In this paper, we consider the following nonlinear second order difference equation

$ \begin{eqnarray}\label{eqn1.1} \left\{\begin{array}{ll} \Delta[a(k)\Delta u(k-1)]-b(k)u(k)+f(k, u(k))=0, \quad k\in\mathbb{Z}, \\ u(k)\rightarrow 0, \quad \quad \quad |k|\rightarrow\infty, \end{array} \right. \end{eqnarray} $

(1.1)

where $a(k)$, $b(k)$ and $f(k, u)$ are $T$-periodic in $k$ $f(k, u):\mathbb{Z}\times \mathbb{R}\rightarrow\mathbb{R}$ is a continuous function on $u$. The forward difference operator $\Delta$ is defined by

$ \Delta u(k)=u(k+1)-u(k)\quad {\rm ~~for~~all~}k\in\mathbb{Z}, $

where $\mathbb{Z}$ and $\mathbb{R}$ denote the sets of all integers and real numbers, respectively.

The solutions of (1.1) are referred to as homoclinic solutions of the equation

$ \begin{eqnarray}\label{eqn1.2} \Delta[a(k)\Delta u(k-1)]-b(k)u(k)+f(k, u(k))=0, \quad k\in\mathbb{Z}. \end{eqnarray} $

(1.2)

In the theory of differential equations, homoclinic orbits play an important role in analyzing the chaos of dynamical systems. If a system has the transversely intersected homoclinic orbits, then it must be chaotic. If a system has the smoothly connected homoclinic orbits, then it can not stand the perturbation, its perturbed system probably produces chaotic. Therefore, it is of practical importance and mathematical significance to study the existence of homoclinic solutions.

Difference equations represent the discrete counterpart of ordinary differential equations. The classical methods are used in difference equations, such as numerical analysis, fixed point methods, linear and nonlinear operator theory, see [1-4]. In recent years, the existence and multiplicity of homoclinic solutions for difference equations have been studied in many papers by variational methods, see [5-12] and the reference therein.

Assume the following conditions hold:

(A) $a(k)>0$ and $a(k+T)=a(k)$ for all $k\in \mathbb{Z}$.

(B) $b(k)>0$ and $b(k+T)=b(k)$ for all $k\in \mathbb{Z}$.

(H1) $f\in C(\mathbb{Z}\times \mathbb{R}, \mathbb{R})$, and there exist $C>0$ and $p\in(2, \infty)$such that

$ \begin{eqnarray} |f(k, u)|\leq C(1+|u|^{p-1})~~{\rm for ~all} ~k\in \mathbb{Z}, u\in \mathbb R.\nonumber \end{eqnarray} $

(H2) $\lim\limits_{|u|\to 0}{f(k, u)/u}=0$uniformly for $ k\in \mathbb{Z}$.

(H3) $\lim\limits_{|u|\to\infty}{F(k, u)/|u|^{2}}=+\infty$uniformly for $ k\in\mathbb{Z}$, where $F(k, u)$ is the primitive function of $f(k, u)$, i.e.,

$ F(k, u)=\int_{0}^{u}f(k, s) ds. $

(H4) $u\mapsto{f(k, u)/|u|}$is strictly increasing on $(-\infty, 0)$ and $(0, \infty)$.

Remark1.1  (H2) implies that $u(k)\equiv 0$ is a trivial solution of (1.1).

Our main result is following.

Theorem1.1  Suppose that conditions (A), (B) and (H1)-(H4) are satisfied. Then equation (1.1) has at least a nontrivial ground state solution, i.e., solution corresponding to the least energy of the action functional of (1.1).

Remark1.2  In [6], the authors also considered (1.1) and assumed that (H2) and the following classical Ambrosetti-Rabinowitz superlinear condition (see [16, 17]): There exists a constant $\mu>2$ such that

$ \begin{eqnarray}\label {eqnAR} 0<\mu F(k, u)\leq f(k, u)u, u\neq 0, \end{eqnarray} $

(1.3)

(1.3) implies that for each $k\in\mathbb{Z}$, there exists a constant $C>0$ such that

$ \begin{eqnarray}\label {eqnAR1} F(k, u)\geq C|u|^{\mu}~ {\rm~for~} |u|\geq 1. \end{eqnarray} $

(1.4)

This implies (H3) holds. There exists a superlinear function, such as

$ f(k, u)=u\ln(1+|u|) $

does not satisfy (1.3). However, it satisfies the conditions (H1)-(H4). So our conditions are weaker than conditions presented in [6]. And in our paper, we do not need periodic approximation technique to obtain homoclinic solutions. Furthermore, the existence of ground state solutions can be obtained.

