Open Access

Homoclinic solutions for a class of nonlinear difference systems with classical \((\phi_{1}, \phi_{2})\)-Laplacian

Advances in Difference Equations20152015:149

https://doi.org/10.1186/s13662-015-0467-x

Received: 4 December 2014

Accepted: 10 April 2015

Published: 8 May 2015

Abstract

In this paper, we consider the existence of homoclinic solutions for a class of nonlinear difference systems involving classical \((\phi_{1}, \phi_{2})\)-Laplacian. First, we improve some inequalities in known literature. Then, by using the variational method, some new existence results are obtained. Finally, some examples are given to verify our results.

Keywords

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

MSC

37J45 58E50 34C25

1 Introduction and main results

Let \(\mathbb{R}\) denote the real numbers and \(\mathbb{Z}\) the integers. Given \(a< b \) in \(\mathbb{Z}\). Let \(\mathbb{Z}[a,b]=\{a,a+1,\ldots,b\}\). Let \(T>1\) and N be fixed positive integers.

In this paper, we investigate the existence of homoclinic solutions for the following nonlinear difference systems involving classical \((\phi _{1},\phi_{2})\)-Laplacian:
$$ \left \{ \begin{array}{l} \Delta\phi_{1} (\Delta u_{1}(t-1) )+\nabla_{u_{1}} V (t,u_{1}(t),u_{2}(t) )=f_{1}(t), \\ \Delta\phi_{2} (\Delta u_{2}(t-1) )+\nabla_{u_{2}} V (t,u_{1}(t),u_{2}(t) )=f_{2}(t), \end{array} \right . $$
(1.1)
where \(t\in\mathbb{Z}\), \(u_{m}(t)\in\mathbb{R}^{N}\), \(m=1,2\), \(V(t,x_{1},x_{2})=-K(t,x_{1},x_{2})+W(t,x_{1},x_{2})\), \(K,W:\mathbb{Z}\times \mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}\) and \(\phi_{m}\), \(m=1,2\), satisfy the following condition:
(\(\mathcal{A}0\)): 

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

Remark 1.1

Assumption (\(\mathcal{A}0\)) is given in [1], which is used to characterize the classical homeomorphism. If, furthermore, \(\Phi_{m}:\mathbb{R}^{N}\rightarrow\mathbb{R}\) is coercive (i.e., \(\Phi_{m}(x)\rightarrow+\infty\) as \(|x|\rightarrow\infty\)), there exists \(\delta_{m}>0\) such that
$$ \Phi_{m}(x)\geq\delta_{m}\bigl(\vert x \vert -1\bigr),\quad x\in\mathbb{R}^{N}, $$
(1.2)
where \(\delta_{m}=\min_{|x|=1}\Phi_{m}(x)\), \(m=1,2\) (see [1]).

We call \(u=(u_{1},u_{2})\) a nontrivial homoclinic solution of system (1.1) if u satisfies system (1.1), \(u\neq0\) and \(u(t)\to0\) as \(t\to \infty\).

It is well known that the variational method has become an important tool to study the existence and multiplicity of solutions for various difference systems. Lots of contributions have been obtained (for example, see [120]). It is remarkable that, to the best of our knowledge, few people investigated system (1.1). Recently, in [1] and [2], by using the variational approach, J Mawhin investigated the following second order nonlinear difference systems with ϕ-Laplacian:
$$ \Delta\phi\bigl[\Delta u(n-1)\bigr]=\nabla_{u} F \bigl[n,u(n)\bigr]+h(n) \quad (n\in \mathbb{Z}), $$
(1.3)
where \(\phi=\nabla\Phi\), Φ strictly convex, is a homeomorphism of \(\mathbb{R}^{N}\) onto the ball \(B_{a}\subset\mathbb{R}^{N}\) or of \(B_{a}\) onto \(\mathbb{R}^{N}\). By using the variational approach, under different conditions, the author obtained that system (1.3) has at least one or \(N+1\) geometrically distinct T-periodic solutions. It is interesting that J Mawhin considered three kinds of ϕ: (1) \(\phi:\mathbb{R}^{N}\to\mathbb{R}^{N}\) is a classical homeomorphism, for example, \(\phi(x)=|x|^{p-1}x\) for some \(p>1\) and all \(x\in\mathbb {R}^{N}\); (2) \(\phi:\mathbb{R}^{N}\to B_{a}\) (\(a<+\infty\)) is a bounded homeomorphism, for example, \(\phi(x)=\frac{x}{\sqrt{1+|x|^{2}}}\in B_{1}\) for all \(x\in\mathbb{R}^{N}\); (3) \(\phi: B_{a}\subset\mathbb {R}^{N}\to\mathbb{R}^{N}\) is a singular homeomorphism, for example, \(\phi(x)=\frac{x}{\sqrt{1-|x|^{2}}}\) for all \(x\in B_{1}\). Recently, in [17], we generalized some results in [2] for classical homeomorphism and bounded homeomorphism to system (1.1), which seem to be the first results for system (1.1).
In 2011, He and Chen [16] investigated the existence of homoclinic solutions for the following discrete p-Laplacian systems:
$$ \Delta\bigl(\bigl\vert \Delta u(t-1)\bigr\vert ^{p-2} \Delta u(t-1)\bigr)=\nabla F\bigl(t,u(t)\bigr)+f(t),\quad t\in\mathbb{Z}, u\in \mathbb{R}^{N}, $$
(1.4)
where \(p>1\). They obtained homoclinic orbits as the limit of the subharmonics for system (1.4).
In this paper, motivated by [1, 2, 15, 16] and [17], we first improve some inequalities in [16] and then investigate the existence of homoclinic solutions for system (1.1) with classical homeomorphism. Next we make the following assumption:
(\(\mathcal{A}1\)): 
Let \(p>1\). Assume that there exist positive constants \(d_{1}\), \(d_{2}\), \(d_{3}\), \(d_{4}\) such that
$$ d_{1}|x|^{p}\le\Phi_{1}(x)\leq d_{3}|x|^{p},\qquad d_{2}|y|^{p}\le \Phi_{2}(y)\leq d_{4}|y|^{p}, \quad \forall x,y\in \mathbb{R}^{N} $$
and
$$ \bigl(\phi_{1}(x),x\bigr)\le p\Phi_{1}(x),\qquad \bigl( \phi_{2}(y),y\bigr)\le p\Phi_{2}(y),\quad \forall x,y\in \mathbb{R}^{N}. $$
For every \(s\in\mathbb{N}\), define
$$l^{s}= \Biggl\{ g:\mathbb{Z}\to\mathbb{R}^{N},\sum _{t=-\infty}^{+\infty} \bigl\vert g(t)\bigr\vert ^{s}< \infty \Biggr\} $$
with the norm
$$\|g\|_{l^{s}}= \Biggl(\sum_{t=-\infty}^{+\infty} \bigl\vert g(t)\bigr\vert ^{s} \Biggr)^{1/s}. $$
Let \(p'>1\) be such that \(\frac{1}{p}+\frac{1}{p'}=1\) and
$$C_{*}= \biggl((2T)^{-p'/p}+\min \biggl\{ \frac {(T+1)^{p'+1}+T^{p'+1}-2}{(2T)^{p'}(p'+1)}, \frac{T}{2^{p'/p}} \biggr\} \biggr)^{1/p'}. $$
Next, we present our main results.

Theorem 1.1

Assume that (\(\mathcal{A}1\)) holds, \(f_{i}\neq0\), \(i=1,2\), W and K satisfy the following conditions:
(\(\mathcal{V}\)): 

\(V(t,x_{1},x_{2})=-K(t,x_{1},x_{2})+W(t,x_{1},x_{2})\), where \(K,W:\mathbb{Z}\times\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb {R}\), \(K(t,x_{1},x_{2})\) and \(W(t,x_{1},x_{2})\) are T-periodic and for every \(t\in\mathbb{Z}\), \(K,W\in C^{1}(\mathbb{Z}\times\mathbb{R}^{N}\times \mathbb{R}^{N},\mathbb{R})\);

(H1): 
there exist \(\gamma\in(1,p)\) and \(a_{1},a_{2}>0\) such that
$$K(t,x_{1},x_{2})\ge a_{1}|x_{1}|^{\gamma}+a_{2}|x_{2}|^{\gamma} \quad \textit{for all } (t,x_{1},x_{2})\in \mathbb{Z}[0,T-1] \times\mathbb{R}^{N}\times\mathbb{R}^{N}; $$
(H2): 
\(K(t,0,0)\equiv0\) and
$$\begin{aligned}& \bigl(x_{1},\nabla_{x_{1}} K(t,x_{1},x_{2}) \bigr)+\bigl(x_{2},\nabla_{x_{2}} K(t,x_{1},x_{2}) \bigr) \\& \quad \le p K(t,x_{1},x_{2})\quad \textit{for all } (t,x_{1},x_{2})\in \mathbb{Z}[0,T-1]\times \mathbb{R}^{N}\times\mathbb{R}^{N}; \end{aligned}$$
(H3): 
  1. (i)
    there exist \(r\in(0,1]\), \(0< b_{1}<a_{1}C_{*}^{\gamma-p}\), and \(0< b_{2}<a_{2}C_{*}^{\gamma-p}\) such that
    $$\begin{aligned}& W(t,x_{1},x_{2}) \le b_{1}|x_{1}|^{p}+b_{2}|x_{2}|^{p}, \\& \quad \forall t\in\mathbb {Z}[0,T-1], |x_{1}|\le rC_{*},|x_{2}| \le rC_{*}; \end{aligned}$$
    (1.5)
     
  2. (ii)

    there exist \(r>1\), \(0< b_{1}<a_{1}(C_{*}r)^{\gamma-p}\), and \(0< b_{2}<a_{2}(C_{*}r)^{\gamma-p}\) such that (1.5) holds;

     
(H4): 
$$\lim_{|x_{1}|+|x_{2}|\to+\infty}\frac {W(t,x_{1},x_{2})}{|x_{1}|^{p}+|x_{2}|^{p}}>d_{3}+d_{4}+2^{p-1}A_{0} \quad \textit{for all } t\in\mathbb{Z}[0,T-1], $$
where
$$A_{0}=\max_{|x_{1}|\le1,|x_{2}|\le1,t\in\mathbb{Z}[0,T-1]}K(t,x_{1},x_{2}); $$
(H5): 
there exist positive constants ξ, \(\eta_{1}\), \(\eta_{2}\) and \(\nu \in[0,\gamma-1)\) such that
$$\begin{aligned} 0 \le& \biggl( p+\frac{1}{\xi+\eta_{1}|x_{1}|^{\nu}+\eta_{2}|x_{2}|^{\nu }} \biggr)W(t,x_{1},x_{2}) \\ \le& \bigl(\nabla_{x_{1}} W(t,x_{1},x_{2}),x_{1} \bigr)+\bigl(\nabla _{x_{2}} W(t,x_{1},x_{2}),x_{2} \bigr) \end{aligned}$$
for all \((t,x_{1},x_{2})\in\mathbb{Z}[0,T-1]\times\mathbb{R}^{N}\times \mathbb{R}^{N}\);
(H6): 
\(f_{1},f_{2}\in l^{p'}\cap l^{\frac{p-\nu}{p-\nu-1}}\) and
  1. (i)
    when \(r\in (0,1]\),
    $$\begin{aligned}& \max\bigl\{ \|f_{1}\|_{l^{p'}},\|f_{2}\|_{l^{p'}}\bigr\} \\& \quad < \frac{1}{2^{p-1}} \min \bigl\{ d_{1},d_{2}, a_{1}C_{*}^{\gamma-p}-b_{1},a_{2}C_{*}^{\gamma -p}-b_{2} \bigr\} r^{p-1}; \end{aligned}$$
     
  2. (ii)
    when \(r\in (1,+\infty)\),
    $$\begin{aligned}& \max\bigl\{ \|f_{1}\|_{l^{p'}},\|f_{2}\|_{l^{p'}}\bigr\} \\& \quad < \frac{1}{2^{p-1}} \min \bigl\{ d_{1},d_{2}, a_{1} (C_{*}r )^{\gamma-p}-b_{1},a_{2} (C_{*}r )^{\gamma-p}-b_{2} \bigr\} r^{p-1}. \end{aligned}$$
     
Then system (1.1) possesses a nontrivial homoclinic solution.

Theorem 1.2

Assume that (\(\mathcal{A}1\)) holds, \(f_{i}\neq0\), \(i=1,2\), W and K satisfy (\(\mathcal{V}\)), (H1)-(H5) and the following conditions:
(H6)′: 
\(f_{1},f_{2}\in l^{1}\) and
  1. (i)
    when \(r\in (0,1]\),
    $$\begin{aligned}& \max\bigl\{ \|f_{1}\|_{l^{1}},\|f_{2}\|_{l^{1}}\bigr\} \\& \quad < \frac{1}{2^{p-1}C_{*}} \min \bigl\{ d_{1},d_{2}, a_{1}C_{*}^{\gamma-p}-b_{1},a_{2}C_{*}^{\gamma -p}-b_{2} \bigr\} r^{p-1}; \end{aligned}$$
     
  2. (ii)
    when \(r\in (1,+\infty)\),
    $$\begin{aligned}& \max\bigl\{ \|f_{1}\|_{l^{1}},\|f_{2}\|_{l^{1}}\bigr\} \\& \quad < \frac{1}{2^{p-1}C_{*}} \min \bigl\{ d_{1},d_{2}, a_{1} (C_{*}r )^{\gamma-p}-b_{1},a_{2} (C_{*}r )^{\gamma-p}-b_{2} \bigr\} r^{p-1}. \end{aligned}$$
     
Then system (1.1) possesses a nontrivial homoclinic solution.

