\vert ^{p} \\& \quad = \sum_{n\in\mathbb {Z}} \bigl\vert \Delta u_{1}^{k}(n)-\Delta u_{1}^{0}(n) \bigr\vert ^{\frac {2p}{2}} \\& \quad \leq \frac{1}{c_{1}^{\frac{p}{2}}}\sum_{n\in\mathbb {Z}}\bigl(\phi _{1}\bigl(\Delta u_{1}^{k}(n)\bigr)- \phi_{1}\bigl(\Delta u_{1}^{0}(n)\bigr),\Delta u_{1}^{k}(n)-\Delta u_{1}^{0}(n) \bigr)^{\frac{p}{2}} \\& \qquad {}\cdot\bigl( \bigl\vert \Delta u_{1}^{k}(n) \bigr\vert + \bigl\vert \Delta u_{1}^{0}(n) \bigr\vert \bigr)^{\frac {p(2-p)}{2}} \\& \quad \leq \frac{1}{c_{1}^{\frac{p}{2}}} \biggl(\sum_{n\in\mathbb {Z}} \bigl(\phi _{1}\bigl(\Delta u_{1}^{k}(n)\bigr)- \phi_{1}\bigl(\Delta u_{1}^{0}(n)\bigr),\Delta u_{1}^{k}(n)-\Delta u_{1}^{0}(n)\bigr) \biggr)^{\frac{p}{2}} \\& \qquad {}\cdot \biggl( \sum_{n\in\mathbb {Z}}\bigl( \bigl\vert \Delta u_{1}^{k}(n) \bigr\vert + \bigl\vert \Delta u_{1}^{0}(n) \bigr\vert \bigr)^{p} \biggr)^{\frac{2-p}{2}} \\& \quad \leq \frac{2^{\frac{p(2-p)}{2}}}{c_{1}^{\frac{p}{2}}} \biggl(\sum_{n\in\mathbb {Z}} \bigl(\phi_{1}\bigl(\Delta u_{1}^{k}(n)\bigr)- \phi_{1}\bigl(\Delta u_{1}^{0}(n)\bigr),\Delta u_{1}^{k}(n)-\Delta u_{1}^{0}(n)\bigr) \biggr)^{\frac {p}{2}} \\& \qquad {}\cdot \biggl( \sum_{n\in\mathbb {Z}}\bigl( \bigl\vert \Delta u_{1}^{k}(n) \bigr\vert ^{p}+ \bigl\vert \Delta u_{1}^{0}(n) \bigr\vert ^{p} \bigr) \biggr)^{\frac{2-p}{2}} \\& \quad \leq \frac{2^{\frac{p(2-p)}{2}}}{c_{1}^{\frac{p}{2}}} \biggl(\sum_{n\in\mathbb {Z}} \bigl(\phi_{1}\bigl(\Delta u_{1}^{k}(n)\bigr)- \phi_{1}\bigl(\Delta u_{1}^{0}(n)\bigr),\Delta u_{1}^{k}(n)-\Delta u_{1}^{0}(n)\bigr) \biggr)^{\frac {p}{2}} \\& \qquad {}\cdot \bigl( \bigl\Vert u_{1}^{k} \bigr\Vert _{p}^{p}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{p}^{p} \bigr)^{\frac{2-p}{2}}. \end{aligned}$$
(3.16)

Similarly, we have

$$\begin{aligned}& \sum_{n\in\mathbb {Z}} \bigl\vert u_{1}^{k}(n)- u_{1}^{0}(n) \bigr\vert ^{p} \\& \quad \leq \frac{2^{\frac{p(2-p)}{2}}}{c_{3}^{\frac{p}{2}}} \biggl(\sum_{n\in\mathbb {Z}} \bigl(\phi_{3}\bigl( u_{1}^{k}(n)\bigr)- \phi_{3}\bigl(u_{1}^{0}(n)\bigr), u_{1}^{k}(n)- u_{1}^{0}(n)\bigr) \biggr)^{\frac{p}{2}} \\& \qquad {}\cdot \bigl( \bigl\Vert u_{1}^{k} \bigr\Vert _{p}^{p}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{p}^{p} \bigr)^{\frac{2-p}{2}}. \end{aligned}$$
(3.17)

If \(p> 2\), then it follows from (\(\mathcal{A}_{2}\)) and the Hölder inequality that

$$\begin{aligned}& \sum_{n\in\mathbb {Z}} \bigl\vert \Delta u_{1}^{k}(n)- \Delta u_{1}^{0}(n) \bigr\vert ^{p} \\& \quad \leq \frac{1}{c_{1}}\sum_{n\in\mathbb {Z}}\bigl( \phi_{1}\bigl(\Delta u_{1}^{k}(n)\bigr)- \phi_{1}\bigl(\Delta u_{1}^{0}(n)\bigr),\Delta u_{1}^{k}(n)-\Delta u_{1}^{0}(n) \bigr), \end{aligned}$$
(3.18)
$$\begin{aligned}& \sum_{n\in\mathbb {Z}} \bigl\vert u_{1}^{k}(n)-u_{1}^{0}(n) \bigr\vert ^{p} \\& \quad \leq \frac{1}{c_{3}}\sum_{n\in\mathbb {Z}}\bigl( \phi_{1}\bigl( u_{1}^{k}(n)\bigr)-\phi _{1}\bigl( u_{1}^{0}(n)\bigr),u_{1}^{k}(n)-u_{1}^{0}(n) \bigr). \end{aligned}$$
(3.19)

By (3.12)-(3.19), it is easy to see that \(u_{1}^{k}\rightarrow u_{1}^{0}\) in \(E_{p}\) for any \(p>1\). Similarly, we can obtain \(u_{2}^{k}\rightarrow u_{2}^{0}\) in \(E_{q}\) for any \(q>1\). So, \(u_{k}\rightarrow u_{0}\) in E, that is, \(\mathcal{J}\) satisfies the PS condition.