Remark1.3  In [13], the authors considered the following difference equation

$ \begin{eqnarray}\label{eqnshi} \Delta(\varphi_{p}(\Delta u(n-1))-q(n)\varphi_{p}(u_{n})+f(n, u_{n+M}, u_{n}, u_{n-M})=0, \quad n\in\mathbb{Z}. \end{eqnarray} $

(1.5)

Let $p=0$ and $M=0$, (1.2) is the special case of (1.5).

The following hypotheses with $p=0$ and $M=0$ are satisfied in [13]:

(F1) $F\in C(\mathbb{Z}\times \mathbb{R}, \mathbb{R})$ with $F(n+T, u)=F(n, u)$ and it satisfies $\frac{\partial F}{\partial u}=f(n, u)$;

(F2) there exist positive constants $\varrho$ and $a<\frac{\underline{q}}{4}(\frac{\kappa_{1}}{\kappa_{2}})^{2}$ such that $|F(n, u)|\leq a|u|^{2}$, for all $n\in \mathbb{Z}$ and $|u|\leq\varrho$;

(F3) there exist constants $\rho, $ $c>\frac{1}{4}(\frac{\kappa_{1}}{\kappa_{2}})^{2}(4+\overline{q})$and $b$ such that $F(n, u)\geq c|u|^{2}+b$, for all $n\in \mathbb{Z}$ and $|u|\geq \rho$;

(F4) $fu-2F>0, $ for all $n\in \mathbb{Z}$ and $|u|\neq 0$;

(F5) $fu-2F\rightarrow\infty, $ as $|u|\rightarrow\infty$.

Note that (H2)-(H4) imply that (F2)-(F4). A nontrivial homoclinic orbit of (1.5) is obtained by Mountain Pass Lemma in combination with periodic approximations in [13]. However, in our paper, we employ the Nehari manifold approach instead of periodic approximation technique to obtain the ground state solutions. Furthermore, we show that the functional is coercive on Nehari manifold (Lemma 3.2), which is weaker than P.S. condition in [13].

The rest of this paper is organized as follows. In Section (1.1), we establish the variational framework associated with (1.1), and transfer the existence of solutions of boundary value problem (1.1) into the existence of critical points of the corresponding functional. By employing the critical point theory, we give proofs of the main results in Section 3. Finally, we give a simple example to demonstrate our results.

In this section, we firstly establish the corresponding variational setting associated with (1.1). Let

$ X=\{u=\{u(k)\}:u(k)\in\mathbb{R}, k\in\mathbb{Z}\} $

be the set of all real sequences

$ u=\{u(k)\}_{k\in\mathbb{Z}}=\left(\cdots, u(-k), u(-k+1), \cdots, u(-1), u(0), u(1), \cdots, u(k), \cdots\right). $

Then $X$ is a vector space with $au+bv=\{au(k)+bu(k)\}$ for $u, v\in X$, $a, b\in\mathbb{R}$.

Define the space

$ \begin{eqnarray*} E:=\{~u\in X: \sum_{k\in\mathbb{Z}}\left[a(k)(\Delta u(k-1))^{2}+b(k)(u(k))^{2}\right]<+\infty\}. \end{eqnarray*} $

Then $E$ is a Banach space equipped with the corresponding norm

$ \begin{eqnarray*} \|u\|^{2}=\sum_{k\in\mathbb{Z}}\left[a(k)(\Delta u(k-1))^{2}+b(k) (u(k))^{2}\right], \quad \forall u\in E. \end{eqnarray*} $

