\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 173, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/173\hfil Existence of solitons] {Existence of solitons for discrete nonlinear Schr\"odinger equations} \author[H. Shi, Y. Zhang \hfil EJDE-2016/173\hfilneg] {Haiping Shi, Yuanbiao Zhang} \address{Haiping Shi \newline Modern Business and Management Department, Guangdong Construction Polytechnic, Guangzhou 510440, China} \email{shp7971@163.com} \address{Yuanbiao Zhang \newline Packaging Engineering Institute, Jinan University, Zhuhai 519070, China} \email{abiaoa@163.com} \thanks{Submitted December 11, 2015. Published July 5, 2016.} \subjclass[2010]{39A12, 39A70, 35C08} \keywords{Existence; soliton; discrete nonlinear Schr\"odinger equation; \hfill\break\indent critical point theory} \begin{abstract} By using the Mountain Pass Lemma, we establish sufficient conditions for the existence of solitons for the discrete nonlinear Schr\"odinger equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The discrete nonlinear Schr\"odinger (DNLS) equation is one of the most important inherently discrete models. DNLS equations play a crucial role in the modeling of a great variety of phenomena, ranging from solid state and condensed matter physics to biology \cite{ChrLS,FlaG,FleSEC}. For example, they have been successfully applied to the modeling of localized pulse propagation optical fibers and wave guides, to the study of energy relaxation in solids, to the behavior of amorphous material, to the modeling of self-trapping of vibrational energy in proteins or studies related to the denaturation of the DNA double strand \cite{FleSEC}. Below $\mathbb{N}$, $\mathbb{z}$ and $\mathbb{R}$ denote the sets of all natural numbers, integers and real numbers respectively. For $a$ and $b$ in $\mathbb{z}$, define $\mathbb{z}(a,b)=\{a,a+1,\dots,b\}$ when $a\leq b$. This article concerns the DNLS equation \begin{equation} \label{e1.1} i\dot{\psi}_n=-\Delta \psi_n+\varepsilon_n\psi_n-f_n(\psi_n),\ n\in \mathbb{z}, \end{equation} where $\Delta \psi_n=\psi_{n+1}+\psi_{n-1}-2\psi_n$ is discrete Laplacian operator, $\varepsilon_n$ is real valued for each $n\in \mathbb{z}$, $f_n\in C(\mathbb{R},\mathbb{R})$, $f_n(0)=0$ and the nonlinearity $f_n(u)$ is gauge invariant, that is, \begin{equation} \label{e1.2} f_n(e^{i\theta}u)=e^{i\theta}f_n(u),\quad \theta\in \mathbb{R}. \end{equation} Since solitons are spatially localized time-periodic solutions and decay to zero at infinity. Thus, $\psi_n$ has the form $$ \psi_n=u_ne^{-i\omega t}, $$ and $$ \lim_{|n|\to \infty}\psi_n=0, $$ where $\psi_n$ is real valued for each $n\in \mathbb{z}$ and $\omega\in\mathbb{R}$ is the temporal frequency. Then \eqref{e1.1} becomes \begin{equation} \label{e1.3} -\Delta u_n+\varepsilon_nu_n-\omega u_n=f_n(u_n),\quad n\in \mathbb{z}, \end{equation} and \begin{equation} \label{e1.4} \lim_{|n|\to \infty}u_n=0 \end{equation} holds. Actually, our methods allow us to consider the following more general equation \begin{equation} \label{e1.5} -\Delta(p_{n}(\Delta u_{n-1})^\delta)+q_nu_n^\delta =f_n(u_{n+T},u_n,u_{n-T}),\ n\in \mathbb{z}, \end{equation} with the same boundary condition \eqref{e1.4}. Here, $\Delta$ is the forward difference operator $\Delta u_n=u_{n+1}-u_n$, $\Delta^2 u_n=\Delta(\Delta u_n)$, $p_n$ and $q_n$ are real valued for each $n\in \mathbb{z}$, $\delta>0$ is the ratio of odd positive integers, $f_n\in C(\mathbb{R}^4,\mathbb{R})$, $T$ is a given nonnegative integer. When $\delta=1, p_n\equiv1, q_n\equiv\varepsilon_n-\omega$ and $T=0$, we obtain \eqref{e1.3}. Naturally, if we look for solitons of \eqref{e1.1}, we just need to get the solutions of \eqref{e1.5} satisfying \eqref{e1.4}. When $f_n(u_{n+T},u_n,u_{n-T})=0,\ n\in \mathbb{z}(0)$, \eqref{e1.5} reduces to