Let \(\varphi=\mathcal{J}\). By Lemma 2.3, \(c=\inf_{E}\mathcal{J}(u)\) is a critical value of \(\mathcal{J}\), that is, there exists a critical point \(u^{*}\in E\) such that \(\mathcal{J}(u^{*})=c\).

Finally, we show that \(u^{*}\neq0\). Let \(u_{*}(n_{0})=(u_{1*}(n_{0}),u_{2*}(n_{0}))\) where \(u_{1*}(n_{0})=(0,\ldots ,1, \ldots,0)^{\tau}\in\mathbb {R}^{N}\) with 1 is the \(i_{0}\)th component of the vector, \(u_{2*}(n_{0})=(0,\ldots,1,\ldots0)^{\tau}\in\mathbb {R}^{N}\) with 1 is the \(j_{0}\)th component of the vector, and \(u_{*}(n)=0\) for \(n\neq n_{0}\), where \(i_{0}, j_{0}\) are defined in assumption (f). Then, by (\(F_{2}\)) and (2.7), we have

$$\begin{aligned} \mathcal{J}(su_{*}) =&\sum_{n\in\mathbb {Z}} \bigl[ \rho_{1}(n)\Phi_{1}\bigl(\Delta su_{1\ast}(n)\bigr)+ \rho _{2}(n)\Phi_{2}\bigl(\Delta su_{2\ast}(n)\bigr) \\ &{} +\rho_{3}(n)\Phi_{3}\bigl(su_{1\ast}(n)\bigr)+ \rho_{4}(n)\Phi_{4}\bigl(su_{2\ast }(n)\bigr) \bigr] \\ &{}-\lambda\sum_{n\in\mathbb {Z}}F\bigl(n,su_{1\ast}(n),su_{2\ast}(n) \bigr)+\sum_{n\in\mathbb {Z}}\bigl(f_{1}(n),u_{1\ast}(n) \bigr)+\sum_{n\in\mathbb {Z}}\bigl(f_{2}(n),u_{2\ast}(n) \bigr) \\ \leq&\overline{\rho_{1}}s^{p}d_{1}\sum _{n\in\mathbb {Z}} \bigl\vert \Delta u_{1\ast }(n) \bigr\vert ^{p}+\overline{\rho_{2}}s^{q}d_{2}\sum _{n\in\mathbb {Z}} \bigl\vert \Delta u_{2\ast }(n) \bigr\vert ^{q}+\overline{\rho_{3}}s^{p}d_{3} \sum_{n\in\mathbb {Z}} \bigl\vert u_{1\ast }(n) \bigr\vert ^{p} \\ &{}+\overline{\rho_{4}}s^{q}d_{4}\sum _{n\in\mathbb {Z}} \bigl\vert u_{2\ast }(n) \bigr\vert ^{q}-\lambda F\bigl(n_{0},su_{1\ast}(n_{0}),su_{2\ast}(n_{0}) \bigr) \\ &{}+\bigl(f_{1}(n_{0}),su_{1\ast}(n_{0}) \bigr)+\bigl(f_{2}(n_{0}),su_{2\ast}(n_{0}) \bigr) \\ \leq&\overline{\rho_{1}}s^{p}d_{1}\bigl( \bigl\vert \Delta u_{1\ast}(n_{0}) \bigr\vert ^{p}+ \bigl\vert \Delta u_{1\ast}(n_{0}-1) \bigr\vert ^{p}\bigr) \\ &{}+\overline{\rho_{2}}s^{q}d_{2}\bigl( \bigl\vert \Delta u_{2\ast}(n_{0}) \bigr\vert ^{q}+ \bigl\vert \Delta u_{2\ast }(n_{0}-1) \bigr\vert ^{q}\bigr) \\ &{}+\overline{\rho_{3}}s^{p}d_{3} \bigl\vert u_{1\ast}(n_{0}) \bigr\vert ^{p}+ \overline{\rho _{4}}s^{q}d_{4} \bigl\vert u_{2\ast}(n_{0}) \bigr\vert ^{q} \\ &{}+\lambda\eta_{1}s^{\gamma_{3}} \bigl\vert u_{1\ast}(n_{0}) \bigr\vert ^{\gamma_{3}}+\lambda\eta _{2}s^{\gamma_{4}} \bigl\vert u_{2\ast}(n_{0}) \bigr\vert ^{\gamma _{4}}+sf_{1i_{0}}(n_{0})+sf_{2j_{0}}(n_{0}) \\ =&(2\overline{\rho_{1}}d_{1}+\overline{ \rho_{3}}d_{3}) s^{p} +(2\overline{ \rho_{2}}d_{2}+\overline{\rho_{4}}d_{4}) s^{q} +\lambda\eta_{1}s^{\gamma_{3}}+\lambda \eta_{2}s^{\gamma_{4}} \\ &{}+s\bigl(f_{1i_{0}}(n_{0})+f_{2j_{0}}(n_{0}) \bigr), \end{aligned}$$
(3.20)

for all \(0< s<\delta_{0}\). Since \(p,q, \gamma_{3},\gamma_{4}\in(1,+\infty)\), it follows from (f) that \(\mathcal{J}(su_{*})<0\) for \(s>0\) small enough. Hence, \(\mathcal{J}(u^{*})=c=\inf_{E}\mathcal{J}(u)<0\), which implies that \(u^{*}\in E\) is a nontrivial critical point of \(\mathcal{J}\) and so \(u^{*}=u^{*}(n)\) is a nontrivial homoclinic solution of system (1.1). The proof is complete. □