For $1\leq p<+\infty$, denote

$ l^{p}=\{u=\{u(k)\}\in X:\sum\limits_{k\in\mathbb{Z}}|u(k)|^{p}<\infty\} $

equipped with the norm

$ \|u{{\|}_{{{l}^{p}}}}={{\left( \sum\limits_{k\in \mathbb{Z}}{|}u(k){{|}^{p}} \right)}^{\frac{1}{p}}}, $

$|\cdot|$ is the usual absolute value in $\mathbb{R}$. Then the following embedding between $l^{p}$ spaces holds

$ \begin{eqnarray}\label{eqn2.1} l^{q}\subset l^{p}, \|u\|_{l^{p}}\leq\|u\|_{l^{q}}, 1\leq q\leq p\leq \infty. \end{eqnarray} $

(2.1)

Now we consider the variational functional $J$ defined on $E$ by

$ \begin{eqnarray*} J(u)&=&\sum_{k\in\mathbb{Z}}\left[\frac{1}{2}a(k)(\Delta u(k-1))^{2}+\frac{1}{2}b(k) (u(k))^{2}-F(k, u(k))\right]\\ &=&\frac{1}{2}\|u\|^{2}-\sum_{k\in\mathbb{Z}}F(k, u(k)). \end{eqnarray*} $

Then $J\in C^{1}(E, \mathbb{R})$, and for all $u, v\in E$, we have

$ \begin{eqnarray}\label{eqn2.4} (J'(u), v)=\sum_{k\in\mathbb{Z}}\left[a(k)\Delta u(k-1)\Delta v(k-1)+b(k)u(k)v(k)-f(k, u(k))v(k)\right]. \end{eqnarray} $

(2.2)

Then, we easily get the variational formulation for (1.1):

Lemma2.1  Every critical point $u\in E$ of $J$ is a solution of (1.1).

Proof  We assume that $u\in E$ is a critical point of $J$, then $J'(u)=0$. According to (2.2), this is equivalent to

$ \begin{eqnarray}\label{eqn2.3} \sum_{k\in\mathbb{Z}}\left[a(k)\Delta u(k-1)\Delta v(k-1)+b(k)u(k)v(k)-f(k, u(k))v(k)\right]=0, ~~~\forall v\in E. \end{eqnarray} $

(2.3)

For any $h\in \mathbb{Z}$, we define $e_h\in E$ by putting $e_h(k)=\delta_{hk}$ for all $k\in \mathbb{Z}$, where $\delta_{hk}=1$ if $h=k$; $\delta_{hk}=0$ if $h\neq k$. If we apply (2.3) with $v=e_h$, then

$ -\Delta[a(k)\Delta u(k-1)]+b(k)u(k)-f(k, u(k))=0, $

i.e., $u$ is a solution of (1.1). The proof is completed.

Therefore, $u\neq 0$ is a critical point of $J$ on $E$ if and only if $u$ is a solution of (1.1) with $u(\pm\infty)=0$, that is to say, $u$ is a homoclinic solution emanating from $0$. Thus, we have reduced the problem of finding homoclinic solutions of (1.1) to that of seeking critical points of the functional $J$ on $E$.

Now, we consider the Nehari manifold $ \mathcal{N}=\{u\in E\setminus\{0\}:J'(u)u=0\}, $ and let $c=_{u\in \mathcal{N}}^{\ \ \inf }J(u).$ By the definition of $\mathcal{N}$, we know $\mathcal{N}$ contains all nontrivial critical points of $J$.

Lemma3.1  Assume that (A), (B) and (H2)-(H4) are satisfied, then $\mathcal{N}$ is homeomorphic to the unit sphere $S$ in $E$, where $S=\{u\in E: \|u\|=1\}$.