Theorem 1.3

Assume that (\(\mathcal{A}1\)) holds, \(f_{i}\neq0\), \(i=1,2\), W and K satisfy (\(\mathcal{V}\)), (H2), (H4), (H5) and the following conditions:
(H1)′: 
there exist \(a_{1}, a_{2}>0\) such that
$$K(t,x_{1},x_{2})\ge a_{1}|x_{1}|^{p}+a_{2}|x_{2}|^{p} \quad \textit{for all } (t,x_{1},x_{2})\in \mathbb{Z}[0,T-1] \times\mathbb{R}^{N}\times\mathbb{R}^{N}; $$
(H3)′: 
there exist \(r>0\) and \(0< b_{1}<a_{1}\), \(0< b_{2}<a_{2}\) such that
$$W(t,x_{1},x_{2})\le b_{1}|x_{1}|^{p}+b_{2}|x_{2}|^{p}, \quad \forall |x_{1}|\le rC_{*},|x_{2}|\le rC_{*} ; $$
(H6)″: 
\(f_{1},f_{2}\in l^{p'}\cap l^{\frac{p-\nu}{p-\nu-1}}\) and
$$\max\bigl\{ \|f_{1}\|_{l^{p'}},\|f_{2}\|_{l^{p'}}\bigr\} < \frac{1}{2^{p-1}} \min \{d_{1},d_{2}, a_{1}-b_{1},a_{2}-b_{2} \}r^{p-1}. $$
Then system (1.1) possesses a nontrivial homoclinic solution.

Theorem 1.4

Assume that (\(\mathcal{A}1\)) holds, \(f_{i}\neq0\), \(i=1,2\), W and K satisfy (\(\mathcal{V}\)), (H1)′, (H2), (H3)′, (H4), (H5) and the following condition:
(H6)‴: 
\(f_{1},f_{2}\in l^{1}\) and
$$\max\bigl\{ \|f_{1}\|_{l^{1}},\|f_{2}\|_{l^{1}}\bigr\} < \frac{1}{2^{p-1}C_{*}} \min \{d_{1},d_{2}, a_{1}-b_{1},a_{2}-b_{2} \}r^{p-1}. $$
Then system (1.1) possesses a nontrivial homoclinic solution.

Remark 1.2

Theorem 1.3 and Theorem 1.4 show that \(f_{1}\), \(f_{2}\) can be large when r is large.

2 Preliminaries

Similar to [15] and [16], we will obtain the homoclinic orbit of system (1.1) as a limit of solutions of a sequence of difference systems:
$$ \left \{ \begin{array}{l} \Delta\phi_{1} (\Delta u_{1}(t-1) )+\nabla_{u_{1}} V (t,u_{1}(t),u_{2}(t) )=f_{1,k}(t), \\ \Delta\phi_{2} (\Delta u_{2}(t-1) )+\nabla_{u_{2}} V (t,u_{1}(t),u_{2}(t) )=f_{2,k}(t), \end{array} \right . $$
(2.1)
where \(f_{m,k}:\mathbb{Z}\to\mathbb{R}^{N}\) is a \(2kT\)-periodic extension of restriction of \(f_{m}\) to the interval \(\mathbb{Z}[-kT,kT-1]\), \(k\in\mathbb{N}\), \(m=1,2\).
Next, we present some basic notations. We use \(|\cdot|\) to denote the usual Euclidean norm in \(\mathbb{R}^{N}\). Define
$$\begin{aligned} \mathcal{V} =&\bigl\{ u=(u_{1},u_{2})^{\tau}=\bigl\{ u(t)\bigr\} |u(t)= \bigl(u_{1}(t),u_{2}(t) \bigr)^{\tau}\in\mathbb{R}^{2N}, \\ &u_{m}=\bigl\{ u_{m}(t)\bigr\} ,u_{m}(t)\in \mathbb{R}^{N},m=1,2,t\in \mathbb{Z}\bigr\} . \end{aligned}$$
\(\mathcal{H}\) is defined as a subspace of \(\mathcal{V}\) by
$$\mathcal{H}_{k}=\bigl\{ u=\bigl\{ u(t)\bigr\} \in\mathcal{V} |u(t+2kT)=u(t),t\in\mathbb {Z}\bigr\} . $$
Define
$$ \mathcal{H}_{m,k}=\bigl\{ u_{m}=\bigl\{ u_{m}(t) \bigr\} |u_{m}(t+2kT)=u_{m}(t),u_{m}(t)\in \mathbb{R}^{N}, t\in\mathbb{Z}\bigr\} ,\quad m=1,2. $$
Then \(\mathcal{H}_{k}=\mathcal{H}_{1,k}\times\mathcal{H}_{2,k}\). For \(u_{m}\in\mathcal{H}_{m,k}\), set
$$\|u_{m}\|_{s,k}= \Biggl(\sum_{t =-kT}^{kT-1} \bigl\vert u_{m}(t)\bigr\vert ^{s} \Biggr)^{1/s}, \quad m=1,2,s>1. $$
Moreover, \(l_{2kT}^{\infty}\) denote the space of all bounded real functions on \(\mathbb{Z}[-kT,kT-1]\) endowed with the norm
$$\|u_{m}\|_{l_{2kT}^{\infty}}=\max_{t\in\mathbb{Z}[-kT,kT-1]}\bigl\vert u_{m}(t)\bigr\vert ,\quad m=1,2. $$
For \(1< p<+\infty\), on \(\mathcal{H}_{m,k}\), we define
$$ \|u_{m}\|_{\mathcal{H}_{m,k}}= \Biggl(\sum_{t=-kT}^{kT-1}\bigl| \Delta u_{m}(t)\bigr|^{p}+\sum_{t=-kT}^{kT-1}\bigl|u_{m}(t)\bigr|^{p} \Biggr)^{1/p}, \quad m=1,2. $$
For \(u=(u_{1},u_{2})^{\tau}\in\mathcal{H}_{k}\), define
$$\|u\|_{\mathcal{H}_{k}}=\|u_{1}\|_{\mathcal{H}_{1,k}}+\|u_{2} \|_{\mathcal {H}_{2,k}}. $$
Then \((\mathcal{H}_{k}, \|u\|_{\mathcal{H}_{k}} )\), \((\mathcal{H}_{1,k}, \|u\|_{\mathcal{H}_{1,k}} )\) and \((\mathcal{H}_{2,k}, \|u\|_{\mathcal{H}_{2,k}} )\) are reflexive Banach spaces.

Lemma 2.1

Let \(a,b\in\mathbb{Z}\), \(a\ge1\), \(b\ge0\), \(\varrho>1\), \(u_{m}\in\mathcal{H}_{m,k}\), \(m=1,2\). Then, for every \(t\in\mathbb{Z}\),
$$\begin{aligned} \bigl\vert u_{m}(t)\bigr\vert \le& (a+b+1)^{-1/\varrho} \Biggl(\sum_{s=t-a}^{t+b} \bigl\vert u_{m}(s)\bigr\vert ^{\varrho} \Biggr)^{1/\varrho} \\ &{} +\min \biggl\{ \frac {[(a+1)^{p'+1}+(b+1)^{p'+1}-2]^{1/p'}}{(a+b+1)^{p'}(p'+1)^{1/p'}},\frac {\max\{a,b\}}{(a+b+1)^{1/p}} \biggr\} \\ &{}\cdot \Biggl(\sum _{s=t-a}^{t+b}\bigl\vert \Delta {u}_{m}(s)\bigr\vert ^{p} \Biggr)^{1/p}, \end{aligned}$$
(2.2)
where \(m=1,2\).

Proof

Fix \(t\in\mathbb{Z}\). For every \(\tau\in\mathbb {Z}[t-a,t-1]\), we have
$$ u_{m}(t) = u_{m}(\tau)+\sum _{s=\tau}^{t-1}\Delta u_{m}(s) $$
(2.3)
and for every \(\tau\in\mathbb{Z}[t, t+b]\),
$$ u_{m}(t) = u_{m}(\tau)-\sum _{s=t}^{\tau-1}\Delta u_{m}(s). $$
(2.4)
Summing (2.3) over \(\mathbb{Z}[t-a,t-1]\) and (2.4) over \(\mathbb{Z}[t, t+b]\), we have
$$\begin{aligned} au_{m}(t) =& \sum_{\tau=t-a}^{t-1}u_{m}( \tau)+\sum_{\tau =t-a}^{t-1}\sum _{s=\tau}^{t-1}\Delta u_{m}(s) \\ =&\sum _{\tau =t-a}^{t-1}u_{m}(\tau)+\sum _{s=t-a}^{t-1}(s-t+a+1)\Delta u_{m}(s) \end{aligned}$$
(2.5)
and
$$\begin{aligned} (b+1)u_{m}(t) =&\sum_{\tau=t}^{t+b} u_{m}(\tau)-\sum_{\tau =t}^{t+b} \sum _{s=t}^{\tau-1}\Delta u_{m}(s) \\ =&\sum _{\tau=t}^{t+b} u_{m}(\tau)-\sum _{s=t}^{t+b-1}(t+b-s)\Delta u_{m}(s). \end{aligned}$$
(2.6)
Set
$$\phi(s)=\left \{ \begin{array}{l@{\quad}l} s-t+a+1, & t-a\le s\le t-1, \\ t+b-s, & t\le s\le t+b. \end{array} \right . $$
Combining (2.5) with (2.6) and using Hölder’s inequality, we obtain
$$\begin{aligned}& (a+b+1)\bigl\vert u_{m}(t)\bigr\vert \\& \quad = \Biggl\vert \sum_{\tau=t-a}^{t+b} u_{m}(\tau)+\sum_{s=t-a}^{t-1}(s-t+a+1) \Delta u_{m}(s)-\sum_{s=t}^{t+b-1}(t+b-s) \Delta u_{m}(s)\Biggr\vert \\& \quad \le \sum_{\tau=t-a}^{t+b}\bigl\vert u_{m}(\tau)\bigr\vert +\sum_{s=t-a}^{t-1}(s-t+a+1) \bigl\vert \Delta u_{m}(s)\bigr\vert +\sum _{s=t}^{t+b-1}(t+b-s)\bigl\vert \Delta u_{m}(s)\bigr\vert \\& \quad = \sum_{\tau=t-a}^{t+b}\bigl\vert u_{m}(\tau)\bigr\vert +\sum_{s=t-a}^{t+b-1} \phi (s)\bigl\vert \Delta u_{m}(s)\bigr\vert = \sum_{\tau=t-a}^{t+b}\bigl\vert u_{m}(\tau)\bigr\vert +\sum_{s=t-a}^{t+b} \phi (s)\bigl\vert \Delta u_{m}(s)\bigr\vert \\& \quad \le (a+b+1)^{(\varrho-1)/\varrho} \Biggl(\sum_{\tau=t-a}^{t+b} \bigl\vert u_{m}(\tau)\bigr\vert ^{\varrho} \Biggr)^{1/\varrho}+ \Biggl(\sum_{s=t-a}^{t+b} \bigl[\phi(s)\bigr]^{p'} \Biggr)^{1/p'} \Biggl(\sum _{s=t-a}^{t+b}\bigl\vert \Delta{u}_{m}(s) \bigr\vert ^{p} \Biggr)^{1/p} \\& \quad = (a+b+1)^{(\varrho-1)/\varrho} \Biggl(\sum_{\tau=t-a}^{t+b} \bigl\vert u_{m}(\tau)\bigr\vert ^{\varrho} \Biggr)^{1/\varrho} \\& \qquad {}+ \Biggl(\sum_{s=t-a}^{t-1}(s-t+a+1)^{p'}+ \sum_{s=t}^{t+b}(t+b-s)^{p'} \Biggr)^{1/p'} \Biggl(\sum_{s=t-a}^{t+b} \bigl\vert \Delta{u}_{m}(s)\bigr\vert ^{q} \Biggr)^{1/q}. \end{aligned}$$
(2.7)
Since
$$ \begin{aligned} &\sum_{s=t-a}^{t-1}(s-t+a+1)^{p'}= \sum_{s=1}^{a}s^{p'}< \frac{(a+1)^{p'+1}-1}{p'+1}, \\ &\sum_{s=t}^{t+b}(t+b-s)^{p'}= \sum_{k=1}^{b}k^{p'}< \frac{(b+1)^{p'+1}-1}{p'+1} \end{aligned} $$
(2.8)
and
$$\begin{aligned}& \sum_{s=t-a}^{t-1}(s-t+a+1)^{p'}+ \sum_{s=t}^{t+b}(t+b-s)^{p'} \\& \quad \le \sum_{s=t-a}^{t-1}a^{p'}+ \sum_{s=t}^{t+b}b^{p'} \le \sum _{s=t-a}^{t+b} \max\bigl\{ a^{p'},b^{p'} \bigr\} = \max\bigl\{ a^{p'},b^{p'}\bigr\} (b+a+1). \end{aligned}$$
(2.9)
Equation (2.7) implies that
$$\begin{aligned}& (a+b+1)\bigl\vert u_{m}(t)\bigr\vert \\& \quad \le (a+b+1)^{(\varrho-1)/\varrho} \Biggl(\sum_{\tau=t-a}^{t+b} \bigl\vert u_{m}(\tau)\bigr\vert ^{\varrho} \Biggr)^{1/\varrho} \\& \qquad {}+ \biggl(\min \biggl\{ \frac{(a+1)^{p'+1}+(b+1)^{p'+1}-2}{p'+1},\max \bigl\{ a^{p'},b^{p'}\bigr\} (b+a+1) \biggr\} \biggr)^{1/p'} \\& \qquad {}\cdot \Biggl(\sum_{s=t-a}^{t+b}\bigl\vert \Delta{u}_{m}(s)\bigr\vert ^{p} \Biggr)^{1/p}, \end{aligned}$$
which implies that (2.2) holds. Thus the proof is complete. □

Corollary 2.1

Let \(u_{m}\in\mathcal{H}_{m,k}\), \(m=1,2\). Then
$$\begin{aligned} \|u_{m}\|_{l_{2kT}^{\infty}} \le& (2T)^{-1/\varrho} \Biggl(\sum_{s=-kT}^{kT-1}\bigl\vert u_{m}(s)\bigr\vert ^{\varrho} \Biggr)^{1/\varrho} \\ &{} +\min \biggl\{ \frac {[(T+1)^{p'+1}+T^{p'+1}-2]^{1/p'}}{(2T)^{p'}(p'+1)^{1/p'}},\frac {T^{1/p'}}{2^{1/p}} \biggr\} \Biggl(\sum _{s=-kT}^{kT-1}\bigl\vert \Delta {u}_{m}(s)\bigr\vert ^{p} \Biggr)^{1/p}, \end{aligned}$$
(2.10)
where \(m=1,2\).