Proof of Theorem 1.2

By the proof of Theorem 1.1, we know that there exists a critical point \(u^{*}\in E\) such that \(\mathcal{J}(u^{*})=c\). Next, we prove that \(u^{*}\neq0\) when \((F_{2})'\) and \((f)'\) hold. We define the same \(u_{*}\) as Theorem 1.1. Then, by \(\lambda>0\), \((F_{2})'\), and \((f)'\), we have

$$\begin{aligned} \mathcal{J}(su_{*}) =&\sum_{n\in\mathbb {Z}} \bigl[ \rho_{1}(n)\Phi_{1}\bigl(\Delta su_{1\ast}(n)\bigr)+ \rho _{2}(n)\Phi_{2}\bigl(\Delta su_{2\ast}(n)\bigr) \\ &{}+\rho_{3}(n)\Phi_{3}\bigl(su_{1\ast}(n)\bigr)+ \rho_{4}(n)\Phi_{4}\bigl(su_{2\ast}(n)\bigr) \bigr] \\ &{}-\lambda\sum_{n\in\mathbb {Z}}F\bigl(n,su_{1\ast}(n),su_{2\ast}(n) \bigr)+\sum_{n\in\mathbb {Z}}\bigl(f_{1}(n),u_{1\ast}(n) \bigr)+\sum_{n\in\mathbb {Z}}\bigl(f_{2}(n),u_{2\ast}(n) \bigr) \\ \leq&\overline{\rho_{1}}s^{p}d_{1}\sum _{n\in\mathbb {Z}} \bigl\vert \Delta u_{1\ast }(n) \bigr\vert ^{p}+\overline{\rho_{2}}s^{q}d_{2}\sum _{n\in\mathbb {Z}} \bigl\vert \Delta u_{2\ast }(n) \bigr\vert ^{q}+\overline{\rho_{3}}s^{p}d_{3} \sum_{n\in\mathbb {Z}} \bigl\vert u_{1\ast }(n) \bigr\vert ^{p} \\ &{}+\overline{\rho_{4}}s^{q}d_{4}\sum _{n\in\mathbb {Z}} \bigl\vert u_{2\ast }(n) \bigr\vert ^{q}-\lambda F\bigl(n_{0},su_{1\ast}(n_{0}),su_{2\ast}(n_{0}) \bigr) \\ &{}+\bigl(f_{1}(n_{0}),su_{1\ast}(n_{0}) \bigr)+\bigl(f_{2}(n_{0}),su_{2\ast}(n_{0}) \bigr) \\ \leq&\overline{\rho_{1}}s^{p}d_{1}\bigl( \bigl\vert \Delta u_{1\ast}(n_{0}) \bigr\vert ^{p}+ \bigl\vert \Delta u_{1\ast}(n_{0}-1) \bigr\vert ^{p}\bigr) \\ &{}+\overline{\rho_{2}}s^{q}d_{2}\bigl( \bigl\vert \Delta u_{2\ast}(n_{0}) \bigr\vert ^{q}+ \bigl\vert \Delta u_{2\ast }(n_{0}-1) \bigr\vert ^{q}\bigr)+\overline{\rho_{3}}s^{p}d_{3} \bigl\vert u_{1\ast}(n_{0}) \bigr\vert ^{p} \\ &{}+\overline{\rho_{4}}s^{q}d_{4} \bigl\vert u_{2\ast}(n_{0}) \bigr\vert ^{q}-\lambda \eta_{1}s^{\gamma _{3}} \bigl\vert u_{1\ast}(n_{0}) \bigr\vert ^{\gamma_{3}} \\ &{}-\lambda\eta_{2}s^{\gamma_{4}} \bigl\vert u_{2\ast}(n_{0}) \bigr\vert ^{\gamma _{4}}+sf_{1i_{0}}(n_{0})+sf_{2j_{0}}(n_{0}) \\ =&(2\overline{\rho_{1}}d_{1}+\overline{ \rho_{3}}d_{3}) s^{p} +(2\overline{ \rho_{2}}d_{2}+\overline{\rho_{4}}d_{4}) s^{q} -\lambda\eta_{1}s^{\gamma_{3}}-\lambda \eta_{2}s^{\gamma_{4}}, \end{aligned}$$
(3.21)

for all \(0< s<\delta_{0}\). Since \(1<\gamma_{3}<p\) and \(1<\gamma_{4}<q\), \(\mathcal{J}(su_{*})<0\) for \(s>0\) small enough. Hence, \(\mathcal{J}(u^{*})=c=\inf_{E}\mathcal{J}(u)<0\), which implies that \(u^{*}\in E\) is a nontrivial critical point of \(\mathcal{J}\) and so \(u^{*}=u^{*}(n)\) is a nontrivial homoclinic solution of system (1.1). The proof is complete. □