Proof  Let $\varphi(u)=\sum\limits_{k\in\mathbb{Z}}F(k, u(k))$. By (H2), we have

$ \begin{eqnarray}\label{eqn3.01} \varphi'(u)=o(\|u\|)~{\rm~as~}~u\rightarrow 0. \end{eqnarray} $

(3.1)

Let $U\subset E\setminus\{0\}$ be a weakly compact subset, we know that

$ \begin{eqnarray}\label{eqn3.03} \varphi(su)/s^{2}\rightarrow\infty {\rm~uniformly ~for~} u {\rm~on~} U, {\rm~as~} s\rightarrow\infty. \end{eqnarray} $

(3.2)

In fact, let $\{u_n\}\subset U$. It needs to show that

$ \begin{eqnarray*} {\rm if~} s_n\rightarrow\infty, \quad \varphi(s_nu_n)/(s_n)^{2}\rightarrow\infty \end{eqnarray*} $

as $n\rightarrow\infty$. Passing to a subsequence if necessary, $u_n\rightharpoonup u\in E\setminus\{0\}$ and $u_{n}(k)\rightarrow u(k)$ for every $k$, as $n\rightarrow\infty$.

Note that from (H2) and (H4), it is easy to get that

$ \begin{eqnarray}\label{eqn3.1} F(k, u)>0, {\rm ~~for ~all~} u\neq 0. \end{eqnarray} $

(3.3)

Since $|s_n u_{n}(k)|\rightarrow\infty$ and $u_n\neq 0$, by (H3) and (3.3), we have

$ \frac{\varphi ({{s}_{n}}{{u}_{n}})}{{{({{s}_{n}})}^{2}}}=\sum\limits_{k\in \mathbb{Z}}{\frac{F(k,{{s}_{n}}{{u}_{n}}(k))}{{{({{s}_{n}}{{u}_{n}}(k))}^{2}}}}{{({{u}_{n}}(k))}^{2}}\to \infty ~~\rm{as}~\mathit{n}\to \infty . $

Thus, we obtain (3.2) holds.

From (H4), for all $u\neq 0$ and $s>0$, we have

$ \begin{eqnarray}\label{eqn3.02} s\mapsto \varphi'(su)u/s {\rm~~is ~strictly~increasing}. \end{eqnarray} $

(3.4)

Let $h(s):=J(sw), s>0$. Then

$ h'(s)=J'(sw)w=s(\|w\|^{2}-s^{-1}\varphi(sw)w) $

from (3.1)-(3.4), then there exists a unique $s_{w}$, such that, when $0<s<s_{w}$, $h'(s)>0 $; and when $s>s_{w}$, $h'(s)<0 $. Therefore, $h'(s_{w})=J'(s_{w}w)w=0$, and $s_{w}w\in \mathcal{N}$.

Therefore, $s_{w}$ is a unique maximum of $h(s)$, and we can define the mapping $\hat{m}:E\setminus\{0\}\rightarrow \mathcal{N}$ by setting $\hat{m}(w):=s_{w}w$. Then the mapping $\hat{m}$ is continuous. Indeed, suppose $w_{n}\rightarrow w\neq 0$. Since $\hat{m}(tu)=\hat{m}(u)$ for each $t>0$, we may assume $w_{n}\in S$ for all $n$. Write $\hat{m}(w_{n})=s_{w_n}w_{n}$. Then $\{s_{w_n}\}$ is bounded. If not, $s_{w_n}\rightarrow\infty$ as $n\rightarrow\infty$.

Note that by (H4), for all $u\neq 0$,

$ \begin{eqnarray*} \frac{1}{2}f(k, u)u-F(k, u)&=&\frac{1}{2} f(k, u)u-\int_{0}^{u} f(k, s)ds\\ &>&\frac{1}{2} f(k, u)u-\frac{f(k, u)}{u}\int_0^u sds=0. \end{eqnarray*} $

So, for all $u\in \mathcal{N}$, we have

$ \begin{eqnarray}\label{eqn4.1} J(u)=J(u)-\frac{1}{2}J'(u)u=\sum_{k\in \mathbb{Z}}\left(\frac{1}{2}f(k, u(k))u(k)-F(k, u(k))\right)>0. \end{eqnarray} $

(3.5)

By (H3), we have

$ 0<\frac{J({{s}_{wn}}w)}{{{({{s}_{wn}})}^{2}}}=\frac{1}{2}\|w{{\|}^{2}}-\sum\limits_{k\in \mathbb{Z}}{\frac{F(k,{{s}_{wn}}w(k))}{{{({{s}_{wn}}w(k))}^{2}}}}{{(w(k))}^{2}}\to -\infty ,~~~\rm{as}~\ \mathit{n}\to \infty , $

which is a contradiction. Therefore, $s_{w_n}\rightarrow s>0$ after passing to a subsequence if needed. Since $\mathcal{N}$ is closed and $\hat{m}(w_{n})=s_{w_n}w_{n}\rightarrow sw, sw\in \mathcal{N}$. Hence $sw=s_{w}w=\hat{m}(w)$ by the uniqueness of $s_{w}$.