Proof

Obviously, there exists \(t^{*}\in\mathbb{Z}[-kT,kT-1]\) such that
$$\bigl\vert u_{m}\bigl(t^{*}\bigr)\bigr\vert =\|u_{m} \|_{l_{2kT}^{\infty}}=\max_{t\in\mathbb{Z}[-kT,kT-1]} \bigl\vert u_{m}(s) \bigr\vert . $$
In Lemma 2.1, let \(a=T\) and \(b=T-1\),
$$\begin{aligned} \bigl\vert u_{m}\bigl(t^{*}\bigr)\bigr\vert \le& (2T)^{-1/\varrho} \Biggl(\sum_{s=t^{*}-T}^{t^{*}+T-1} \bigl\vert u_{m}(s)\bigr\vert ^{\varrho} \Biggr)^{1/\varrho} \\ &{} +\min \biggl\{ \frac {[(T+1)^{p'+1}+T^{p'+1}-2]^{1/p'}}{(2T)^{p'}(p'+1)^{1/p'}},\frac {T}{(2T)^{1/p}} \biggr\} \Biggl(\sum _{s=t^{*}-T}^{t^{*}+T-1}\bigl\vert \Delta {u}_{m}(s)\bigr\vert ^{p} \Biggr)^{1/p} \\ \le& (2T)^{-1/\varrho} \Biggl(\sum_{s=t^{*}-kT}^{t^{*}+kT-1} \bigl\vert u_{m}(s)\bigr\vert ^{\varrho} \Biggr)^{1/\varrho} \\ &{} +\min \biggl\{ \frac {[(T+1)^{p'+1}+T^{p'+1}-2]^{1/p'}}{(2T)^{p'}(p'+1)^{1/p'}},\frac {T}{(2T)^{1/p}} \biggr\} \Biggl(\sum _{s=t^{*}-kT}^{t^{*}+kT-1}\bigl\vert \Delta {u}_{m}(s)\bigr\vert ^{p} \Biggr)^{1/p} \\ = & (2T)^{-1/\varrho} \Biggl(\sum_{s=-kT}^{kT-1} \bigl\vert u_{m}(s)\bigr\vert ^{\varrho } \Biggr)^{1/\varrho} \\ &{} +\min \biggl\{ \frac {[(T+1)^{p'+1}+T^{p'+1}-2]^{1/p'}}{(2T)^{p'}(p'+1)^{1/p'}},\frac {T^{1/p'}}{2^{1/p}} \biggr\} \Biggl(\sum _{s=-kT}^{kT-1}\bigl\vert \Delta {u}_{m}(s)\bigr\vert ^{p} \Biggr)^{1/p}. \end{aligned}$$
The proof is complete. □

Corollary 2.2

Let \(u_{m}\in\mathcal{H}_{m,k}\), \(m=1,2\). Then
$$\begin{aligned} \|u_{m}\|_{l_{2kT}^{\infty}} \le & \biggl((2T)^{-p'/p}+\min \biggl\{ \frac {(T+1)^{p'+1}+T^{p'+1}-2}{(2T)^{p^{\prime2}}(p'+1)^{1/p'}},\frac {T}{2^{p'/p}} \biggr\} \biggr)^{1/p'} \\ &{} \cdot \Biggl(\sum_{s=-kT}^{kT-1}\bigl\vert u_{m}(s)\bigr\vert ^{p}+\sum _{s=-kT}^{kT-1}\bigl\vert \Delta{u}_{m}(s) \bigr\vert ^{p} \Biggr)^{1/p}, \end{aligned}$$
(2.11)
where \(m=1,2\).

Proof

In Corollary 2.1, let \(\varrho=p\) and then use Hölder’s inequality. Then the proof is completed easily. □

Remark 2.1

As \(a\ge1\), Lemma 2.1, Corollary 2.1, and Corollary 2.2 improve Lemma 3.1, Corollary 3.1, and Corollary 3.2 in [16], respectively.

By Lemma 2.3 and Lemma 2.4 in [17], we have the following two lemmas.

Lemma 2.2

(see [17])

For any \(u=(u_{1},u_{2}),v=(v_{1},v_{2})\in\mathcal{H}_{k} \), the following two equalities hold:
$$\begin{aligned}& -\sum_{t = -kT}^{kT-1}\bigl( \Delta\phi_{1}\bigl(\Delta u_{1}(t-1)\bigr),v_{1}(t) \bigr)=\sum_{t = -kT}^{kT-1}\bigl(\Delta \phi_{1}\bigl(\Delta u_{1}(t)\bigr),\Delta v_{1}(t) \bigr), \end{aligned}$$
(2.12)
$$\begin{aligned}& -\sum_{t = -kT}^{kT-1}\bigl( \Delta\phi_{2}\bigl(\Delta u_{2}(t-1)\bigr),v_{2}(t) \bigr)=\sum_{t = -kT}^{kT-1}\bigl(\Delta \phi_{2}\bigl(\Delta u_{2}(t)\bigr),\Delta v_{2}(t) \bigr). \end{aligned}$$
(2.13)

Lemma 2.3

(see [17])

Let \(L:\mathbb{Z}[-kT,kT-1]\times\mathbb{R}^{N}\times\mathbb {R}^{N}\times\mathbb{R}^{N}\times\mathbb{R}^{N}\rightarrow \mathbb{R}\), \((t,x_{1},x_{2},y_{1},y_{2})\rightarrow L(t,x_{1},x_{2},y_{1},y_{2})\) and assume that L is continuously differential in \((x_{1},x_{2},y_{1},y_{2})\) for all \(t\in\mathbb{Z}[-kT,kT-1]\). Then the function \(\varphi_{k}:\mathcal{H}_{k}\to\mathbb{R} \) defined by
$$\varphi_{k}(u)= \varphi_{k}(u_{1},u_{2})= \sum_{t=-kT}^{kT-1}L \bigl(t,u_{1}(t),u_{2}(t), \Delta u_{1}(t),\Delta u_{2}(t) \bigr) $$
is continuously differentiable on \(\mathcal{H}_{k}\) and
$$\begin{aligned} \bigl\langle \varphi_{k}'(u),v\bigr\rangle = & \bigl\langle \varphi_{k}'(u_{1},u_{2}),(v_{1},v_{2}) \bigr\rangle \\ = & \sum_{t=-kT}^{kT-1}\bigl[ \bigl(D_{x_{1}}L\bigl(t,u_{1}(t),u_{2}(t),\Delta u_{1}(t),\Delta u_{2}(t)\bigr),v_{1}(t) \bigr) \\ &{}+ \bigl(D_{y_{1}}L\bigl(t,u_{1}(t),u_{2}(t), \Delta u_{1}(t),\Delta u_{2}(t)\bigr),\Delta v_{1}(t) \bigr) \\ &{} + \bigl(D_{x_{2}}L\bigl(t,u_{1}(t),u_{2}(t), \Delta u_{1}(t),\Delta u_{2}(t)\bigr),v_{2}(t) \bigr) \\ &{} + \bigl(D_{y_{2}}L\bigl(t,u_{1}(t),u_{2}(t), \Delta u_{1}(t),\Delta u_{2}(t)\bigr),\Delta v_{2}(t) \bigr)\bigr], \end{aligned}$$
where \(u,v\in\mathcal{H}_{k}\).
Let
$$ L(t,x_{1},x_{2},y_{1},y_{2}) = \Phi_{1}(y_{1})+\Phi_{2}(y_{2}) +K(t,x_{1},x_{2})-W(t,x_{1},x_{2})+ \bigl(f_{1,k}(t),x_{1}\bigr)+\bigl(f_{2,k}(t),x_{2} \bigr) $$
and define \(\eta_{k}:\mathcal{H}_{k}\to[0,+\infty)\) by
$$ \eta_{k}(u)=\eta_{k}(u_{1},u_{2})=\sum _{t=-kT}^{kT-1}\bigl[\Phi_{1}\bigl( \Delta u_{1}(t)\bigr)+\Phi_{2}\bigl(\Delta u_{2}(t) \bigr)+K\bigl(t,u_{1}(t),u_{2}(t)\bigr)\bigr]. $$
Then
$$\begin{aligned} \varphi_{k}(u) = & \varphi_{k}(u_{1},u_{2}) \\ = & \sum_{t=-kT}^{kT-1}\bigl[ \Phi_{1}\bigl(\Delta u_{1}(t)\bigr)+\Phi_{2}\bigl( \Delta u_{2}(t)\bigr)+K\bigl(t,u_{1}(t),u_{2}(t) \bigr) \\ & {}-W\bigl(t,u_{1}(t),u_{2}(t)\bigr)+ \bigl(f_{1,k}(t),u_{1}(t)\bigr)+\bigl(f_{2,k}(t),u_{2}(t) \bigr)\bigr] \\ = & \eta_{k}(u)+\sum_{t=-kT}^{kT-1} \bigl[-W\bigl(t,u_{1}(t),u_{2}(t)\bigr)+\bigl(f_{1,k}(t),u_{1}(t) \bigr)+\bigl(f_{2,k}(t),u_{2}(t)\bigr)\bigr]. \end{aligned}$$
(2.14)
It follows from (\(\mathcal{A}0\)), (\(\mathcal{V}\)) and Lemma 2.3 that
$$\begin{aligned} \bigl\langle \varphi_{k}'(u),v\bigr\rangle = & \bigl\langle \varphi_{k}'(u_{1},u_{2}),(v_{1},v_{2}) \bigr\rangle \\ = & \sum_{t=-kT}^{kT-1}\bigl[\bigl( \phi_{1}\bigl(\Delta u_{1}(t)\bigr),\Delta v_{1}(t) \bigr)+\bigl(\phi _{2}\bigl(\Delta u_{2}(t)\bigr),\Delta v_{2}(t)\bigr) \\ &{} +\bigl(\nabla_{u_{1}} K\bigl(t,u_{1}(t),u_{2}(t) \bigr),v_{1}(t)\bigr)+\bigl(\nabla _{u_{2}}K \bigl(t,u_{1}(t),u_{2}(t)\bigr),v_{2}(t)\bigr) \\ &{} -\bigl(\nabla_{u_{1}} W\bigl(t,u_{1}(t),u_{2}(t) \bigr),v_{1}(t)\bigr)-\bigl(\nabla _{u_{2}}W \bigl(t,u_{1}(t),u_{2}(t)\bigr),v_{2}(t)\bigr) \\ &{} +\bigl(f_{1,k}(t),v_{1}(t)\bigr)+\bigl(f_{2,k}(t),v_{2}(t) \bigr)\bigr], \quad \forall u,v\in\mathcal{H}_{k}. \end{aligned}$$
(2.15)
By Lemma 2.2, it is easy to see that critical points of \(\varphi_{k}\) in \(\mathcal{H}_{k}\) are \(2kT\)-periodic solutions of system (2.1).
We shall use one linking method in [21] to obtain the critical points of φ (the details can be seen in [21]). Let \((E,\|\cdot\| )\) be a Banach space. Define a continuous map \(\Gamma:[0,1]\times E\to E\) by \(\Gamma(t,x)=\Gamma (t)x\), where \(\Gamma(t)\) satisfies the following conditions:
  1. (1)

    \(\Gamma(0)=I\), the identity map.

     
  2. (2)

    For each \(t\in[0,1)\), \(\Gamma(t)\) is a homeomorphism of E onto E and \(\Gamma^{-1}(t)\in C(E\times[0,1),E)\).

     
  3. (3)

    \(\Gamma(1)E\) is a single point in E and \(\Gamma(t)A\) converges uniformly to \(\Gamma(1)E\) as \(t\to1\) for each bounded set \(A\subset E\).

     
  4. (4)
    For each \(t_{0}\in[0,1)\) and each bounded set \(A\subset E\),
    $$\sup_{\substack{0\le t\le t_{0}\\ u\in A}}\bigl\{ \bigl\Vert \Gamma(t)u\bigr\Vert +\bigl\Vert \Gamma^{-1}(t)u\bigr\Vert \bigr\} < \infty. $$
     
Let Φ be the set of all continuous maps Γ as defined above.

Definition 2.1

(see [21], Definition 3.2)

We say that A links \(B[\mathrm{hm}]\) if A and B are subsets of E such that \(A\cap B=\emptyset\), and for each \(\Gamma\in\Phi\), there is \(t'\in(0,1]\) such that \(\Gamma(t')A\cap B\neq\emptyset\).

Example 1

(see [21], p.21)

Let B be an open set in E, and let A consist of two points \(e_{1}\), \(e_{2}\) with \(e_{1}\in B\) and \(e_{2}\notin\bar{B}\). Then A links \(\partial B[\mathrm{hm}]\).

We use the following theorem to prove our main results.

Theorem 2.1

(see [21], Theorem 3.4 and Theorem 2.12)

Let E be a Banach space, \(\varphi\in C^{1}(E, \mathbb{R})\) and A and B be two subsets of E such that A links \(B[\mathrm{hm}]\). Assume that
$$\sup_{A} \varphi\le\inf_{B}\varphi $$
and
$$c:=\inf_{\Gamma\in\Phi}\sup_{\substack{s\in[0,1]\\ u\in A}} \varphi\bigl( \Gamma (s)u\bigr)< \infty. $$
Let \(\psi(t)\) be a positive, nonincreasing, locally Lipschitz continuous function on \([0,\infty)\) satisfying \(\int_{0}^{\infty}\psi (r)\, dr=\infty\). Then there exists a sequence \(\{u_{n}\}\subset E\) such that \(\varphi(u_{n})\to c \) and \(\varphi'(u_{n})/\psi(\|u_{n}\|)\to 0\), as \(n\to\infty\). Moreover, if \(c=\sup_{A} \varphi\), then there is a sequence \(\{u_{n}\}\subset E\) satisfying \(\varphi(u_{n})\to c\), \(\varphi'(u_{n})\to0\), and \(d(u_{n},B)\to0\), as \(n\to\infty\).