Proof of Theorem 1.3

By Lemma 2.2, \(\mathcal{J}\in C^{1}(E,\mathbb {R})\). Similar to (3.1), it follows from (ρ), (\(\mathcal{A}_{1}\)), \((F_{1})'\), and (2.11), by replacing \(\gamma_{1}\), \(\gamma_{2}\), \(\|a_{1}\|_{l^{p/(p-\gamma_{1})}}\), and \(\|a_{2}\|_{l^{q/(q-\gamma _{2})}}\) with p, q, \(\|a_{1}\|_{l^{\infty}}\), and \(\|a_{2}\|_{l^{\infty}}\), respectively, that

$$\begin{aligned} \mathcal{J}(u) \geq& \min\{\underline{\rho_{1}}b_{1}, \underline{\rho_{3}}b_{3}\}\|u_{1}\| _{p}^{p}+\min\{\underline{\rho_{2}}b_{2}, \underline{\rho_{4}}b_{4}\}\|u_{2}\| _{q}^{q} \\ &{} -\frac{\lambda}{p}\|a_{1}\|_{\infty}\|u_{1} \|_{p}^{p} -\frac{\lambda}{q}\|a_{2} \|_{\infty}\|u_{2}\|_{q}^{q} \\ &{}-\lambda\|b_{1}\|_{l^{p/(p-1)}}\|u_{1} \|_{p} -\lambda\|b_{2}\| _{l^{q/(q-1)}}\|u_{2} \|_{q} \\ &{}- \|f_{1}\|_{l^{\frac{p}{p-1}}}\|u_{1}\|_{p} - \|f_{2}\|_{l^{\frac {q}{q-1}}}\|u_{2}\|_{q}. \end{aligned}$$
(3.22)

Note that \(\lambda<\min \{\frac{p\min\{\underline{\rho _{1}}b_{1},\underline{\rho_{3}}b_{3}\}}{\|a_{1}\|_{\infty}},\frac{q\min\{ \underline{\rho_{2}}b_{2},\underline{\rho_{4}}b_{4}\}}{\|a_{2}\|_{\infty}} \} \). Then (3.22) shows that \(\mathcal{J}(u)\rightarrow+\infty\) as \(\| u\|\rightarrow+\infty\), which implies that \(\mathcal{J}\) is bounded from below.

Next, we show that \(\mathcal{J}\) satisfies the PS condition. Suppose that \(\{u_{k}=(u_{1}^{k},u_{2}^{k})\}_{k\in\mathbb {N}}\subset E\) is a sequence such that \(\{\mathcal{J}(u_{k})\}_{k\in\mathbb {N}}\) is bounded and \(\mathcal {J}'(u_{k})\rightarrow0\) as \(k\rightarrow+\infty\). Similar to the proof of Theorem 1.1, by (3.22), there exists a constant \(M_{0}>0\) such that (3.2)-(3.4) hold. Note that \(a_{1}(n)\to0\) as \(n\to\infty\) and \(b_{1}\in l^{\frac {p}{p-1}}(\mathbb {Z},[0,+\infty))\). Then, for any given \(\varepsilon>0\), there exists an integer \(M_{1}>0\) such that

$$ \sup_{ \vert n \vert >M_{1}} \bigl\vert a_{1}(n) \bigr\vert < \varepsilon,\qquad \biggl(\sum_{ \vert n \vert >M_{1}} \bigl\vert b_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \biggr)^{\frac{p-1}{p}}< \varepsilon. $$
(3.23)

It follows from (3.2)-(3.4) and \((F_{1})'\) that (3.6) holds. On the other hand, it follows from (3.2), (3.4), (3.23), \((F_{1})'\), and Young’s inequality that

$$\begin{aligned}& \sum_{|n|>M_{1}} \bigl\vert \nabla_{u_{1}}F\bigl(n,u_{1}^{k}(n),u_{2}^{k}(n) \bigr)-\nabla _{u_{1}}F\bigl(n,u_{1}^{0}(n),u_{2}^{0}(n) \bigr) \bigr\vert \bigl\vert u_{1}^{k}(n)-u_{1}^{0}(n) \bigr\vert \\& \quad \leq\sum_{|n|>M_{1}}\bigl[ \bigl\vert a_{1}(n) \bigr\vert \bigl( \bigl\vert u_{1}^{k}(n) \bigr\vert ^{p-1}+ \bigl\vert u_{1}^{0}(n) \bigr\vert ^{p-1}\bigr)+2b_{1}(n)\bigr]\bigl( \bigl\vert u_{1}^{k}(n) \bigr\vert + \bigl\vert u_{1}^{0}(n) \bigr\vert \bigr) \\& \quad \leq3\sum_{|n|>M_{1}} \bigl\vert a_{1}(n) \bigr\vert \bigl( \bigl\vert u_{1}^{k}(n) \bigr\vert ^{p}+ \bigl\vert u_{1}^{0}(n) \bigr\vert ^{p}\bigr)+2\sum_{|n|>M_{1}}b_{1}(n) \bigl( \bigl\vert u_{1}^{k}(n) \bigr\vert + \bigl\vert u_{1}^{0}(n) \bigr\vert \bigr) \\& \quad \leq3\sup_{|n|>M_{1}} \bigl\vert a_{1}(n) \bigr\vert \bigl( \bigl\Vert u_{1}^{k} \bigr\Vert _{l^{p}}^{p}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{l^{p}}^{p}\bigr) \\& \qquad {} +2 \biggl(\sum_{|n|>M_{1}} \bigl\vert b_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \biggr)^{\frac{p-1}{p}} \bigl( \bigl\Vert u_{1}^{k} \bigr\Vert _{l^{p}}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{l^{p}}\bigr) \\& \quad \leq3\sup_{|n|>M_{1}} \bigl\vert a_{1}(n) \bigr\vert \bigl( \bigl\Vert u_{1}^{k} \bigr\Vert _{p}^{p}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{p}^{p}\bigr) \\& \qquad {} +2 \biggl(\sum_{|n|>M_{1}} \bigl\vert b_{1}(n) \bigr\vert ^{\frac{p}{p-1}} \biggr)^{\frac{p-1}{p}} \bigl( \bigl\Vert u_{1}^{k} \bigr\Vert _{p}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{p}\bigr) \\& \quad \leq3\varepsilon\bigl(M_{0}^{p}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{p}^{p} \bigr)+2\varepsilon\bigl(M_{0}+ \bigl\Vert u_{1}^{0} \bigr\Vert _{p}\bigr), \quad \forall k\in N. \end{aligned}$$