Therefore, we define a mapping $m:S\rightarrow \mathcal{N}$ by setting $m:=\hat{m}|_{S}$, then $m$ is a homeomorphism between $S$ and $\mathcal{N}$.

We also consider the functional $\hat{\Psi}:E\setminus \{0\}\rightarrow \mathbb{R}$ and $\Psi:S\rightarrow \mathbb{R}$ by

$ \begin{eqnarray*} \hat{\Psi}(w):=J(\hat{m}(w))~~{\rm~and}~~\Psi(w):=\hat{\Psi}|_{S}. \end{eqnarray*} $

Then we have $\hat{\Psi}\in C^{1}(E\setminus\{0\}, \mathbb{R})$ and

$ \begin{eqnarray*} \hat{\Psi}'(w)z=\frac{\|\hat{m}(w)\|}{\|w\|}J'(\hat{m}(w))z ~~{\rm~for~all}~w, z\in E, w\neq 0. \end{eqnarray*} $

In fact, let $w\in E\setminus\{0\}$ and $z\in E$. By Lemma 3.1 and the mean value theorem, we obtain

$ \begin{eqnarray*} \hat{\Psi}(w+tz)-\hat{\Psi}(w)&=&J(s_{w+tz}(w+tz))-J(s_{w}w)\\ &\leq&J(s_{w+tz}(w+tz))-J(s_{w+tz}(w))\\ &=& J'(s_{w+tz}(w+\tau_{t}tz))s_{w+tz}tz, \end{eqnarray*} $

where $|t|$ is small enough and $\tau_{t}\in(0, 1)$. Similarly,

$ \begin{eqnarray*} \hat{\Psi}(w+tz)-\hat{\Psi}(w)&=&J(s_{w+tz}(w+tz))-J(s_{w}w)\\ &\geq&J(s_{w}(w+tz))-J(s_{w}(w))\\ &=& J'(s_{w}(w+\eta_{t}tz))s_{w}tz, \end{eqnarray*} $

where $\eta_{t}\in(0, 1)$. Combining these two inequalities and the continuity of function $w\mapsto s_{w}$, we have

$ \begin{eqnarray*} \lim_{t\rightarrow 0}\frac{\hat{\Psi}(w+tz)-\hat{\Psi}(w)}{t}=s_{w}J'(s_{w}w)z=\frac{\|\hat{m}(w)\|}{\|w\|}J'(\hat{m}(w))z. \end{eqnarray*} $

Hence the G$\hat{a}$teaux derivative of $\hat{\Psi}$ is bounded linear in $z$ and continuous in $w$. It follows that $\hat{\Psi}$ is a class of $C^{1}$ (see [15], Proposition1.3).

Since $w\in S$, $m(w)=\hat{m}(w)$, so we have $\Psi\in C^{1}(S, \mathbb{R})$ and

$ \ {\Psi }'(w)z=\|m(w)\|{J}'(m(w))z~~~\rm{for}~\rm{all}~z\in {{T}_{w}}(S)=\{v\in E:(w,v)=0\}. $

Lemma3.2  Assume that (A), (B) and (H1)-(H3) are satisfied, for $u\in \mathcal{N}$ then $J(u)\rightarrow \infty$ as $\|u\|\rightarrow \infty$.

Proof  By way of contradiction, we assume that there exists a sequence $\{u_n\}\subset\mathcal{N}$ such that $J(u_n)\leq d$, as $\|{u_n}\|\rightarrow \infty$. Set $v_n=\frac{u_n}{\|u_n\|}$, and then there exists a subsequence, still denoted by $v_n$, such that $v_n\rightharpoonup v$, and therefore $v_{n}(k)\rightarrow v(k)$ for every $k$, as $n\rightarrow \infty$.