Remark 2.2

Since A links B, by Definition 2.1, it is easy to know that \(c\ge\inf_{B}\varphi\). By [21], if we let \(\psi(r)=\frac{1}{1+r}\), the sequence \(\{u_{n}\} \) is the Cerami sequence that is \(\{u_{n}\}\) satisfying
$$\varphi(u_{n})\to c,\qquad \bigl(1+\Vert u_{n}\Vert \bigr)\bigl\Vert \varphi'(u_{n})\bigr\Vert \to0, \quad \mbox{as } n\to \infty. $$

3 Proofs

Lemma 3.1

Suppose that (H2) holds. Then
$$\begin{aligned}& K(t,x_{1},x_{2})\le2^{p-1}K \biggl(t, \frac{x_{1}}{|x|},\frac {x_{2}}{|x|} \biggr) \bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr) \quad \textit{for all } t\in\mathbb {Z}[0,T-1], |x|\ge 1; \\& K(t,x_{1},x_{2})\ge\frac{1}{2}K \biggl(t, \frac{x_{1}}{|x|},\frac {x_{2}}{|x|} \biggr) \bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr) \quad \textit{for all } t\in\mathbb {Z}[0,T-1], |x|\le 1. \end{aligned}$$

Proof

Define the function \(\xi\in(0,+\infty)\to K(t,\xi^{-1}x_{1},\xi^{-1}x_{2})(\xi^{p}+\xi^{p})\). Then we have
$$\begin{aligned}& \bigl(K\bigl(t,\xi^{-1}x_{1},\xi^{-1}x_{2} \bigr) \bigl(\xi^{p}+\xi^{p}\bigr)\bigr)_{\xi}' \\& \quad = - \bigl(\nabla_{x_{1}} K\bigl(t,\xi^{-1}x_{1}, \xi^{-1}x_{2}\bigr),\xi ^{-2}x_{1} \bigr) \bigl(\xi^{p}+\xi^{p}\bigr) \\& \qquad {}- \bigl(\nabla_{x_{2}} K \bigl(t,\xi ^{-1}x_{1},\xi^{-1}x_{2} \bigr),\xi^{-2}x_{2} \bigr) \bigl(\xi^{p}+ \xi^{p}\bigr) +K\bigl(t,\xi^{-1}x_{1},\xi^{-1}x_{2} \bigr) \bigl(p\xi^{p-1}+p\xi^{p-1}\bigr) \\& \quad \ge - \bigl(\nabla_{x_{1}} K\bigl(t,\xi^{-1}x_{1}, \xi^{-1}x_{2}\bigr),\xi ^{-1}x_{1} \bigr) \bigl(\xi^{p-1}+\xi^{p-1}\bigr) \\& \qquad {}- \bigl(\nabla_{x_{2}} K \bigl(t,\xi ^{-1}x_{1},\xi^{-1}x_{2} \bigr),\xi^{-1}x_{2} \bigr) \bigl(\xi^{p-1}+ \xi^{p-1}\bigr) +pK\bigl(t,\xi^{-1}x_{1},\xi^{-1}x_{2} \bigr) \bigl(\xi^{p-1}+\xi^{p-1}\bigr) \\& \quad \ge 0. \end{aligned}$$
Hence the function \(\xi\in(0,+\infty)\to K(t,\xi^{-1}x_{1},\xi^{-1}x_{2})(\xi^{p}+\xi^{p})\) is nondecreasing. Moreover, note that
$$\frac{|x_{1}|^{p}+|x_{2}|^{p}}{2}\le|x|^{p}\le\bigl(\vert x_{1}\vert +|x_{2}|\bigr)^{p}\le 2^{p-1}\bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr). $$
Then the proof can be completed easily. □

Lemma 3.2

Suppose that (H1) holds. Then, for any \(u\in \mathcal{H}_{k}\),
$$\begin{aligned} \eta_{k}(u) \ge& \min\bigl\{ d_{1}\|u_{1} \|_{\mathcal {H}_{1,k}}^{p},a_{1}C_{*}^{\gamma-p} \|u_{1}\|_{\mathcal{H}_{1,k}}^{\gamma }\bigr\} \\ &{}+\min\bigl\{ d_{2}\|u_{2}\|_{\mathcal{H}_{1,k}}^{p},a_{2}C_{*}^{\gamma-p} \|u_{2}\| _{\mathcal{H}_{2,k}}^{\gamma}\bigr\} ,\quad \forall k\in \mathbb{N}. \end{aligned}$$

Proof

It follows from (\(\mathcal{A}1\)), (H1), \(\gamma\in (1,p)\) and Corollary 2.2 that
$$\begin{aligned} \eta_{k}(u) = & \sum_{t=-kT}^{kT-1} \bigl[\Phi_{1}\bigl(\Delta u_{1}(t)\bigr)+\Phi_{2} \bigl(\Delta u_{2}(t)\bigr)+K\bigl(t,u_{1}(t),u_{2}(t) \bigr)\bigr] \\ \ge& \sum_{t=-kT}^{kT-1} \bigl[d_{1}\bigl\vert \Delta u_{1}(t)\bigr\vert ^{p}+d_{2}\bigl\vert \Delta u_{2}(t)\bigr\vert ^{p}+a_{1}\bigl\vert u_{1}(t)\bigr\vert ^{\gamma}+a_{2}\bigl\vert u_{2}(t)\bigr\vert ^{\gamma}\bigr] \\ \ge& \sum_{t=-kT}^{kT-1} \bigl[d_{1}\bigl\vert \Delta u_{1}(t)\bigr\vert ^{p}+d_{2}\bigl\vert \Delta u_{2}(t)\bigr\vert ^{p}+a_{1}\|u_{1}\|_{l_{2kT}^{\infty}}^{\gamma-p} \bigl\vert u_{1}(t)\bigr\vert ^{p}+a_{2} \|u_{2}\| _{l_{2kT}^{\infty}}^{\gamma-p}\bigl\vert u_{2}(t) \bigr\vert ^{p} \bigr] \\ \ge& \sum_{t=-kT}^{kT-1} \bigl[d_{1}\bigl\vert \Delta u_{1}(t)\bigr\vert ^{p}+d_{2}\bigl\vert \Delta u_{2}(t)\bigr\vert ^{p}+ a_{1}\bigl(C_{*}\Vert u_{1} \Vert _{\mathcal{H}_{1,k}}\bigr)^{\gamma -p}\bigl\vert u_{1}(t)\bigr\vert ^{p} \\ &{}+ a_{2}\bigl(C_{*}\Vert u_{2}\Vert _{\mathcal{H}_{2,k}}\bigr)^{\gamma -p}\bigl\vert u_{2}(t)\bigr\vert ^{p} \bigr] \\ \ge& \min\bigl\{ d_{1},a_{1}\bigl(C_{*}\Vert u_{1}\Vert _{\mathcal{H}_{1,k}}\bigr)^{\gamma-p}\bigr\} \Vert u_{1}\Vert _{\mathcal{H}_{1,k}}^{p}+ \min\bigl\{ d_{2},a_{2}\bigl(C_{*}\Vert u_{2} \Vert _{\mathcal {H}_{1,k}}\bigr)^{\gamma-p}\bigr\} \Vert u_{2}\Vert _{\mathcal{H}_{2,k}}^{p} \\ = & \min\bigl\{ d_{1}\Vert u_{1}\Vert _{\mathcal{H}_{1,k}}^{p},a_{1}C_{*}^{\gamma-p} \Vert u_{1}\Vert _{\mathcal{H}_{1,k}}^{\gamma}\bigr\} +\min\bigl\{ d_{2}\Vert u_{2}\Vert _{\mathcal {H}_{2,k}}^{p},a_{2}C_{*}^{\gamma-p} \Vert u_{2}\Vert _{\mathcal{H}_{2,k}}^{\gamma }\bigr\} . \end{aligned}$$
 □

Proof of Theorem 1.1

We divide the proof into the following Lemmas 3.3-3.5.

Lemma 3.3

Under the assumptions of Theorem  1.1, for every \(k\in\mathbb{N}\), system (2.1) has a nontrivial solution \(u_{k}\) in \(\mathcal{H}_{k}\).

Proof

We first construct A and B which satisfy the assumptions in Theorem 2.1.