Then arbitrariness of ε, together with (3.6), implies that

$$\begin{aligned}& \sum_{n\in\mathbb {Z}}\bigl(\nabla _{u_{1}}F \bigl(n,u_{1}^{k}(n),u_{2}^{k}(n)\bigr) \\& \quad {} -\nabla _{u_{1}}F\bigl(n,u_{1}^{0}(n),u_{2}^{0}(n) \bigr),u_{1}^{k}(n)-u_{1}^{0}(n)\bigr) \rightarrow0,\quad \mbox{as } k\rightarrow+\infty. \end{aligned}$$

Similarly, we have

$$\begin{aligned}& \sum_{n\in\mathbb {Z}}\bigl(\nabla _{u_{2}}F \bigl(n,u_{1}^{k}(n),u_{2}^{k}(n)\bigr) \\& \quad {}-\nabla _{u_{2}}F\bigl(n,u_{1}^{0}(n),u_{2}^{0}(n) \bigr),u_{2}^{k}(n)-u_{2}^{0}(n)\bigr) \rightarrow0, \quad \mbox{as } k\rightarrow+\infty. \end{aligned}$$

Following the argument of Theorem 1.1, we can obtain \(u_{k}\rightarrow u_{0}\) in E, that is, \(\mathcal{J}\) satisfies the PS condition.

Let \(\varphi=\mathcal{J}\). By Lemma 2.3, \(c=\inf_{E}\mathcal{J}(u)\) is a critical value of \(\mathcal{J}\), that is, there exists a critical point \(u^{*}\in E\) such that \(\mathcal{J}(u^{*})=c\).

Finally, with the same argument as Theorem 1.1, we know that \(u^{*}\neq 0\). The proof is complete. □

Proof of Theorem 1.4

In view of Lemma 2.1 and the proof of Theorem 1.1, \(\mathcal{J}\in C^{1}(E,\mathbb {R})\) is bounded from below and satisfies the PS condition. It follows from (\(\mathcal {A}_{0}\)), (\(F_{1}\)), (\(F_{3}\)), and \((f)''\) that \(\mathcal{J}\) is even and \(\mathcal{J}(0)=0\). In order to apply Lemma 2.4, let \(\varphi=\mathcal{J}\). We prove now that there is a set \(K\subset E\) such that K is homeomorphic to \(S^{m-1}\) by an odd map and \(\sup_{K}\mathcal{J}<0\). The proof is motivated by [7] and [19]. Let

$$J=\{n_{1},n_{2},\ldots,n_{m}\}, $$

where \(n_{1}< n_{2}<\cdots<n_{m}\). Note that \(m\le N\). Define

$$\begin{aligned}& u_{j}^{i}(n)= \textstyle\begin{cases} (0,\ldots,0,\underset{\underset{i}{\downarrow}}{1},0,\ldots,0)^{\tau}\in \mathbb {R}^{N}, & n=n_{i}, \\ 0, & n\neq n_{i}, \end{cases}\displaystyle \quad i=1,2, \ldots,m, j=1,2, \\& u^{i}(n)=\bigl(u_{1}^{i}(n),u_{2}^{i}(n) \bigr)^{\tau},\quad i=1,2,\ldots,m, \end{aligned}$$

and

$$ E_{m}=\operatorname{span}\bigl\{ u^{1},u^{2}, \ldots,u^{m}\bigr\} , \qquad K_{m}=\bigl\{ u\in E_{m}: \Vert u \Vert _{(2)}=\delta_{0}\bigr\} , $$
(3.24)

where \(\|u\|_{(2)}\) is defined by \(\|u\|_{(2)}=\|u_{1}\|_{l^{2}}+\|u_{2}\|_{l^{2}}\). For any \(u\in E_{m}\), there exist \(\lambda_{i}\in\mathbb {R}\), \(i=1,2,\ldots,m\), such that

$$ u=\sum_{i=1}^{m} \lambda_{i}u^{i}\quad \mbox{and}\quad \bigl(u_{1}(n),u_{2}(n) \bigr)=\sum_{i=1}^{m}\lambda _{i}\bigl(u_{1}^{i}(n),u_{2}^{i}(n) \bigr), \quad \mbox{for } n\in\mathbb {Z}. $$
(3.25)