First we can prove that there exist $\delta>0$ and $k_j\in \mathbb{Z}$ such that

$ \begin{eqnarray}\label{eqnv} |v_n(k_j)|\geq \delta. \end{eqnarray} $

(3.6)

In fact, if not, then $v_n\rightarrow 0$ in $l^{\infty}$ as $n\rightarrow \infty$. For $q>2$,

$ \begin{eqnarray*} \|v_n\|^{q}_{l^{q}}\leq \|v_n\|^{q-2}_{l^{\infty}}\|v_n\|^{2}_{l^{2}} \end{eqnarray*} $

so we have $v_n\rightarrow 0$ in all $l^{q}, ~q>2$.

Note that from (H1) and (H2), we have for any $\varepsilon>0$, there exists $c_\varepsilon>0$ such that

$ \begin{eqnarray}\label{eqnf} |f(k, u)|\leq \varepsilon|u|+c_{\varepsilon}|u|^{p-1} ~~{\rm~and~}~~ |F(k, u)|\leq \varepsilon|u|^{2}+c_{\varepsilon}|u|^{p}. \end{eqnarray} $

(3.7)

Then for each $s>0$, we have

$ \sum\limits_{k\in \mathbb{Z}}{F}(k, s{{v}_{n}}(k))\le \varepsilon {{s}^{2}}\|{{v}_{n}}\|_{{{l}^{2}}}^{2}+{{c}_{\varepsilon }}{{s}^{p}}\|{{v}_{n}}\|_{{{l}^{p}}}^{p}, $

which implies that $\sum\limits_{k\in\mathbb{Z}}F(k, sv_{n}(k))\rightarrow 0 {\rm~as~} n\rightarrow\infty$.So

$ \begin{eqnarray}\label{eqnd} d\geq J(u_n)\geq J(sv_n)=\frac{s^{2}}{2}\|v^{(k)}\|^{2}-\sum_{k\in\mathbb{Z}}F(k, sv_{n}(k))\rightarrow \frac{s^{2}}{2}, \end{eqnarray} $

(3.8)

as $n\rightarrow \infty$. This is a contradiction if $s>\sqrt{2d}$.

By periodicity of coefficients, we know $J$ and $\mathcal{N}$ are both invariant under $ \mathit{T}$-translation. Making such shifts, we can assume that $1\leq k_j\leq T-1$ in (3.6). Moreover, passing to a subsequence, we can assume that $k_{j}=k_{0}$ is independent of $j$.

Next we can extract a subsequence, still denoted by $\{v_n\}$, such that $v_{n}(k)\rightarrow v(k)$ for all $k\in\mathbb{Z}$. Specially, for $k=k_{0}$, inequality (3.6) shows that $|v(k_0)|\geq \delta$, so $v\neq 0$. Since $|u_{n}(k)|\rightarrow \infty$ as $n\rightarrow \infty$, it follows again from (H3) that

$ 0\le \frac{J({{u}_{n}})}{\|{{u}_{n}}{{\|}^{2}}}=\frac{1}{2}-\sum\limits_{k\in \mathbb{Z}}{\frac{F(k, {{u}_{n}}(k))}{{{({{u}_{n}}(k))}^{2}}}}{{({{v}_{n}}(k))}^{2}}\to -\infty ~~~~~~~as~n\to \infty, $

a contradiction again.

From above, we have the following lemma, which is important in this paper.

Lemma3.3   $\{w_{n}\}$ is a Palais-Smale sequence for $\Psi$ if and only if $\{m(w_{n})\}$ is a Palais-Smale sequence for $J$.