(i) When \(r\in(0,1]\), by Corollary 2.2, (H1), (H3)(i), Hölder’s inequality and \(\gamma< p\), for \(u\in\mathcal{H}_{k}\) with \(\|u\|_{\mathcal {H}_{k}}= r\), we have \(\|u_{1}\|_{l_{2kT}^{\infty}}\le C_{*}\|u_{1}\|_{\mathcal{H}_{1,k}}\le rC_{*} \) and \(\|u_{2}\|_{l_{2kT}^{\infty}}\le C_{*}\|u_{2}\|_{\mathcal{H}_{2,k}}\le rC_{*}\),
$$\begin{aligned} \varphi_{k}(u) \ge& \eta_{k}(u)-b_{1} \sum_{t=-kT}^{kT-1}\bigl\vert {u}_{1}(t)\bigr\vert ^{p}-b_{2}\sum _{t=-kT}^{kT-1}\bigl\vert {u}_{2}(t)\bigr\vert ^{p} \\ &{} - \Biggl(\sum_{t=-kT}^{kT-1}\bigl\vert f_{1,k}(t)\bigr\vert ^{p'} \Biggr)^{1/{p'}} \Biggl( \sum_{t=-kT}^{kT-1}\bigl\vert u_{1}(t)\bigr\vert ^{p} \Biggr)^{1/p} \\ &{}- \Biggl( \sum_{t=-kT}^{kT-1}\bigl\vert f_{2,k}(t)\bigr\vert ^{p'} \Biggr)^{1/p'} \Biggl( \sum_{t=-kT}^{kT-1}\bigl\vert u_{2}(t)\bigr\vert ^{p} \Biggr)^{1/p} \\ \ge& \sum_{t=-kT}^{kT-1} \bigl[d_{1}\bigl\vert \Delta u_{1}(t)\bigr\vert ^{p}+d_{2}\bigl\vert \Delta u_{2}(t)\bigr\vert ^{p}+a_{1}\bigl\vert u_{1}(t)\bigr\vert ^{\gamma}+a_{2}\bigl\vert u_{2}(t)\bigr\vert ^{\gamma}\bigr]-b_{1}\sum_{t=-kT}^{kT-1} \bigl\vert {u}_{1}(t)\bigr\vert ^{p} \\ & {}-b_{2}\sum_{t=-kT}^{kT-1}\bigl\vert {u}_{2}(t)\bigr\vert ^{p}- \Biggl(\sum _{t=-kT}^{kT-1}\bigl\vert f_{1,k}(t)\bigr\vert ^{p'} \Biggr)^{1/{p'}} \Biggl(\sum _{t=-kT}^{kT-1}\bigl\vert u_{1}(t)\bigr\vert ^{p} \Biggr)^{1/p} \\ &{} - \Biggl(\sum_{t=-kT}^{kT-1}\bigl\vert f_{2,k}(t)\bigr\vert ^{p'} \Biggr)^{1/{p'}} \Biggl( \sum_{t=-kT}^{kT-1}\bigl\vert u_{2}(t)\bigr\vert ^{p} \Biggr)^{1/p} \\ \ge& \sum_{t=-kT}^{kT-1}d_{1}\bigl\vert \Delta u_{1}(t)\bigr\vert ^{p}+\sum _{t=-kT}^{kT-1}d_{2}\bigl\vert \Delta u_{2}(t)\bigr\vert ^{p} \\ &{}+a_{1} \bigl(C_{*}\Vert u_{1}\Vert _{\mathcal {H}_{1,k}} \bigr)^{\gamma-p}\sum_{t=-kT}^{kT-1}\bigl\vert u_{1}(t)\bigr\vert ^{p} -b_{1}\sum _{t=-kT}^{kT-1}\bigl\vert {u}_{1}(t) \bigr\vert ^{p} \\ &{} -b_{2}\sum_{t=-kT}^{kT-1}\bigl\vert {u}_{2}(t)\bigr\vert ^{p}+a_{2} \bigl(C_{*}\Vert u_{2}\Vert _{\mathcal {H}_{2,k}} \bigr)^{\gamma-p}\sum_{t=-kT}^{kT-1}\bigl\vert u_{2}(t)\bigr\vert ^{p} \\ &{}-\Vert f_{1} \Vert _{l^{p'}}\Vert u_{1}\Vert _{\mathcal{H}_{1,k}}-\Vert f_{2}\Vert _{l^{p'}}\Vert u_{2}\Vert _{\mathcal {H}_{2,k}} \\ \ge& \min \bigl\{ d_{1},a_{1} (C_{*}r )^{\gamma-p}-b_{1} \bigr\} \Vert u_{1}\Vert _{\mathcal{H}_{1,k}}^{p}+\min \bigl\{ d_{2},a_{2} (C_{*}r )^{\gamma-p}-b_{2} \bigr\} \Vert u_{2}\Vert _{\mathcal {H}_{2,k}}^{p} \\ &{} -\Vert f_{1}\Vert _{l^{p'}}\Vert u_{1} \Vert _{\mathcal{H}_{1,k}}-\Vert f_{2}\Vert _{l^{p'}}\Vert u_{2}\Vert _{\mathcal{H}_{2,k}} \\ \ge& \min \bigl\{ d_{1},a_{1}C_{*}^{\gamma-p}-b_{1} \bigr\} \Vert u_{1}\Vert _{\mathcal{H}_{1,k}}^{p}+\min \bigl\{ d_{2},a_{2}C_{*}^{\gamma-p}-b_{2} \bigr\} \Vert u_{2}\Vert _{\mathcal{H}_{2,k}}^{p} \\ &{} -\Vert f_{1}\Vert _{l^{p'}}\Vert u_{1} \Vert _{\mathcal{H}_{1,k}}-\Vert f_{2}\Vert _{l^{p'}}\Vert u_{2}\Vert _{\mathcal{H}_{2,k}} \\ \ge& \min \bigl\{ d_{1},d_{2}, a_{1}C_{*}^{\gamma-p}-b_{1},a_{2}C_{*}^{\gamma -p}-b_{2} \bigr\} \frac{1}{2^{p-1}} \bigl(\Vert u_{1}\Vert _{\mathcal{H}_{1,k}}+ \Vert u_{2}\Vert _{\mathcal{H}_{2,k}} \bigr)^{p} \\ &{} -\max\bigl\{ \Vert f_{1}\Vert _{l^{p'}},\Vert f_{2}\Vert _{l^{p'}}\bigr\} \bigl(\Vert u_{1}\Vert _{\mathcal {H}_{1,k}}+\Vert u_{2}\Vert _{\mathcal{H}_{2,k}}\bigr). \end{aligned}$$
(3.1)
(H6)(i) implies that there exists \(\alpha>0\) such that
$$\varphi_{k}(u)\ge\alpha>0\quad \mbox{for all } u\in \mathcal{H}_{k} \mbox { with } \|u\|_{\mathcal{H}_{k}}=r, \forall k\in \mathbb{N}. $$
(ii) When \(r\in(1,+\infty)\), by Corollary 2.2, (H1), (H3)(ii), Hölder’s inequality and \(\gamma< p\), for \(u\in \mathcal{H}_{k}\) with \(\|u\|_{\mathcal{H}_{k}}= r\), we have
$$\begin{aligned} \varphi_{k}(u) \ge& \min \bigl\{ d_{1},a_{1} (C_{*}r )^{\gamma-p}-b_{1} \bigr\} \|u_{1} \|_{\mathcal{H}_{1,k}}^{p}+\min \bigl\{ d_{2},a_{2} (C_{*}r )^{\gamma-p}-b_{2} \bigr\} \|u_{2} \|_{\mathcal {H}_{2,k}}^{p} \\ &{} -\|f_{1}\|_{l^{p'}}\|u_{1}\|_{\mathcal{H}_{1,k}}- \|f_{2}\|_{l^{q'}}\| u_{2}\|_{\mathcal{H}_{2,k}} \\ \ge& \min \bigl\{ d_{1},d_{2}, a_{1} (C_{*}r )^{\gamma -p}-b_{1},a_{2} (C_{*}r )^{\gamma-p}-b_{2} \bigr\} \frac {1}{2^{p-1}} \bigl(\Vert u_{1}\Vert _{\mathcal{H}_{1,k}}+\|u_{2} \|_{\mathcal {H}_{2,k}} \bigr)^{p} \\ &{} -\max\bigl\{ \Vert f_{1}\Vert _{l^{p'}},\Vert f_{2}\Vert _{l^{p'}}\bigr\} \bigl(\Vert u_{1}\Vert _{\mathcal {H}_{1,k}}+\Vert u_{2}\Vert _{\mathcal{H}_{2,k}}\bigr). \end{aligned}$$
(3.2)
(H6)(ii) implies that there exists \(\alpha>0\) such that
$$\varphi_{k}(u)\ge\alpha>0 \quad \mbox{for all } u\in \mathcal{H}_{k} \mbox { with } \|u\|_{\mathcal{H}_{k}}=r, \forall k\in \mathbb{N}. $$
By Lemma 3.1 and the T-periodicity of K, there exists a constant \(B_{0}>0\) such that
$$ K(t,x_{1},x_{2})\le2^{p-1} A_{0}\bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr)+B_{0}\quad \mbox{for all } (t,x_{1},x_{2}) \in\mathbb{Z}\times\mathbb{R}^{N}\times\mathbb{R}^{N}, $$
(3.3)
where
$$A_{0}=\max_{|x_{1}|\le1, |x_{2}|\le1,t\in\mathbb{Z}[0,T-1]}K(t,x_{1},x_{2}). $$
By (H4), we know that there exist \(\varepsilon_{0}>0\) and \(L>0\) such that
$$\begin{aligned}& W(t,x_{1},x_{2})\ge \bigl(d_{3}+d_{4}+2^{p-1}A_{0}+ \varepsilon _{0} \bigr) \bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr) \\& \quad \mbox{for all } t\in\mathbb{Z}[0,T-1] \mbox{ and } \forall|x|\ge L. \end{aligned}$$
(3.4)
By (3.4) and the T-periodicity of W, there exists a constant \(B_{1}>0\) such that
$$ W(t,x_{1},x_{2})\ge \bigl(d_{3}+d_{4}+2^{p-1}A_{0}+ \varepsilon _{0} \bigr) \bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr)-B_{1} $$
(3.5)
for all \((t,x_{1},x_{2})\in\mathbb{Z}[0,T-1]\times\mathbb{R}^{N}\times \mathbb{R}^{N}\). For any \(k\in\mathbb{N}\), define \({w}^{(k)}\in \mathcal{H}_{k}\) by
$$ {w}^{(k)}(t)=\bigl({w}_{1}^{(k)}(t),{w}_{2}^{(k)}(t) \bigr)= \left \{ \begin{array}{l@{\quad}l} (1,0,\ldots,0,1,0,\ldots,0)& \mbox{if }t=0, \\ 0& \mbox{if } t\in\mathbb{Z}[-kT,kT-1]/\{0\}, \end{array} \right . $$
where
$$ {w}_{i}^{(k)}(t)= \left \{ \begin{array}{l@{\quad}l} (1,0,\ldots,0) &\mbox{if }t=0, \\ 0 &\mbox{if } t\in\mathbb{Z}[-kT,kT-1]/\{0\}, \end{array} \right .\quad i=1,2. $$
Since \(K(t,0,0)\equiv0\) and \(W(t,0,0)\equiv0\), which are implied by (H2) and (H5), then by (3.3) and (3.5) we have
$$\begin{aligned} \varphi_{k}\bigl(\xi{w}^{(k)}\bigr) = & \sum_{t=-kT}^{kT-1}\bigl[\Phi_{1} \bigl(\xi\Delta w_{1}^{(k)}(t)\bigr)+\Phi _{2}\bigl( \xi\Delta w_{2}^{(k)}(t)\bigr)+K\bigl(t,\xi w_{1}^{(k)}(t),\xi w_{2}^{(k)}(t)\bigr) \\ &{} -W\bigl(t,\xi w_{1}^{(k)}(t),\xi w_{2}^{(k)}(t) \bigr)+\xi \bigl(f_{1,k}(t),w_{1}^{(k)}(t)\bigr)+\xi \bigl(f_{2,k}(t),w_{2}^{(k)}(t)\bigr)\bigr] \\ \le& d_{3}\vert \xi \vert ^{p}\sum _{t=-kT}^{kT-1}\bigl\vert \Delta w_{1}^{(k)}(t) \bigr\vert ^{p}+d_{4}\vert \xi \vert ^{p} \sum_{t=-kT}^{kT-1}\bigl\vert \Delta w_{2}^{(k)}(t)\bigr\vert ^{p} \\ &{}+K\bigl(0,\xi w_{1}^{(k)}(0),\xi w_{2}^{(k)}(0)\bigr) -W\bigl(0,\xi w_{1}^{(k)}(0),\xi w_{2}^{(k)}(0) \bigr) \\ &{}+\xi \bigl(f_{1,k}(0),w_{1}^{(k)}(0)\bigr)+\xi \bigl(f_{2,k}(0),w_{2}^{(k)}(0)\bigr) \\ = & d_{3}\vert \xi \vert ^{p} \bigl(\bigl\vert \Delta w_{1}^{(k)}(-1)\bigr\vert ^{p}+\bigl\vert \Delta w_{1}^{(k)}(0)\bigr\vert ^{p} \bigr)+d_{4}\vert \xi \vert ^{p} \bigl(\bigl\vert \Delta w_{2}^{(k)}(-1)\bigr\vert ^{p}+\bigl\vert \Delta w_{2}^{(k)}(0)\bigr\vert ^{p} \bigr) \\ &{} +K\bigl(0,\xi w_{1}^{(k)}(0),\xi w_{2}^{(k)}(0) \bigr)-W\bigl(0,\xi w_{1}^{(k)}(0),\xi w_{2}^{(k)}(0) \bigr) \\ &{} +\xi\bigl(f_{1,k}(0),w_{1}^{(k)}(0)\bigr)+\xi \bigl(f_{2,k}(0),w_{2}^{(k)}(0)\bigr) \\ \le& 2d_{3}\vert \xi \vert ^{p}+2d_{4} \vert \xi \vert ^{p}+2^{p-1}A_{0}\vert \xi \vert ^{p}+2^{p-1}A_{0}\vert \xi \vert ^{p} \\ &{}+B_{0}- \bigl(d_{3}+d_{4}+2^{p-1}A_{0}+ \varepsilon _{0} \bigr) \bigl(\vert \xi \vert ^{p}+ \vert \xi \vert ^{p}\bigr) \\ &{} +B_{1}+\vert \xi \vert \bigl\vert f_{1,k}(0)\bigr\vert +\vert \xi \vert \bigl\vert f_{2,k}(0)\bigr\vert \\ \le& -2\varepsilon_{0}\vert \xi \vert ^{p} +\vert \xi \vert \bigl\vert f_{1,k}(0)\bigr\vert +\vert \xi \vert \bigl\vert f_{2,k}(0)\bigr\vert +B_{0}+B_{1}. \end{aligned}$$
(3.6)
So there exists \(\xi_{0}\in\mathbb{R}\) such that \(\|\xi_{0} {w}^{(k)}\| >r\) and \(\varphi_{k}(\xi_{0} {w}^{(k)})<0\). Moreover, it is clear that \(\varphi _{k}(0)=0\). Let \(e_{1}=\xi_{0} {w}^{(k)}\) and
$$ A=\{0,e_{1}\},\qquad B=\bigl\{ u\in\mathcal{H}_{k}:\Vert u \Vert _{\mathcal{H}_{k}}< r\bigr\} . $$
Then \(0\in B\) and \(e_{1}\notin\bar{B}\). So by Example 1 in Section 2, we know that A links \(\partial B[\mathrm{hm}]\). So by Theorem 2.1 and Remark 2.2, we have
$$ c_{k}=\inf_{\Gamma\in\Phi}\sup _{\substack{s\in[0,1]\\ u\in A}} \varphi_{k}\bigl(\Gamma(s)u\bigr)\ge\inf _{\partial B}\varphi_{k}>\alpha>0, $$
(3.7)
and there exists a sequence \(\{u_{n}=(u_{1}^{(n)},u_{2}^{(n)})\}_{n=1}^{\infty}\subset \mathcal{H}_{k}\) such that
$$ \varphi_{k}(u_{n})\to c_{k}, \qquad \bigl(1+ \Vert u_{n}\Vert _{\mathcal{H}_{k}} \bigr)\bigl\Vert \varphi_{k}'(u_{n})\bigr\Vert \to0. $$
Then there exists a constant \(C_{1k}>0\) such that
$$ \bigl\vert \varphi_{k}(u_{n})\bigr\vert \le C_{1k},\qquad \bigl(1+\Vert u_{n}\Vert _{\mathcal {H}_{k}} \bigr)\bigl\Vert \varphi_{k}'(u_{n}) \bigr\Vert \le C_{1k}\quad \mbox{for all } n\in \mathbb{N}. $$
(3.8)
It follows from (H5) and the T-periodicity and continuity of W, \(\nabla_{x_{1}} W\) and \(\nabla_{x_{2}} W\) that