Then

$$\begin{aligned} \begin{aligned} &\|u_{1}\|_{l^{\gamma_{3}}}= \biggl(\sum _{n\in\mathbb {Z}} \bigl\vert u_{1}(n) \bigr\vert ^{\gamma _{3}} \biggr)^{1/\gamma_{3}}= \Biggl(\sum_{i=1}^{m} \vert \lambda_{i} \vert ^{\gamma_{3}} \bigl\vert u_{1}^{i}(n_{i}) \bigr\vert ^{\gamma_{3}} \Biggr)^{1/\gamma _{3}}, \\ &\|u_{2}\|_{l^{\gamma_{4}}}= \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert u_{2}(n) \bigr\vert ^{\gamma _{4}} \biggr)^{1/\gamma_{4}}= \Biggl(\sum_{i=1}^{m} \vert \lambda_{i} \vert ^{\gamma_{4}} \bigl\vert u_{2}^{i}(n_{i}) \bigr\vert ^{\gamma_{4}} \Biggr)^{1/\gamma_{4}}. \end{aligned} \end{aligned}$$
(3.26)

Note that \(|u_{1}^{i}(n_{i})|^{2}=|u_{2}^{i}(n_{i})|^{2}=1\), \(i=1,2,\ldots,m\). Hence

$$\begin{aligned} \Vert u \Vert _{(2)}^{2} =&\bigl( \Vert u_{1} \Vert _{l^{2}}+ \Vert u_{2} \Vert _{l^{2}}\bigr)^{2} \\ =& \Vert u_{1} \Vert _{l^{2}}^{2}+2 \Vert u_{1} \Vert _{l^{2}} \Vert u_{2} \Vert _{l^{2}}+ \Vert u_{2} \Vert _{l^{2}}^{2} \\ =&\sum_{n\in\mathbb {Z}} \bigl\vert u_{1}(n) \bigr\vert ^{2}+2 \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert u_{1}(n) \bigr\vert ^{2} \biggr)^{1/2} \biggl(\sum_{n\in\mathbb {Z}} \bigl\vert u_{2}(n) \bigr\vert ^{2} \biggr)^{1/2} +\sum_{n\in\mathbb {Z}} \bigl\vert u_{2}(n) \bigr\vert ^{2} \\ =&\sum_{n\in\mathbb {Z}} \Biggl(\sum _{i=1}^{m}\lambda _{i}u_{1}^{i}(n), \sum_{i=1}^{m}\lambda_{i}u_{1}^{i}(n) \Biggr) \\ &{} +\sum_{n\in\mathbb {Z}} \Biggl(\sum _{i=1}^{m}\lambda _{i}u_{2}^{i}(n), \sum_{i=1}^{m}\lambda_{i}u_{2}^{i}(n) \Biggr) \\ &{}+2 \Biggl(\sum_{n\in\mathbb {Z}} \Biggl(\sum _{i=1}^{m}\lambda _{i}u_{1}^{i}(n), \sum_{i=1}^{m}\lambda_{i}u_{1}^{i}(n) \Biggr) \Biggr)^{1/2} \\ &{}\cdot \Biggl(\sum_{n\in\mathbb {Z}} \Biggl(\sum _{i=1}^{m}\lambda _{i}u_{2}^{i}(n), \sum_{i=1}^{m}\lambda_{i}u_{2}^{i}(n) \Biggr) \Biggr)^{1/2} \\ =&\sum_{i=1}^{m}\lambda_{i}^{2} \bigl\vert u_{1}^{i}(n_{i}) \bigr\vert ^{2}+\sum_{i=1}^{m}\lambda _{i}^{2} \bigl\vert u_{2}^{i}(n_{i}) \bigr\vert ^{2} \\ &{}+2 \Biggl(\sum_{i=1}^{m} \lambda_{i}^{2} \bigl\vert u_{1}^{i}(n_{i}) \bigr\vert ^{2} \Biggr)^{1/2} \Biggl(\sum _{i=1}^{m}\lambda_{i}^{2} \bigl\vert u_{2}^{i}(n_{i}) \bigr\vert ^{2} \Biggr)^{1/2} \\ =&4\sum_{i=1}^{m}\lambda_{i}^{2}. \end{aligned}$$
(3.27)

Since all the norms of a finite dimensional normed space are equivalent, there are constants \(R_{i}>0, i=1,2,3,4\), such that

$$ \begin{aligned} &\|u_{1}\|_{p}\leq R_{1}\|u_{1}\|_{l^{2}}, \qquad \|u_{2} \|_{q}\leq R_{2}\|u_{2}\| _{l^{2}}, \\ &R_{3}\|u_{1}\|_{l^{2}}\leq\|u_{1} \|_{l^{\gamma_{3}}},\qquad R_{4}\|u_{2}\| _{l^{2}}\leq \|u_{2}\|_{l^{\gamma_{4}}}, \quad \mbox{for } u_{1},u_{2} \in E_{m}. \end{aligned} $$
(3.28)

Note that \(\delta_{0}\in(0,1)\). Then, for all \(u\in K_{m}\), we have

$$\begin{aligned} \begin{aligned}[b] &\min\bigl\{ \lambda\eta_{1}(sR_{3})^{\gamma_{3}},\lambda \eta _{2}(sR_{4})^{\gamma_{4}}\bigr\} \bigl( \Vert u_{1} \Vert _{l^{2}}+ \Vert u_{2} \Vert _{l^{2}}\bigr)^{\max\{\gamma _{3},\gamma_{4}\}} \\ &\quad \le 2^{\max\{\gamma_{3},\gamma_{4}\}}\min\bigl\{ \lambda\eta_{1}(sR_{3})^{\gamma _{3}}, \lambda\eta_{2}(sR_{4})^{\gamma_{4}}\bigr\} \bigl( \Vert u_{1} \Vert _{l^{2}}^{\gamma_{3}}+ \Vert u_{2} \Vert _{l^{2}}^{\gamma_{4}}\bigr) \\ &\quad \le 2^{\max\{\gamma_{3},\gamma_{4}\}} \bigl[\lambda\eta_{1}(sR_{3})^{\gamma _{3}} \Vert u_{1} \Vert _{l^{2}}^{\gamma_{3}}+\lambda \eta_{2}(sR_{4})^{\gamma_{4}} \Vert u_{2} \Vert _{l^{2}}^{\gamma_{4}} \bigr]. \end{aligned} \end{aligned}$$
(3.29)