Proof  Let $\{w_{n}\}$ be a Palais-Smale sequence for $\Psi$, and let $u_{n}=m(w_{n})\in\mathcal{ N}$. Since for every $w_{n}\in S$ we have an orthogonal splitting $E=T_{w_{n}}S\oplus\mathbb{R}w_{n}$, we have

$ \begin{eqnarray*} \|\Psi'(w_{n})\|=\sup_{\substack {z\in T_{w_{n}}S \\ \|z\|=1}}\Psi'(w_{n})z=\|m(w_{n})\|\sup_{\substack {z\in T_{w_{n}}S \\ \|z\|=1}}J'(m(w_{n}))z=\|u_{n}\|\sup_{\substack {z\in T_{w_{n}}S \\ \|z\|=1}}J'(u_{n})z. \end{eqnarray*} $

Then

$ \begin{eqnarray*} \|\Psi'(w_{n})\|&\leq& \|u_{n}\|\|J'(u_{n})\|=\|u_{n}\|\sup_{\substack {z\in T_{w_{n}}S, ~t\in\mathbb{R} \\ z+tw\neq 0}}\frac{J'(u_{n})(z+tw)}{\|z+tw\|}\\ &\leq&\|u_{n}\|\sup_{z\in T_{w_{n}}S\backslash\{0\}}\frac{J'(u_{n})(z)}{\|z\|}=\|\Psi'(w_{n})\|. \\ \end{eqnarray*} $

Therefore

$ \begin{eqnarray}\label{eqn3.7} \|\Psi'(w_{n})\|=\|u_{n}\|\|J'(u_{n})\|. \end{eqnarray} $

(3.9)

By (3.5), for $u_{n}\in\mathcal{ N}$, $J(u_{n})>0$, so there exists a constant $c_0>0$ such that $J(u_{n})>c_0$. And since $c_0\leq J(u_{n})=\frac{1}{2}~\|u_{n}\|^{2}-I(u_{n})\leq \frac{1}{2}~\|u_{n}\|^{2}$, $\|u_{n}\|\geq \sqrt{2c_0}$. Together with Lemma3.2, $\sqrt{2c_0}\leq \|u_{n}\|\leq\sup_{n}\|u_{n}\|<\infty$. Hence $\{w_{n}\}$ is a Palais-Smale sequence for $\Psi$ if and only if $\{u_{n}\}$ is a Palais-Smale sequence for $J$.

Now, we give the detailed proof of Theorem 1.1.

Proof  From(3.9), $\Psi'(w)=0$ if and only if $J'(m(w))=0$. So $w$ is a critical point of $\Psi$ if and only if $m(w)$ is a nontrivial critical point of $J$. Moreover, the corresponding values of $\Psi$ and $J$ coincide and $\inf_{S}\Psi=\inf_{\mathcal{N}}J$.

Let $u_{0}\in \mathcal{N}$ such that $J(u_{0})=c$, then $m^{-1}(u_{0})\in S$ is a minimizer of $\Psi$ and therefore a critical point of $\Psi$, so $u_{0}$ is a critical point of $J$. It needs to show that there exists a minimizer $u\in \mathcal{N}$ of $J|_\mathcal{N}$.

Let $\{w_n\}\subset S$ be a minimizing sequence for $\Psi$. By Ekeland's variational principle we may assume $\Psi(w_n)\rightarrow c, ~\Psi'(w_n)\rightarrow 0$ as $n\rightarrow \infty$, hence $J(u_n)\rightarrow c, J'(u_n)\rightarrow 0$ as $n\rightarrow \infty$, where $u_n:=m(w_n)\in \mathcal{N}$.

It follows from Lemma 3.2 that $\{u_n\}$ is bounded in $\mathcal{N}$, then there exists a subsequence, still denoted by the same notation, such that $u_n$ weakly converges to some $u \in E$. We know that there exist $\delta>0$ and $k_j\in \mathbb{Z}$ such that

$ \begin{eqnarray}\label{eqnu} |u_n(k_j)|\geq \delta. \end{eqnarray} $

(3.10)

If not, then $u_n\rightarrow 0$ in $l^{\infty}$ as $n\rightarrow \infty$. Note that, for $q>2$,