$$\begin{aligned}& \bigl[\bigl(\nabla_{x_{1}} W(t,x_{1},x_{2}),x_{1} \bigr)+\bigl(\nabla_{x_{2}} W(t,x_{1},x_{2}),x_{2} \bigr)-pW(t,x_{1},x_{2})\bigr]\bigl(\zeta+ \eta_{1}|x_{1}|^{\nu}+\eta _{2}|x_{2}|^{\nu} \bigr) \\& \quad \ge W(t,x_{1},x_{2})\ge0,\quad \forall (t, x_{1},x_{2})\in\mathbb{Z}\times \mathbb{R}^{N} \times\mathbb{R}^{N}. \end{aligned}$$
(3.9)
So by (3.5) and \(p-\nu>1\), there exists \(C_{2}>0\) such that
$$\begin{aligned}& \bigl[\bigl(\nabla_{x_{1}} W(t,x_{1},x_{2}),x_{1} \bigr)+\bigl(\nabla_{x_{2}} W(t,x_{1},x_{2}),x_{2} \bigr)-pW(t,x_{1},x_{2})\bigr] \\& \quad \ge \frac{W(t,x_{1},x_{2})}{\zeta+\eta_{1}|x_{1}|^{\nu}+\eta _{2}|x_{2}|^{\nu}} \\& \quad \ge \frac{ (d_{3}+d_{4}+2^{p-1}A_{0}+\varepsilon_{0} )(|x_{1}|^{p}+|x_{2}|^{p})-B_{1}}{\zeta+\eta_{1}|x_{1}|^{\nu}+\eta_{2}|x_{2}|^{\nu }} \\& \quad \ge \frac{ (d_{3}+d_{4}+2^{p-1}A_{0}+\varepsilon_{0} )\frac {1}{2^{p-1}}(|x_{1}|+|x_{2}|)^{p}-B_{1}}{\zeta+2\max\{\eta_{1},\eta_{2}\} (|x_{1}|+|x_{2}|)^{\nu}} \\& \quad \ge \frac{ (d_{3}+d_{4}+2^{p-1}A_{0}+\varepsilon_{0} )\frac {1}{2^{p-1}}}{4\max\{\eta_{1},\eta_{2}\}}\bigl(\vert x_{1}\vert +|x_{2}|\bigr)^{p-\nu }-C_{2} \\& \quad \ge \frac{ (d_{3}+d_{4}+2^{p-1}A_{0}+\varepsilon_{0} )\frac {1}{2^{p-1}}}{4\max\{\eta_{1},\eta_{2}\}}\bigl(|x_{1}|^{p-\nu}+|x_{2}|^{p-\nu } \bigr)-C_{2},\quad \forall x\in \mathbb{R}^{N}. \end{aligned}$$
(3.10)
Hence, it follows from (H2), (3.8) and (3.10) that
$$\begin{aligned}& pC_{1k}+C_{1k} \\& \quad \ge p\varphi_{k}(u_{n})-\bigl\langle \varphi_{k}'(u_{n}), u_{n}\bigr\rangle \\& \quad = p\varphi_{k}\bigl(u_{1}^{(n)},u_{2}^{(n)} \bigr)-\bigl\langle \varphi _{k}'\bigl(u_{1}^{(n)},u_{2}^{(n)} \bigr), \bigl(u_{1}^{(n)},u_{2}^{(n)}\bigr) \bigr\rangle \\& \quad \ge \sum_{t=-kT}^{kT-1}\bigl[\bigl( \nabla_{u_{1}} W\bigl(t,u_{1}^{(n)}(t),u_{2}^{(n)}(t) \bigr),u_{1}^{(n)}(t)\bigr)+\bigl(\nabla_{u_{2}} W \bigl(t,u_{1}^{(n)}(t),u_{2}^{(n)}(t) \bigr),u_{2}^{(n)}(t)\bigr) \\& \qquad {}-pW\bigl(t,u_{1}^{(n)}(t),u_{2}^{(n)}(t) \bigr)\bigr] \\& \qquad {}+( p-1)\sum_{t=-kT}^{kT-1} \bigl(f_{1,k}(t),u_{1}^{(n)}(t)\bigr)+( p-1)\sum _{t=-kT}^{kT-1}\bigl(f_{2,k}(t),u_{2}^{(n)}(t) \bigr) \\& \quad \ge \frac{ (d_{3}+d_{4}+2^{p-1}A_{0}+\varepsilon_{0} )\frac {1}{2^{p-1}}}{4\max\{\eta_{1},\eta_{2}\}} \sum_{t=-kT}^{kT-1} \bigl(\bigl\vert u_{1}^{(n)}(t)\bigr\vert ^{p-\nu}+ \bigl\vert u_{2}^{(n)}(t)\bigr\vert ^{p-\nu } \bigr)-2kTC_{2} \\& \qquad {}-(p-1)\sum_{t=-kT}^{kT-1}\bigl\vert f_{1,k}(t)\bigr\vert \bigl\vert u_{1}^{(n)}(t) \bigr\vert -(p-1)\sum_{t=-kT}^{kT-1}\bigl\vert f_{2,k}(t)\bigr\vert \bigl\vert u_{2}^{(n)}(t) \bigr\vert \\& \quad \ge \frac{ (d_{3}+d_{4}+2^{p-1}A_{0}+\varepsilon_{0} )\frac {1}{2^{p-1}}}{4\max\{\eta_{1},\eta_{2}\}} \sum_{t=-kT}^{kT-1} \bigl(\bigl\vert u_{1}^{(n)}(t)\bigr\vert ^{p-\nu}+ \bigl\vert u_{2}^{(n)}(t)\bigr\vert ^{p-\nu } \bigr)-2kTC_{2} \\& \qquad {}-(p-1) \Biggl(\sum_{t=-kT}^{kT-1} \bigl\vert f_{1,k}(t)\bigr\vert ^{\frac {p-\nu}{p-\nu-1}} \Biggr)^{\frac{p-\nu-1}{p-\nu}} \Biggl(\sum_{t=-kT}^{kT-1}\bigl\vert u_{1}^{(n)}(t)\bigr\vert ^{p-\nu} \Biggr)^{1/(p-\nu)} \\& \qquad {}-(p-1) \Biggl(\sum_{t=-kT}^{kT-1} \bigl\vert f_{2,k}(t)\bigr\vert ^{\frac{p-\nu}{p-\nu -1}} \Biggr)^{\frac{p-\nu-1}{p-\nu}} \Biggl(\sum_{t=-kT}^{kT-1}\bigl\vert u_{2}^{(n)}(t)\bigr\vert ^{p-\nu} \Biggr)^{1/(p-\nu)}. \end{aligned}$$
(3.11)
The fact \(p-\nu>1\) and the above inequality show that \(\sum_{t=-kT}^{kT-1} |u_{1}^{(n)}(t)|^{p-\nu}\) and \(\sum_{t=-kT}^{kT-1} |u_{2}^{(n)}(t)|^{p-\nu}\) are bounded. By (\(\mathcal{A}1\)), (H1), (H6), (3.8), (3.9), (3.11), Hölder’s inequality and Corollary 2.2, we have
$$\begin{aligned}& d_{1}\bigl\Vert u_{1}^{(n)}\bigr\Vert _{\mathcal{H}_{1,k}}^{p}+d_{2}\bigl\Vert u_{2}^{(n)}\bigr\Vert _{\mathcal {H}_{2,k}}^{p} \\& \quad = d_{1}\sum_{t=-kT}^{kT-1} \bigl\vert \Delta u_{1}^{(n)}(t)\bigr\vert ^{p}+d_{1}\sum_{t=-kT}^{kT-1} \bigl\vert u_{1}^{(n)}(t)\bigr\vert ^{p}+d_{2} \sum_{t=-kT}^{kT-1} \bigl\vert \Delta u_{2}^{(n)}(t)\bigr\vert ^{p}+d_{2}\sum _{t=-kT}^{kT-1} \bigl\vert u_{2}^{(n)}(t)\bigr\vert ^{p} \\& \quad \le \varphi_{k}\bigl(u^{(n)}\bigr)-\sum _{t=-kT}^{kT-1} K\bigl(t,u_{1}^{(n)}(t),u_{2}^{(n)}(t) \bigr) \\& \qquad {}+\sum_{t=-kT}^{kT-1} W \bigl(t,u_{1}^{(n)}(t),u_{2}^{(n)}(t) \bigr)+d_{1}\sum_{t=-kT}^{kT-1} \bigl\vert u_{1}^{(n)}(t)\bigr\vert ^{p} \\& \qquad {}+d_{2}\sum_{t=-kT}^{kT-1} \bigl\vert u_{2}^{(n)}(t)\bigr\vert ^{p}-\sum _{t=-kT}^{kT-1}\bigl(f_{1k}(t),u_{1}^{(n)}(t) \bigr)-\sum_{t=-kT}^{kT-1}\bigl(f_{2k}(t),u_{2}^{(n)}(t) \bigr) \\& \quad \le \varphi_{k}\bigl(u^{(n)}\bigr) \\& \qquad {}+\sum_{t=-kT}^{kT-1} \bigl[\bigl( \nabla_{u_{1}} W\bigl(t,u_{1}^{(n)}(t), u_{2}^{(n)}(t)\bigr), u_{1}^{(n)}(t) \bigr)+\bigl(\nabla_{u_{2}} W\bigl(t,u_{1}^{(n)}(t),u_{2}^{(n)}(t) \bigr), u_{2}^{(n)}(t)\bigr) \\& \qquad {}-pW\bigl(t,u_{1}^{(n)}(t), u_{2}^{(n)}(t) \bigr)\bigr]\bigl(\zeta+\eta _{1}\bigl\vert u_{1}^{(n)}(t) \bigr\vert ^{\nu}+\eta_{2}\bigl\vert u_{2}^{(n)}(t) \bigr\vert ^{\nu}\bigr) \\& \qquad {}+d_{1}\sum_{t=-kT}^{kT-1} \bigl\vert u_{1}^{(n)}(t)\bigr\vert ^{p}+d_{2} \sum_{t=-kT}^{kT-1} \bigl\vert u_{2}^{(n)}(t)\bigr\vert ^{p}+ \Biggl(\sum _{t=-kT}^{kT-1}\bigl\vert u_{1}^{(n)}(t) \bigr\vert ^{p} \Biggr)^{\frac{1}{p}} \biggl(\sum _{t\in\mathbb{Z}} \bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{\frac{1}{p'}} \\& \qquad {}+ \Biggl(\sum_{t=-kT}^{kT-1}\bigl\vert u_{2}^{(n)}(t)\bigr\vert ^{p} \Biggr)^{\frac {1}{p}} \biggl(\sum_{t\in\mathbb{Z}} \bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{\frac {1}{p'}} \\& \quad \le C_{1k}+d_{1}\sum_{t=-kT}^{kT-1} \bigl\vert u_{1}^{(n)}(t)\bigr\vert ^{p}+d_{2} \sum_{t=-kT}^{kT-1} \bigl\vert u_{2}^{(n)}(t)\bigr\vert ^{p}+\bigl\Vert u_{1}^{(n)}\bigr\Vert _{\mathcal {H}_{1,k}} \biggl(\sum _{t\in\mathbb{Z}} \bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{\frac {1}{p'}} \\& \qquad {}+\bigl\Vert u_{2}^{(n)}\bigr\Vert _{\mathcal{H}_{2,k}} \biggl(\sum_{t\in\mathbb{Z}} \bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{\frac{1}{p'}} \\& \qquad {}+ \bigl(\zeta+\eta_{1}\bigl\Vert u_{1}^{(n)} \bigr\Vert _{l_{2kT}^{\infty}}^{\nu}+\eta_{2}\bigl\Vert u_{2}^{(n)}\bigr\Vert _{l_{2kT}^{\infty}}^{\nu}\bigr) \\& \qquad {}\cdot\sum_{t=-kT}^{kT-1}\bigl[\bigl( \nabla_{u_{1}} W\bigl(t,u_{1}^{(n)}(t),u_{2}^{(n)}(t) \bigr),u_{1}^{(n)}(t)\bigr)+\bigl(\nabla_{u_{2}} W \bigl(t,u_{1}^{(n)}(t), u_{2}^{(n)}(t) \bigr),u_{2}^{(n)}(t)\bigr) \\& \qquad {}-pW\bigl(t,u_{1}^{(n)}(t),u_{2}^{(n)}(t) \bigr)\bigr] \\& \quad \le C_{1k}+d_{1}\bigl\Vert u_{1}^{(n)} \bigr\Vert _{l_{2kT}^{\infty}}^{\nu}\sum_{t=-kT}^{kT-1} \bigl\vert u_{1}^{(n)}(t)\bigr\vert ^{p-\nu}+d_{2} \bigl\Vert u_{2}^{(n)}\bigr\Vert _{l_{2kT}^{\infty}}^{\nu} \sum_{t=-kT}^{kT-1} \bigl\vert u_{2}^{(n)}(t)\bigr\vert ^{p-\nu } \\& \qquad {}+\bigl\Vert u_{1}^{(n)}\bigr\Vert _{\mathcal{H}_{1,k}} \biggl(\sum_{t\in\mathbb{Z}} \bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{\frac{1}{p'}}+\bigl\Vert u_{2}^{(n)}\bigr\Vert _{\mathcal {H}_{2,k}} \biggl(\sum _{t\in\mathbb{Z}} \bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{\frac {1}{p'}} \\& \qquad {}+\bigl(\zeta+\eta_{1}\bigl\Vert u_{1}^{(n)} \bigr\Vert _{l_{2kT}^{\infty}}^{\nu}+\eta_{2}\bigl\Vert u_{2}^{(n)}\bigr\Vert _{l_{2kT}^{\infty}}^{\nu}\bigr) \\& \qquad {}\cdot \biggl[(p+1)C_{1k}+(p-1)\bigl\Vert u_{1}^{(n)}\bigr\Vert _{\mathcal {H}_{1,k}} \biggl(\sum _{t\in\mathbb{Z}} \bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{\frac{1}{p'}} \\& \qquad {}+(p-1)\bigl\Vert u_{2}^{(n)}\bigr\Vert _{\mathcal{H}_{2,k}} \biggl(\sum _{t\in\mathbb{Z}} \bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{\frac{1}{p'}} \biggr] \\& \quad \le C_{1k}+d_{1}C_{*}^{\nu}\bigl\Vert u_{1}^{(n)}\bigr\Vert _{\mathcal{H}_{1,k}}^{\nu } \sum_{t=-kT}^{kT-1} \bigl\vert u_{1}^{(n)}(t)\bigr\vert ^{p-\nu}+d_{2}C_{*}^{\nu} \bigl\Vert u_{2}^{(n)}\bigr\Vert _{\mathcal{H}_{2,k}}^{\nu} \sum_{t=-kT}^{kT-1} \bigl\vert u_{2}^{(n)}(t)\bigr\vert ^{p-\nu} \\& \qquad {}+\bigl\Vert u_{1}^{(n)}\bigr\Vert _{\mathcal{H}_{1,k}} \biggl(\sum_{t\in\mathbb{Z}} \bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{\frac{1}{p'}}+\bigl\Vert u_{2}^{(n)}\bigr\Vert _{\mathcal {H}_{2,k}} \biggl(\sum _{t\in\mathbb{Z}} \bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{\frac {1}{p'}} \\& \qquad {}+\bigl(\zeta+\eta_{1}C_{*}^{\nu}\bigl\Vert u_{1}^{(n)}\bigr\Vert _{\mathcal{H}_{1,k}}^{\nu}+\eta_{2}C_{*}^{\nu}\bigl\Vert u_{2}^{(n)}\bigr\Vert _{\mathcal{H}_{2,k}}^{\nu}\bigr) \\& \qquad {}\cdot\biggl[(p+1)C_{1k}+(p-1)\bigl\Vert u_{1}^{(n)}\bigr\Vert _{\mathcal {H}_{1,k}} \biggl(\sum _{t\in\mathbb{Z}} \bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{\frac {1}{p'}} \\& \qquad {}+(p-1)\bigl\Vert u_{2}^{(n)}\bigr\Vert _{\mathcal{H}_{2,k}} \biggl(\sum_{t\in \mathbb{Z}} \bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{\frac{1}{p'}}\biggr]. \end{aligned}$$
(3.12)
Since \(\nu< p-1\), (3.12) and the boundedness of \(\sum_{t=-kT}^{kT-1} |u_{1}^{(n)}(t)|^{p-\nu}\) and \(\sum_{t=-kT}^{kT-1} |u_{2}^{(n)}(t)|^{p-\nu}\) imply that \(\|u_{1}^{(n)}\|_{\mathcal {H}_{1,k}}\) and \(\|u_{2}^{(n)}\|_{\mathcal{H}_{2,k}}\) are bounded. Since \(\mathcal{H}\) is a finite-dimensional space, \(\{ u^{(n)}=(u_{1}^{(n)},u_{2}^{(n)})\}\) has a convergence subsequence, still denoted by \(\{u^{(n)}=(u_{1}^{(n)},u_{2}^{(n)})\}\), such that \(u^{(n)}=(u_{1}^{(n)},u_{2}^{(n)})\to u_{k}=(u_{1k},u_{2k})\) as \(n\to\infty \). Moreover, by the continuity of \(\varphi_{k}\) and \(\varphi_{k}'\), we obtain \(\varphi_{k}'(u_{k})=0\) and \(\varphi_{k}(u_{k})=c_{k}>0\). It is clear that \(u_{k}\neq0\) and so \(u_{k}\) is a desired nontrivial solution of system (2.1). The proof is complete. □