Note that \(F(n,0,0)=0\) for all \(n\in\mathbb {Z}\) and \(\lambda>0\). Then, by (\(\mathcal{A}_{1}\)), \((F_{2})'''\), \((f)''\), (2.7), (3.24), (3.26), (3.28), and (3.29), we have

$$\begin{aligned} \mathcal{J}(su) =&\sum_{n\in\mathbb {Z}}\rho_{1}(n) \Phi_{1}\bigl(\Delta su_{1}(n)\bigr)+\sum _{n\in\mathbb {Z}}\rho_{2}(n)\Phi_{2}\bigl(\Delta su_{2}(n)\bigr) \\ &{}+\sum_{n\in\mathbb {Z}}\rho_{3}(n) \Phi_{3}\bigl(su_{1}(n)\bigr)+\sum _{n\in\mathbb {Z}}\rho_{4}(n)\Phi_{4} \bigl(su_{2}(n)\bigr) \\ &{} -\lambda\sum_{n\in\mathbb {Z}}F\bigl(n,su_{1}(n),su_{2}(n) \bigr) \\ \leq&\overline{\rho_{1}}d_{1}s^{p}\sum _{n\in\mathbb {Z}} \bigl\vert \Delta u_{1}(n) \bigr\vert ^{p}+\overline{\rho_{2}}d_{2}s^{q}\sum _{n\in\mathbb {Z}} \bigl\vert \Delta u_{2}(n) \bigr\vert ^{q} \\ &{}+\overline{\rho_{3}}d_{3}s^{p}\sum _{n\in\mathbb {Z}} \bigl\vert u_{1}(n) \bigr\vert ^{p} \\ &{}+\overline{\rho_{4}}d_{4}s^{q}\sum _{n\in\mathbb {Z}} \bigl\vert u_{2}(n) \bigr\vert ^{q}-\lambda \sum_{i=1}^{m}F \bigl(n_{i},s\lambda_{i}u_{1}^{i}(n_{i}),s \lambda _{i}u_{2}^{i}(n_{i})\bigr) \\ \leq&\max\{\overline{\rho_{1}}d_{1},\overline{ \rho_{3}}d_{3}\}s^{p} \Vert u_{1} \Vert _{p}^{p}+\max\{\overline{\rho_{2}}d_{2}, \overline{\rho_{4}}d_{4}\}s^{q} \Vert u_{2} \Vert _{q}^{q} \\ &{}-\lambda\sum_{i=1}^{m}\bigl[ \eta_{1} \bigl\vert \lambda_{i}su_{1}^{i}(n_{i}) \bigr\vert ^{\gamma _{3}}+\eta_{2} \bigl\vert \lambda_{i}su_{2}^{i}(n_{i}) \bigr\vert ^{\gamma_{4}}\bigr] \\ = &\max\{\overline{\rho_{1}}d_{1},\overline{ \rho_{3}}d_{3}\}s^{p} \Vert u_{1} \Vert _{p}^{p}+\max\{\overline{\rho_{2}}d_{2}, \overline{\rho_{4}}d_{4}\}s^{q} \Vert u_{2} \Vert _{q}^{q} \\ &{}-\lambda\eta_{1}s^{\gamma_{3}}\sum _{i=1}^{m} \vert \lambda_{i} \vert ^{\gamma _{3}} \bigl\vert u_{1}^{i}(n_{i}) \bigr\vert ^{\gamma_{3}} -\lambda\eta_{2}s^{\gamma_{4}}\sum _{i=1}^{m} \vert \lambda_{i} \vert ^{\gamma _{4}} \bigl\vert u_{2}^{i}(n_{i}) \bigr\vert ^{\gamma_{4}} \\ =&\max\{\overline{\rho_{1}}d_{1},\overline{ \rho_{3}}d_{3}\}s^{p} \Vert u_{1} \Vert _{p}^{p}+\max\{\overline{\rho_{2}}d_{2}, \overline{\rho_{4}}d_{4}\}s^{q} \Vert u_{2} \Vert _{q}^{q} \\ &{}-\lambda\eta_{1}s^{\gamma_{3}} \Vert u_{1} \Vert _{l^{\gamma_{3}}}^{\gamma _{3}}-\lambda\eta_{2}s^{\gamma_{4}} \Vert u_{2} \Vert _{l^{\gamma_{4}}}^{\gamma _{4}} \\ \leq&\max\{\overline{\rho_{1}}d_{1},\overline{ \rho_{3}}d_{3}\}(sR_{1})^{p} \Vert u_{1} \Vert _{l^{2}}^{p}+\max\{\overline{ \rho_{2}}d_{2},\overline{\rho_{4}}d_{4} \} (sR_{2})^{q} \Vert u_{2} \Vert _{l^{2}}^{q} \\ &{}-\lambda\eta_{1}(s R_{3})^{\gamma_{3}} \Vert u_{1} \Vert _{l^{2}}^{\gamma_{3}}-\lambda \eta_{2}(s R_{4})^{\gamma_{4}} \Vert u_{2} \Vert _{l^{2}}^{\gamma_{4}} \\ \leq&\max\{\overline{\rho_{1}}d_{1},\overline{ \rho_{3}}d_{3}\}(sR_{1})^{p} \Vert u_{1} \Vert _{l^{2}}^{p}+\max\{\overline{ \rho_{2}}d_{2},\overline{\rho_{4}}d_{4} \} (sR_{2})^{q} \Vert u_{2} \Vert _{l^{2}}^{q} \\ &{}-\frac{1}{ 2^{\max\{\gamma_{3},\gamma_{4}\}}}\min\bigl\{ \lambda\eta _{1}(sR_{3})^{\gamma_{3}}, \lambda\eta_{2}(sR_{4})^{\gamma_{4}}\bigr\} \\ &{}\cdot\bigl( \Vert u_{1} \Vert _{l^{2}}+ \Vert u_{2} \Vert _{l^{2}}\bigr)^{\max\{\gamma_{3},\gamma_{4}\} } \\ \leq&\max\{\overline{\rho_{1}}d_{1},\overline{ \rho_{3}}d_{3}\} (sR_{1})^{p} \delta_{0}^{p}+\max\{\overline{\rho_{2}}d_{2}, \overline{\rho _{4}}d_{4}\}(sR_{2})^{q} \delta_{0}^{q} \\ &{}-\frac{\lambda}{ 2^{\max\{\gamma_{3},\gamma_{4}\}}}\min\bigl\{ \eta _{1}(sR_{3})^{\gamma_{3}}, \eta_{2}(sR_{4})^{\gamma_{4}}\bigr\} \delta_{0}^{\max\{\gamma _{3},\gamma_{4}\}} \\ \leq&\max\bigl\{ \overline{\rho_{1}}d_{1}R_{1}^{p} \delta_{0}^{p},\overline{\rho _{3}}d_{3}R_{1}^{p} \delta_{0}^{p},\overline{\rho_{2}}d_{2}R_{2}^{q} \delta _{0}^{q},\overline{\rho_{4}}d_{4}R_{2}^{q} \delta_{0}^{q}\bigr\} s^{\min\{p,q\}} \\ &{}-\frac{\lambda}{ 2^{\max\{\gamma_{3},\gamma_{4}\}}}\min\bigl\{ \eta _{1}(sR_{3})^{\gamma_{3}}, \eta_{2}(sR_{4})^{\gamma_{4}}\bigr\} \delta_{0}^{\max\{\gamma _{3},\gamma_{4}\}}, \end{aligned}$$
(3.30)

