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. □