Lemma 3.4

Let \(\{u_{k}=(u_{1k}, u_{2k})\}_{k\in\mathbb {N}}\) be the solutions of system (2.1). Then there exists \(M_{1}>0\) such that \(\|u_{1k}\|_{l_{2kT}^{\infty}}\le M_{1}\) and \(\|u_{2k}\| _{l_{2kT}^{\infty}}\le M_{1}\).

Proof

First, we prove that the sequence \(\{c_{k}\}_{k\in \mathbb{N}}\) is bounded. For every \(k\in\mathbb{N}\), define \(\Gamma _{k}:[0,1]\times\mathcal{H}_{k}\to\mathcal{H}_{k}\) by
$$\Gamma_{k}(s)v=(1-s)v, \quad v\in\mathcal{H}_{k}. $$
Then \(\Gamma\in\Phi\). Note that the set \(A=\{0,e_{1}\}\). So (3.7) and the argument of (3.6) imply that
$$\begin{aligned} \varphi_{k}(u_{k})=c_{k} \le& \sup_{s\in[0,1] u\in A} \varphi_{k}\bigl((1-s)u\bigr) \\ = & \sup_{s\in[0,1]} \varphi_{k}\bigl((1-s)e_{1} \bigr) \\ = & \sup_{s\in[0,1]} \varphi_{k}\bigl((1-s)e_{1} \bigr) \\ = & \sup_{s\in[0,1]}\Biggl\{ \sum_{t=-kT}^{kT-1} \bigl[\Phi _{1}\bigl((1-s)\Delta w_{1}^{(k)}(t) \bigr)+\Phi_{2}\bigl((1-s)\Delta w_{2}^{(k)}(t)\bigr) \\ &{} +K\bigl(t,(1-s) w_{1}^{(k)}(t),(1-s) w_{2}^{(k)}(t)\bigr)-W\bigl(t,(1-s) w_{1}^{(k)}(t),(1-s) w_{2}^{(k)}(t)\bigr) \\ &{} +(1-s) \bigl(f_{1,k}(t),w_{1}^{(k)}(t) \bigr)+(1-s) \bigl(f_{2,k}(t),w_{2}^{(k)}(t)\bigr) \bigr] \Biggr\} \\ \le& \sup_{s\in[0,1]} \bigl\{ -2\varepsilon_{0}\vert 1-s\vert ^{p} +\vert 1-s\vert \bigl\vert f_{1,k}(0) \bigr\vert +\vert 1-s\vert \bigl\vert f_{2,k}(0)\bigr\vert +B_{0}+B_{1} \bigr\} \\ \le& \bigl\vert f_{1}(0)\bigr\vert + \bigl\vert f_{2}(0)\bigr\vert +B_{0}+B_{1}:=M_{2}, \end{aligned}$$
(3.13)
where \(M_{2}\) is independent of \(k\in\mathbb{N}\), which implies that the sequence \(\{c_{k}\}_{k\in\mathbb{N}}\) is bounded. Moreover, \(\varphi_{k}'(u_{k})=0\). Then it follows from (\(\mathcal{A}1\)), (H2), and (3.10) that
$$\begin{aligned} pM_{2}\ge pc_{k} = & p\varphi_{k}(u_{k})- \bigl\langle \varphi_{k}'(u_{k}), u_{k}\bigr\rangle \\ = & p\varphi_{k}(u_{1k},u_{2k})-\bigl\langle \varphi_{k}'(u_{1k},u_{2k}), (u_{1k},u_{2k})\bigr\rangle \\ \ge& \sum_{t=-kT}^{kT-1}\bigl[\bigl( \nabla_{u_{1}} W\bigl(t,u_{1k}(t),u_{2k}(t) \bigr),u_{1k}(t)\bigr)+\bigl(\nabla_{u_{2}} W \bigl(t,u_{1k}(t),u_{2k}(t)\bigr),u_{2k}(t)\bigr) \\ &{} -pW\bigl(t,u_{1k}(t),u_{2k}(t)\bigr)\bigr] \\ &{} +( p-1)\sum_{t=-kT}^{kT-1} \bigl(f_{1k}(t),u_{1k}(t)\bigr)+( p-1)\sum _{t=-kT}^{kT-1}\bigl(f_{2k}(t),u_{2k}(t) \bigr) \\ \ge&\sum_{t=-kT}^{kT-1}\frac{W(t,u_{1k}(t),u_{2k}(t))}{\xi+\eta _{1}\vert u_{1k}(t)\vert ^{\nu}+\eta_{2}\vert u_{2k}(t)\vert ^{\nu}}+( p-1)\sum_{t=-kT}^{kT-1}\bigl(f_{1k}(t),u_{1k}(t) \bigr) \\ &{} +(p-1)\sum_{t=-kT}^{kT-1} \bigl(f_{2k}(t),u_{2k}(t)\bigr). \end{aligned}$$
So
$$\begin{aligned}& \sum_{t=-kT}^{kT-1}\frac{W(t,u_{1k}(t),u_{2k}(t))}{\xi+\eta _{1}|u_{1k}(t)|^{\nu}+\eta_{2}|u_{2k}(t)|^{\nu}} \\& \quad \le pM_{2}- (p-1)\sum_{t=-kT}^{kT-1} \bigl(f_{1k}(t),u_{1k}(t)\bigr)-(p-1)\sum _{t=-kT}^{kT-1}\bigl(f_{2k}(t),u_{2k}(t) \bigr). \end{aligned}$$
Then
$$\begin{aligned} \eta_{k}(u_{k}) = & \varphi_{k}(u_{k})+ \sum_{t=-kT}^{kT-1}\frac {W(t,u_{1k}(t),u_{2k}(t))}{\xi+\eta_{1}\vert u_{1k}(t)\vert ^{\nu}+\eta _{2}\vert u_{2k}(t)\vert ^{\nu}}\bigl(\xi+ \eta_{1}\bigl\vert u_{1k}(t)\bigr\vert ^{\nu}+ \eta_{2}\bigl\vert u_{2k}(t)\bigr\vert ^{\nu}\bigr) \\ &{} -\sum_{t=-kT}^{kT-1} \bigl(f_{1,k}(t),u_{1k}(t)\bigr)-\sum _{t=-kT}^{kT-1}\bigl(f_{2,k}(t),u_{2k}(t) \bigr) \\ \le& \varphi_{k}(u_{k})+\bigl(\xi+\eta_{1} \Vert u_{1k}\Vert _{l_{2kT}^{\infty}}^{\nu}+ \eta_{2}\Vert u_{2k}\Vert _{l_{2kT}^{\infty}}^{\nu}\bigr)\sum_{t=-kT}^{kT-1}\frac {W(t,u_{1k}(t),u_{2k}(t))}{\xi+\eta_{1}\vert u_{1k}(t)\vert ^{\nu}+\eta _{2}\vert u_{2k}(t)\vert ^{\nu}} \\ &{} -\sum_{t=-kT}^{kT-1} \bigl(f_{1,k}(t),u_{1k}(t)\bigr)-\sum _{t=-kT}^{kT-1}\bigl(f_{2,k}(t),u_{2k}(t) \bigr) \\ \le& \varphi_{k}(u_{k}) +\bigl(\xi+\eta_{1}C_{*}^{\nu} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu}+\eta _{2}C_{*}^{\nu} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu}\bigr) \\ &{} \cdot \Biggl[ pM_{2}- (p-1)\sum_{t=-kT}^{kT-1} \bigl(f_{1,k}(t),u_{1k}(t)\bigr)-(p-1)\sum _{t=-kT}^{kT-1}\bigl(f_{2,k}(t),u_{2k}(t) \bigr) \Biggr] \\ &{} -\sum_{t=-kT}^{kT-1} \bigl(f_{1,k}(t),u_{1k}(t)\bigr)-\sum _{t=-kT}^{kT-1}\bigl(f_{2,k}(t),u_{2k}(t) \bigr) \\ = & M_{2}+ p\xi M_{2}+p\eta_{1} C_{*}^{\nu}M_{2}\Vert u_{1k}\Vert _{\mathcal {H}_{1,k}}^{\nu}+p\eta_{2} C_{*}^{\nu}M_{2}\Vert u_{2k}\Vert _{\mathcal {H}_{2,k}}^{\nu} \\ &{}- (p-1)\xi\sum_{t=-kT}^{kT-1} \bigl(f_{1,k}(t),u_{1k}(t)\bigr)- (p-1)\xi\sum_{t=-kT}^{kT-1} \bigl(f_{2,k}(t),u_{2k}(t)\bigr) \\ &{} -(p-1)\eta_{1} C_{*}^{\nu} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu}\sum_{t=-kT}^{kT-1} \bigl(f_{1,k}(t),u_{1k}(t)\bigr) \\ &{} -(p-1)\eta_{2} C_{*}^{\nu} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu}\sum _{t=-kT}^{kT-1}\bigl(f_{2,k}(t),u_{2k}(t) \bigr) \\ &{}-(p-1)\eta_{1} C_{*}^{\nu} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu}\sum _{t=-kT}^{kT-1}\bigl(f_{2,k}(t),u_{2k}(t) \bigr) \\ &{} -(p-1)\eta_{2} C_{*}^{\nu} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu}\sum _{t=-kT}^{kT-1}\bigl(f_{1,k}(t),u_{1k}(t) \bigr) \\ &{} -\sum_{t=-kT}^{kT-1} \bigl(f_{1,k}(t),u_{1k}(t)\bigr)-\sum _{t=-kT}^{kT-1}\bigl(f_{2,k}(t),u_{2k}(t) \bigr) \\ \le& (1+p\xi)M_{2}+\bigl[(p-1)\xi+1\bigr] \biggl(\sum _{t\in\mathbb {Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \Biggl(\sum_{t=-kT}^{kT-1} \bigl\vert u_{1k}(t)\bigr\vert ^{p} \Biggr)^{1/p} \\ &{} +\bigl[(p-1)\xi+1\bigr] \biggl(\sum_{t\in\mathbb{Z}} \bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \Biggl(\sum_{t=-kT}^{kT-1}\bigl\vert u_{2k}(t)\bigr\vert ^{p} \Biggr)^{1/p} \\ &{} +p\eta_{1} C_{*}^{\nu}M_{2} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu}+p\eta _{2} C_{*}^{\nu}M_{2}\Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu} \\ &{} +(p-1)\eta_{1} C_{*}^{\nu} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu}\Biggl( \biggl(\sum _{t\in\mathbb{Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \Biggl(\sum_{t=-kT}^{kT-1} \bigl\vert u_{1k}(t)\bigr\vert ^{p} \Biggr)^{1/p} \\ &{} + \biggl(\sum_{t\in\mathbb{Z}}\bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \Biggl(\sum _{t=-kT}^{kT-1}\bigl\vert u_{2k}(t) \bigr\vert ^{p} \Biggr)^{1/p} \Biggr) \\ &{} +(p-1)\eta_{2} C_{*}^{\nu} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu}\Biggl( \biggl(\sum _{t\in\mathbb{Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \Biggl(\sum_{t=-kT}^{kT-1} \bigl\vert u_{1k}(t)\bigr\vert ^{p} \Biggr)^{1/p} \\ &{} + \biggl(\sum_{t\in\mathbb{Z}}\bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \Biggl(\sum _{t=-kT}^{kT-1}\bigl\vert u_{2k}(t) \bigr\vert ^{p} \Biggr)^{1/p}\Biggr) \\ \le& (1+p\xi)M_{2}+\bigl[(p-1)\xi+1\bigr] \biggl(\sum _{t\in\mathbb {Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}}\Vert u_{1k}\Vert _{\mathcal {H}_{1,k}} \\ &{} +\bigl[(p-1)\xi+1\bigr] \biggl(\sum_{t\in\mathbb{Z}} \bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}} \\ &{} +p\eta_{1} C_{*}^{\nu}M_{2} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu}+p\eta _{2} C_{*}^{\nu}M_{2}\Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu} \\ &{} +(p-1)\eta_{1} C_{*}^{\nu} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu} \\ &{}\cdot\biggl( \biggl(\sum _{t\in\mathbb{Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}}\Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}+ \biggl(\sum_{t\in\mathbb {Z}}\bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}}\Vert u_{2k}\Vert _{\mathcal{H}_{2,k}} \biggr) \\ &{} +(p-1)\eta_{2} C_{*}^{\nu} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu} \\ &{}\cdot \biggl( \biggl(\sum _{t\in\mathbb{Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}}\Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}+ \biggl(\sum_{t\in\mathbb {Z}}\bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}}\Vert u_{2k}\Vert _{\mathcal{H}_{2,k}} \biggr) \\ \le& (1+p\xi)M_{2}+\bigl[(p-1)\xi+1\bigr] \biggl(\sum _{t\in\mathbb {Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}}\Vert u_{1k}\Vert _{\mathcal {H}_{1,k}} \\ &{} +\bigl[(p-1)\xi+1\bigr] \biggl(\sum_{t\in\mathbb{Z}} \bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}} \\ &{} +p\eta_{1} C_{*}^{\nu}M_{2} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu}+p\eta _{2} C_{*}^{\nu}M_{2}\Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu} \\ &{} +(p-1)\eta_{1} C_{*}^{\nu} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu +1} \biggl(\sum _{t\in\mathbb{Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \\ &{}+(p-1)\eta_{1} C_{*}^{\nu} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu }\Vert u_{2k}\Vert _{\mathcal{H}_{2,k}} \biggl(\sum_{t\in\mathbb {Z}}\bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \\ &{} +(p-1)\eta_{2} C_{*}^{\nu} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu +1} \biggl(\sum _{t\in\mathbb{Z}}\bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \\ &{}+(p-1)\eta_{2} C_{*}^{\nu} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}} \biggl(\sum_{t\in\mathbb {Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \\ \le& (1+p\xi)M_{2}+\bigl[(p-1)\xi+1\bigr] \biggl(\sum _{t\in\mathbb {Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}}\Vert u_{1k}\Vert _{\mathcal {H}_{1,k}} \\ &{} +\bigl[(p-1)\xi+1\bigr] \biggl(\sum_{t\in\mathbb{Z}} \bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}} \\ &{}+p\eta_{1} C_{*}^{\nu}M_{2}\Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu}+p\eta_{2} C_{*}^{\nu}M_{2}\Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu} \\ &{} +(p-1)\eta_{1} C_{*}^{\nu} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu +1} \biggl(\sum _{t\in\mathbb{Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \\ &{} +(p-1)\eta_{1} C_{*}^{\nu}\biggl( \frac{\nu}{\nu+1}\Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu+1}+ \frac{1}{\nu+1}\Vert u_{2k}\Vert _{\mathcal {H}_{2,k}}^{\nu+1} \biggr) \biggl(\sum_{t\in\mathbb {Z}}\bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \\ &{} +(p-1)\eta_{2} C_{*}^{\nu} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu +1} \biggl(\sum _{t\in\mathbb{Z}}\bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \\ &{} +(p-1)\eta_{2} C_{*}^{\nu}\biggl( \frac{\nu}{\nu+1}\Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu+1}+ \frac{1}{\nu+1}\Vert u_{1k}\Vert _{\mathcal {H}_{1,k}}^{\nu+1} \biggr) \biggl(\sum_{t\in\mathbb {Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}}. \end{aligned}$$
(3.14)
Thus (3.14) and Lemma 3.2 imply that
$$\begin{aligned}& (1+p\xi)M_{2}+\bigl[(p-1)\xi+1\bigr] \biggl(\sum _{t\in\mathbb {Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}}\Vert u_{1k}\Vert _{\mathcal {H}_{1,k}} \\ & \qquad {}+\bigl[(p-1)\xi+1\bigr] \biggl(\sum_{t\in\mathbb {Z}} \bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \Vert u_{2k}\Vert _{\mathcal {H}_{2,k}} \\& \qquad {}+p\eta_{1} C_{*}^{\nu}M_{2} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu}+p\eta _{2} C_{*}^{\nu}M_{2}\Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu}\\& \qquad {}+(p-1)\eta_{1} C_{*}^{\nu} \Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu+1} \biggl(\sum_{t\in \mathbb{Z}} \bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \\& \qquad {}+(p-1)\eta_{1} C_{*}^{\nu}\biggl( \frac{\nu}{\nu+1}\Vert u_{1k}\Vert _{\mathcal{H}_{1,k}}^{\nu+1}+ \frac{1}{\nu+1}\Vert u_{2k}\Vert _{\mathcal {H}_{2,k}}^{\nu+1} \biggr) \biggl(\sum_{t\in\mathbb {Z}}\bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \\& \qquad {}+(p-1)\eta_{2} C_{*}^{\nu} \Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu +1} \biggl(\sum _{t\in\mathbb{Z}}\bigl\vert f_{2}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \\& \qquad {}+(p-1)\eta_{2} C_{*}^{\nu}\biggl( \frac{\nu}{\nu+1}\Vert u_{2k}\Vert _{\mathcal{H}_{2,k}}^{\nu+1}+ \frac{1}{\nu+1}\Vert u_{1k}\Vert _{\mathcal {H}_{1,k}}^{\nu+1} \biggr) \biggl(\sum_{t\in\mathbb {Z}}\bigl\vert f_{1}(t)\bigr\vert ^{p'} \biggr)^{1/{p'}} \\& \quad \ge \min\bigl\{ d_{1}\Vert u_{1}\Vert _{\mathcal{H}_{1,k}}^{p},a_{1}C_{*}^{\gamma-p} \Vert u_{1}\Vert _{\mathcal{H}_{1,k}}^{\gamma}\bigr\} +\min\bigl\{ d_{2}\Vert u_{2}\Vert _{\mathcal {H}_{2,k}}^{p},a_{2}C_{*}^{\gamma-p} \Vert u_{2}\Vert _{\mathcal{H}_{2,k}}^{\gamma }\bigr\} . \end{aligned}$$
Note that \(p>\gamma>\nu+1\). So (H6) implies there exists \(M_{3}>0\) (independent of k) such that
$$\|u_{1k}\|_{\mathcal{H}_{1,k}}\le M_{3},\qquad \|u_{2k}\|_{\mathcal {H}_{2,k}}\le M_{3} \quad \mbox{for every } k\in\mathbb{N}. $$
By Corollary 2.1,
$$\|u_{1k}\|_{l_{2kT}^{\infty}}\le C_{*}M_{3},\qquad \|u_{2k}\|_{l_{2kT}^{\infty}}\le C_{*}M_{3} \quad \mbox{for every } k\in\mathbb{N}. $$
Let \(M_{1}=\max\{ C_{1*}M_{3}, C_{2*}M_{3}\}\). Thus the proof is complete. □