$ \|{{u}_{n}}\|_{{{l}^{q}}}^{q}\le \|{{u}_{n}}\|_{{{l}^{\infty }}}^{q-2}\|{{u}_{n}}\|_{{{l}^{2}}}^{2}, $

then $u_n\rightarrow 0$ in all $l^{q}, ~q>2$. By (3.7), we have

$ \begin{eqnarray*} \sum_{k\in\mathbb{Z}}f(k, u_{n}(k))u_{n}(k) &\leq& \varepsilon\sum_{k\in \mathbb{Z}}|u_{n}(k)|\cdot|u_{n}(k)|+c_{\varepsilon}\sum_{k\in\mathbb{Z}} |u_{n}(k)|^{p-1}\cdot|u_{n}(k)|\\ &\leq& \varepsilon \|u_n\|_{l^{2}}\cdot\|u_n\|_{l^{2}}+c_{\varepsilon}\|u_n\|^{p-1}_{l^{p}}\cdot\|u_n\|_{l^{p}}\\ &\leq& \varepsilon \|u_n\|_{l^{2}}\cdot\|u_n\|+c_{\varepsilon}\|u_n\|^{p-1}_{l^{p}}\cdot\|u_n\|, \\ \end{eqnarray*} $

which implies that $\sum\limits_{k\in\mathbb{Z}}f(k, u_{n}(k))u_{n}(k)=o(\|u_n\|){\rm~as~} n\rightarrow\infty$. Therefore

$ \begin{eqnarray*} o(\|u_n\|)=\left(J'(u_n), u_n\right) = \|u_n\|^{2}-\sum_{k\in \mathbb{Z}}f(k, u_{n}(k))u_{n}(k)=\|u_n\|^{2}-o(\|u_n\|). \end{eqnarray*} $

So $\|u_n\|^{2}\rightarrow 0$, as $n\rightarrow\infty$, which contradicts with $u_n\in\mathcal{N}$.

Since $J$ and $J'$ are both invariant under $T$-translation. Making such shifts, we assume that $1\leq k_j\leq T-1$ in (3.10). Moreover passing to a subsequence, we assume that $k_j=k_{0}$ is independent of $j$. Extracting a subsequence, still denoted by $\{u_n\}$, we have $u_n\rightharpoonup u$ and $u_{n}(k)\rightarrow u(k)$ for all $k\in\mathbb{Z}$. Specially, for $k=k_{0}$, inequality (3.10) shows that $|u(k_0)|\geq \delta$, so $u\neq 0$. Hence $u\in \mathcal{N}$.

Now, we prove that $J(u)=c$. By Fatou's lemma,

$ \begin{eqnarray*} c&=&\lim_{n\rightarrow\infty}\left(J(u_n)-\frac{1}{2}J'(u_n)u_n\right)=\lim_{n\rightarrow\infty}\sum_{k\in \mathbb{Z}}\left(\frac{1}{2}f(k, u_{n}(k))u_{n}(k)-F(k, u_{n}(k))\right)\\ &\geq& \sum_{k\in \mathbb{Z}} \left(\frac{1}{2}f(k, u(k))u(k)-F(k, u(k))\right)=J(u)-\frac{1}{2}J'(u)u=J(u)\geq c. \end{eqnarray*} $

So $J(u)=c$. The proof of Theorem 1.1 is completed.

Example1  Consider the difference equation

$ \begin{eqnarray} &&\Delta[|\sin k|\Delta u(k-1)]-|\cos k|u(k)\nonumber\label{eqnex} \\ &&+\left[c(\alpha+2)u(k)|u(k)|^{\alpha}+d(\beta+2)u(k)|u(k)|^{\beta}\right](\phi(k)+M)=0, \end{eqnarray} $

(3.11)

where $c>0, d>0, \alpha\geq\beta>0, M>0, \phi(k)$ is a bounded continuous $\pi-$periodic function and $|\phi(k)|<M$, $k\in\mathbb{Z}$. Let $a(k)=|\sin k|$, $b(k)=|\cos k|$,

$ f(k, u)=\left[c(\alpha+2)u|u|^{\alpha}+d(\beta+2)u|u|^{\beta}\right](\phi(k)+M) $

and

$ F(k, u)=\int_{0}^{u}f(k, s) ds=\left[c|u|^{\alpha+2}+d|u|^{\beta+2}\right](\phi(k)+M). $

It is easy to show that all the assumptions of Theorem 1.1 are satisfied. Therefore, equation (3.11) has at least one homoclinic solution.