for all \(u=(u_{1},u_{2})^{\tau}\in K_{m}\) and \(0< s<\min\{1, \delta_{0}(\sum_{i=1}^{m}|\lambda_{i}|)^{-1}\}\). Note that \(\gamma_{3},\gamma_{4}\in(1,\min \{p,q\})\). Then (3.30) implies that, for any given \(\lambda>0\), there exist sufficiently small \(s_{0\lambda}\in(0,1)\) and \(\varepsilon>0\) such that

$$ \mathcal{J}(s_{0\lambda}u)< -\varepsilon,\quad \forall u\in K_{m}. $$
(3.31)

Let

$$ K_{m}^{s_{0\lambda}}=\{s_{0\lambda}u:u\in K_{m}\} $$

and

$$ S^{m-1} = \Biggl\{ (\lambda_{1}, \lambda_{2},\ldots,\lambda_{m})^{\tau}\in R^{m}:\sum_{i=1}^{m} \lambda_{i}^{2}=1 \Biggr\} . $$
(3.32)

Then

$$ K_{m}^{s_{0\lambda}}= \Biggl\{ \sum _{i=1}^{m}\lambda_{i}u^{i}:\sum _{i=1}^{m}\lambda_{i}^{2}= \frac{s_{0\lambda}^{2}\delta_{0}^{2}}{4} \Biggr\} . $$
(3.33)

Define the map \(\psi:K_{m}^{s_{0\lambda}}\rightarrow S^{m-1}\) by

$$ \psi(u)=\frac{4}{s_{0\lambda}^{2}\delta_{0}^{2}}(\lambda_{1},\lambda _{2},\ldots,\lambda_{m})^{\tau},\quad \forall u\in K_{m}^{s_{0\lambda}}. $$
(3.34)

It is easy to verify that \(\psi:K_{m}^{s_{0\lambda}}\rightarrow S^{m-1}\) is an odd homeomorphic map. On the other hand, by (3.31), we have

$$ \mathcal{J}(u)< -\varepsilon,\quad \mbox{for } u\in K_{m}^{s_{0\lambda}}, $$
(3.35)

and so \(\sup_{K_{m}^{s_{0\lambda}}}\mathcal{J}\leq-\varepsilon<0\). Therefore, by Lemma 2.4, \(\mathcal{J}\) has at least m distinct pairs of critical points, so system (1.1) possesses at least m distinct pairs of nontrivial homoclinic solutions. The proof is complete. □

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Acknowledgements

This project is supported by the National Natural Science Foundation of China (No: 11301235) and the Tianyuan Fund for Mathematics of the National Natural Science Foundation of China (No: 11226135).

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Zhang, X., Zong, C., Deng, H. et al. Existence and multiplicity of homoclinic solutions for difference systems involving classical \((\phi_{1},\phi_{2})\)-Laplacian and a parameter. Adv Differ Equ 2017, 380 (2017). https://doi.org/10.1186/s13662-017-1419-4

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Keywords

  • difference systems
  • classical \((\phi_{1},\phi_{2})\)-Laplacian
  • homoclinic solutions
  • variational method