Lemma 3.5

Let \(\{u_{k}\}\) be determined by Lemma  3.4. Then there exists a subsequence \(\{u_{k_{j}}=(u_{1k_{j}},u_{2k_{j}})\}\) of \(\{u_{k}\}_{k\in\mathbb {N}}\) convergent to a certain function \(u_{\infty}=(u_{1\infty },u_{2\infty})\) and when \(f_{1}\neq0\) and \(f_{2}\neq0\), \(u_{\infty}\) is a nontrivial solution of system (1.1) such that \(u_{\infty}(t)\to0\) and \(\Delta{u}_{\infty}(t-1)\to0\) as \(t\to\pm\infty\).

Proof

Note that
$$\|u_{1k}\|_{\mathcal{H}_{1,k}}\le M_{1},\qquad \|u_{2k}\|_{\mathcal {H}_{2,k}}\le M_{1}\quad \mbox{for every } k \in\mathbb{N}. $$
Then, similar to the argument in [15] or [16], one can prove that \(\{u_{mk}\}_{k\in\mathbb{N}}\) has a convergent subsequence \(\{u_{mk_{j}}\}\) such that \(u_{mk_{j}}\to u_{m\infty}\) and \(u_{m\infty}(t)\to0\) and \(\Delta{u}_{m\infty}(t-1)\to0\) as \(t\to \pm\infty\), where \(m=1,2\). Let \(u_{\infty}=(u_{1\infty},u_{2\infty})\). By (3.13) and the continuity of \(\Phi_{m}\), \(K(t,\cdot,\cdot)\), \(W(t,\cdot,\cdot)\) and \(\varphi_{k}'\), similar to the argument in [15] or [16], the proof is easy to be completed. □

 □

Proof of Theorem 1.2

The proof is easy to be completed by replacing
$$\begin{aligned} \sum_{t=-kT}^{kT-1}\bigl(f_{m}(t),u_{m}(t) \bigr) \le& \Biggl(\sum_{t=-kT}^{kT-1}\bigl\vert f_{m}(t)\bigr\vert ^{p'} \Biggr)^{1/p'} \Biggl( \sum_{t=-kT}^{kT-1}\bigl\vert u_{m}(t)\bigr\vert ^{p} \Biggr)^{1/p} \\ \le& \|u_{m}\|_{\mathcal {H}_{m,k}} \biggl(\sum_{t\in\mathbb{Z}} \bigl\vert f_{m}(t)\bigr\vert ^{p'} \biggr)^{1/p'} \end{aligned}$$
with
$$\sum_{t=-kT}^{kT-1}\bigl(f_{m}(t),u_{m}(t) \bigr)\le \|u_{m}\|_{l_{2kT}^{\infty}}\sum_{t=-kT}^{kT-1} \bigl\vert f_{m}(t)\bigr\vert \le C_{*} \|u_{m}\|_{\mathcal{H}_{m,k}}\sum_{t\in\mathbb{Z}}\bigl\vert f_{m}(t)\bigr\vert ,\quad m=1,2, $$
in the proofs of Lemma 3.3 and Lemma 3.4. □

Proofs of Theorem 1.3 and Theorem 1.4

We only note that in the proof of Lemma 3.3, when \(\gamma=p\), we do not need to consider the case that \(r\in(0,1]\) alone and it is sufficient that \(r>0\). Other proofs are the same as those of Theorem 1.1 and Theorem 1.2, respectively. □

4 Examples

We first give two examples about Φ which satisfy assumption (\(\mathcal{A}1\)).

(I) An example with \(N=1\). Define \(\Phi_{m}:\mathbb{R}\to\mathbb {R}^{N}\), \(m=1,2\), by
$$\Phi_{1}(x)=\left \{ \begin{array}{l@{\quad}l} \alpha_{1}|x|^{p},& x\ge0, \\ \alpha_{2}|x|^{p},& x< 0, \end{array} \right . \qquad \Phi_{2}(y)=\left \{ \begin{array}{l@{\quad}l} \beta_{1}|y|^{p},& y\ge0, \\ \beta_{2}|y|^{p}, &y< 0, \end{array} \right . $$
where \(\alpha_{1},\alpha_{2}\in[d_{1},d_{3}]\), \(\beta_{1},\beta_{2}\in [d_{2},d_{4}]\). Then it is easy to verify that \(\Phi_{m}\), \(m=1,2\), satisfies (\(\mathcal{A}1\)).
(II) As described in [1], the following classical case with p-Laplacian also satisfies the assumption (\(\mathcal{A}1\)). Define \(\Phi_{m}:\mathbb{R}^{N}\to\mathbb{R}^{N}\), \(m=1,2\), by
$$\Phi_{1}(x)=\alpha|x|^{p},\qquad \Phi_{2}(y)= \beta|y|^{p}, $$
where \(\alpha\in[d_{1},d_{3}]\), \(\beta\in[d_{2},d_{4}]\).
Next, we present some examples of K and W which satisfy those assumptions in Theorem 1.1. There are lots of examples of K. For example, let
$$K(t,x_{1},x_{2})=a_{1}(t)|x_{1}|^{\gamma}+a_{2}(t)|x_{2}|^{\gamma}, \quad (t,x_{1},x_{2})\in\mathbb{Z}[0,T-1]\times \mathbb{R}^{N}\times\mathbb{R}^{N}, $$
where \(\gamma\in(1,p)\), \(a_{i}, i=1,2:\mathbb{Z}\to\mathbb{R}^{+}\) are T-periodic. Let \(a_{i}=\min_{t\in\mathbb{Z}[0,T-1]} a_{i}(t)\). Then it is easy to see that K satisfies (H1) and (H2).
For W, we assume that
$$W(t,x_{1},x_{2})=b(t) \bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr)\ln\bigl(|x_{1}|^{p}+|x_{2}|^{p}+1 \bigr),\quad (t,x_{1},x_{2})\in\mathbb{Z}[0,T-1]\times \mathbb{R}^{N}\times\mathbb{R}^{N}, $$
where \(b:\mathbb{Z}\to\mathbb{R}^{+}\) is T-periodic. Let \(b^{+}=\max_{t\in\mathbb{Z}[0,T-1]}\{b(t)\}\). Then
$$\begin{aligned}& W(t,x_{1},x_{2})\le b^{+} \bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr)\ln\bigl(|rC_{*}|^{p}+|rC_{*}|^{p}+1\bigr) \\& \quad \mbox {for all } t\in\mathbb{Z}[0,T-1], |x_{1}|\le rC_{*}, |x_{2}|\le rC_{*}. \end{aligned}$$
Let \(b_{1}=b_{2}=b^{+}\ln(|rC_{*}|^{p}+|rC_{*}|^{p}+1)\). If r is sufficiently small, then (H3)(i) holds. It is easy to see that
$$\lim_{|x_{1}|+|x_{2}|\to+\infty}\frac {W(t,x_{1},x_{2})}{|x_{1}|^{p}+|x_{2}|^{p}}=+\infty\quad \mbox{for all }t\in \mathbb{Z}[0,T-1]. $$
So (H4) holds. Let \(\nu\in(0,\gamma-1)\). Note that
$$p\xi\bigl(|x_{1}|^{p}+|x_{2}|^{p}\bigr) \ge\ln\bigl(|x_{1}|^{p}+|x_{2}|^{p}+1 \bigr),\qquad p\bigl(\eta_{1}|x_{1}|^{\nu}+ \eta_{2}|x_{2}|^{\nu}\bigr)\ge \ln \bigl(|x_{1}|^{p}+|x_{2}|^{p}+1\bigr) $$
for all \((x_{1},x_{2})\in \mathbb{R}^{N}\times\mathbb{R}^{N}\), when we choose sufficiently large ξ, \(\eta_{1}\) and \(\eta_{2}\). Hence
$$\begin{aligned}& p\xi \bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr)+p\bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr) \bigl(\eta_{1}|x_{1}|^{\nu}+\eta _{2}|x_{2}|^{\nu}\bigr) \\& \quad \ge \ln\bigl(|x_{1}|^{p}+|x_{2}|^{p}+1 \bigr)+\ln \bigl(|x_{1}|^{p}+|x_{2}|^{p}+1 \bigr) \bigl(|x_{1}|^{p}+|x_{2}|^{p}\bigr) \\& \quad \Longleftrightarrow \quad p\bigl(\xi+\eta_{1}|x_{1}|^{\nu}+ \eta_{2}|x_{2}|^{\nu }\bigr) \bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr) \\& \hphantom{\quad \Longleftrightarrow\quad}\quad \ge \ln\bigl(|x_{1}|^{p}+|x_{2}|^{p}+1 \bigr) \bigl(|x_{1}|^{p}+|x_{2}|^{p}+1 \bigr) \\& \quad \Longleftrightarrow\quad p\bigl(\xi+\eta_{1}|x_{1}|^{\nu}+ \eta_{2}|x_{2}|^{\nu }\bigr) \bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr)^{2} \\& \hphantom{\quad \Longleftrightarrow\quad}\quad \ge \bigl(|x_{1}|^{p}+|x_{2}|^{p} \bigr) \ln \bigl(|x_{1}|^{p}+|x_{2}|^{p}+1 \bigr) \bigl(|x_{1}|^{p}+|x_{2}|^{p}+1 \bigr) \\& \quad \Longleftrightarrow \quad \frac{p(|x_{1}|^{p}+|x_{2}|^{p})^{2}}{|x_{1}|^{p}+|x_{2}|^{p}+1} \ge \frac{(|x_{1}|^{p}+|x_{2}|^{p}) \ln(|x_{1}|^{p}+|x_{2}|^{p}+1)}{\xi+\eta _{1}|x_{1}|^{\nu}+\eta_{2}|x_{2}|^{\nu}} \\& \quad \Longleftrightarrow\quad \bigl(\nabla_{x_{1}} W(t,x_{1},x_{2}), x_{1}\bigr)+\bigl(\nabla _{x_{2}} W(t,x_{1},x_{2}), x_{2}\bigr)-pW(t,x_{1},x_{2}) \\& \hphantom{\quad \Longleftrightarrow\quad}\quad \ge \frac{W(t,x_{1},x_{2})}{\xi+\eta_{1}|x_{1}|^{\nu}+\eta_{2}|x_{2}|^{\nu}} \end{aligned}$$
for all \((t,x_{1},x_{2})\in\mathbb{Z}[0,T-1]\times\mathbb{R}^{N}\times \mathbb{R}^{N}\), which implies (H5) holds.

Declarations

Acknowledgements

This project is supported by the National Natural Science Foundation of China (No. 11301235), Tianyuan Fund for Mathematics of the National Natural Science Foundation of China (No. 11226135) and the Fund for Fostering Talents in Kunming University of Science and Technology (No. KKSY201207032).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Kunming University of Science and Technology

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