the equation \begin{equation} \label{e1.6}\numberwithin{equation}{section} \Delta(p_{n}(\Delta u_{n-1})^\delta)+q_nu_n^\delta=0, \end{equation} which has been studied in \cite{He} for results on oscillation, asymptotic behavior and the existence of positive solutions. In 2008, Cai and Yu \cite{CaY} obtained some sufficient conditions for the existence of periodic solutions of the nonlinear difference equation \begin{equation} \label{e1.7} \Delta(p_{n}(\Delta u_{n-1})^\delta)+q_nu_n^\delta=f_n(u_n),\quad n\in \mathbb{z}. \end{equation} It is well known that critical point theory is an effective approach to study the behavior of differential equations \cite{GuRW,GuRWA,GuRXA1,GuRXA2,MaW,Ra}. Only since 2003, critical point theory has been employed to establish sufficient conditions on the existence of periodic solutions for second order difference equations \cite{GuY1,GuY2}. Along this direction, Ma and Guo \cite{MaG1} (without periodicity assumption) and \cite{MaG2} (with periodicity assumption) applied variational methods to prove the existence of homoclinic orbits for the special form of \eqref{e1.5} (with $\delta=1$ and $T=0$). Chen and Wang \cite{ChW} studied the existence infinitely many homoclinic orbits of the following nonlinear difference equation \begin{equation} \label{e1.8} \Delta(p_{n}(\Delta u_{n-1})^\delta)-q_nu_n^\delta+f_n(u_n)=0,\ n\in \mathbb{z}, \end{equation} by using the Symmetric Mountain Pass Lemma. In the past decade, the existence of solitons of the DNLS equations has drawn a great deal of interest \cite{HuZ1,HuZ2,MaZ1,MaZ2,Pa1,Pa2,Zh,ZhL,ZhM,ZhY, ZhYC}. The existence for the periodic DNLS equations with superlinear nonlinearity \cite{MaZ1,MaZ2,Pa1,Pa2}, and with saturable nonlinearity \cite{ZhY,ZhYC} has been studied. And the existence results of solitons of the DNLS equations without periodicity assumptions were established in \cite{HuZ1,HuZ2,Zh, ZhL,ZhM}. As for the existence of the homoclinic orbits of nonlinear Schr\"odinger equations, we refer to \cite{ChTi,Ta1,Ta2,Ta3}. Our main results are the following theorems. \begin{theorem} \label{thm1.1} Suppose that the following hypotheses are satisfied: \begin{itemize} %(p), (q) \item[(A1)] for any $n\in\mathbb{Z}$, $p_n>0$; \item[(A2)] for any $n\in\mathbb{Z}$, $\underline{q}=\inf_{n\in\mathbb{Z}}q_n>0$ and $\lim_{|n|\to +\infty}q_n=+\infty$; \item[(A3)] there exists a function $F_n(v_1,v_2)\in C^1(\mathbb{R}^3,\mathbb{R})$ satisfies \begin{gather*} \frac{\partial F_{n-T}(v_2,v_3)}{\partial v_2} +\frac{\partial F_n(v_1,v_2)}{\partial v_2} =f_n(v_1,v_2,v_3), \\ \lim_{\beta_1\to 0} \frac{F_n(v_1,v_2)} {\beta_1^{\delta+1}}=0\quad\text{ uniformly for }n\in\mathbb{Z}\setminus M,\; \beta_1=(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}, \\ \lim_{\beta_2\to 0} \frac{f_n(v_1,v_2,v_3)} {\beta_2^\delta}=0\quad\text{uniformly for }n\in\mathbb{Z}\setminus M,\; \beta_2=(v_1^{\delta+1}+v_2^{\delta+1}+v_3^{\delta+1})^{\frac{1}{\delta+1}}; \end{gather*} \item[(A4)] for each $n\in\mathbb{Z}$, $F_n(v_1,v_2)=W_n(v_2)-H_n(v_1,v_2)$, $W, H$ are continuously differentiable in $v_2$ and $v_1,v_2$ respectively. Moreover, there is a bounded set $M\subset\mathbb{Z}$ such that $H_n(v_1,v_2)\geq0$; \item[(A5)] there is a constant $\mu>\delta+1$ such that $$ 0<\mu W_n(v_2)\leq \frac{\partial W_n(v_2)}{\partial v_2}v_2,\quad \forall (n, v_2)\in \mathbb{Z}\times(\mathbb{R}\setminus\{0\}); $$ \item[(A6)] $H_n(0,0)=0$ and there is a constant $\varrho\in(\delta+1,\mu)$ such that $$ \frac{\partial H_n(v_1,v_2)}{\partial v_1}v_1 +\frac{\partial H_n(v_1,v_2)}{\partial v_2}v_2\leq\varrho H_n(v_1,v_2); $$ \item[(A7)] there exists a constant $c$ such that \[ H_n(v_1,v_2)\leq c(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{\varrho}{{\delta+1}}}\quad \text{for }n\in\mathbb{Z},\; v_1^{\delta+1}+v_2^{\delta+1}>1. \] \end{itemize} Then \eqref{e1.5} has a nontrivial solution satisfying \eqref{e1.4}. \end{theorem} \begin{theorem} \label{thm1.2} Suppose that {\rm (A1)--(A3), (A5)--(A8)}, and the following hypothesis are satisfied: \begin{itemize} \item[(A4')] for each $n\in\mathbb{Z}$, $F_n(v_1,v_2)=W_n(v_2)-H_n(v_1,v_2)$, $W, H$ are continuously differentiable in $v_2$ and $v_1,v_2$ respectively; \end{itemize} or \[ \lim_{\beta_1\to 0} \frac{F_n(v_1,v_2)} {\beta_1^{\delta+1}}=0\quad\text{uniformly for } n\in\mathbb{Z},\; \beta_1=(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}. \] Then \eqref{e1.5} has a nontrivial solution satisfying \eqref{e1.4}. \end{theorem} \begin{remark} \label{rmk1.3} \rm Equations similar in structure to \eqref{e1.5} are discussed by Zhang et al \cite{Zh,ZhL} under the assumption that $f$ satisfies: $$ 0<(q-1)f(u)u\leq f'(u)u^2,\quad \forall u\neq0 $$ holds for some constant $q\in(2,+\infty)$. This is a stronger condition than the classical Ambrosetti- Rabinowitz superlinear condition, i.e., there exist constants $q>2$ and $r>0$ such that $$ 00$ such that $J|_{\partial B_\rho}\geq \alpha$, and \item there exists $e\in E\setminus B_\rho$ such that $J(e)\leq 0$. \end{enumerate} Then $J$ possesses a critical value $c\geq \alpha$ given by \begin{equation} \label{e2.5} c=\inf_{g\in \Gamma}\max_{s\in[0,1]} J(g(s)), \end{equation} where \begin{equation} \label{e2.6} \Gamma =\{g\in C([0,1],E)|g(0)=0,\ g(1)=e\}. \end{equation} \end{lemma} \begin{lemma} \label{lem2.2} For $u\in E$, \begin{equation} \label{e2.7} \underline{q}\|u\|_\infty^{\delta+1} \leq\underline{q}\|u\|_{\delta+1}^{\delta+1}\leq\|u\|^{\delta+1}. \end{equation} \end{lemma} \begin{proof} Since $u\in E$, it follows that $\lim_{|n|\to \infty}|u_n|=0$. Hence, there exists $n^*\in\mathbb{Z}$ such that $$ \|u\|_\infty=|u_{n^*}|=\max_{n\in\mathbb{Z}}|u_n|. $$ By (A2) and \eqref{e2.2}, we have $$ \|u\|^{\delta+1}=\sum_{n\in\mathbb{Z}}\big[p_{n}(\Delta u_{n-1})^{\delta+1}+ q_nu_n^{\delta+1}\big] \geq\underline{q}\sum_{n\in\mathbb{Z}}u_n^{\delta+1} \geq\underline{q}\|u\|_\infty^{\delta+1}. $$ The proof is complete. \end{proof} \begin{lemma} \label{lem2.3} Suppose that {\rm (A5)} holds. Then for each $(n,u)\in\mathbb{Z}\times\mathbb{R}$, $s^{-\mu}W_n(su)$ is nondecreasing on $(0,+\infty)$. \end{lemma} The proof of the above lemma is routine and so we omit it. \begin{lemma} \label{lem2.4} Suppose that {\rm (A1)--(A8)} are satisfied. Then $J$ satisfies the (PS) condition. \end{lemma} \begin{proof} Let $\{u^{(k)}\}_{k\in\mathbb{N}}\subset E$ be such that $\{J(u^{(k)})\}_{k\in\mathbb{N}}$ is bounded and $J'(u^{(k)})\to 0$ as $k\to \infty$. Then there is a positive constant $K$ such that \[ |J(u^{(k)})|\leq K,\quad \|J'(u^{(k)})\|_{E^*}\leq \rho K\quad \text{for } k\in\mathbb{N}. \] Thus, by \eqref{e2.3}, (A5) and (A6), we have \begin{align*} &(\delta+1)K+(\delta+1)K\|u^{(k)} \| \\ & \geq (\delta+1)J(u^{(k)})-\frac{(\delta+1)}{\varrho} \langle J'(u^{(k)}),u^{(k)}\rangle \\ &=\frac{\varrho-(\delta+1)}{\varrho}\|u^{(k)}\|^{\delta+1} -(\delta+1)\sum^{+\infty}_{n=-\infty}\big[W_n(u_n^{(k)}) -\frac{1}{\varrho}\frac{\partial W_n(u_n^{(k)})}{\partial v_2}u_n^{(k)}\big] \\ &\quad +(\delta+1)\sum^{+\infty}_{n=-\infty}H_n(u_{n+T}^{(k)},u_n^{(k)}) \\ &\quad -\frac{(\delta+1)}{\varrho}\sum^{+\infty}_{n=-\infty} \Big[\frac{\partial H_n(u_{n+T}^{(k)},u_n^{(k)})}{\partial v_1}u_{n+T}^{(k)} +\frac{\partial H_n(u_{n+T}^{(k)},u_n^{(k)})}{\partial v_2}u_n^{(k)}\Big] \\ &\geq\frac{\varrho-(\delta+1)}{\varrho}\|u^{(k)}\|^{\delta+1}. \end{align*} Since $\varrho>\delta+1$, it is not difficult to know that $\{u^{(k)}\}_{k\in\mathbb{N}}$ is a bounded sequence in $E$, i.e., there exists a constant $K_1>0$ such that \begin{equation} \label{e2.8} \|u^{(k)}\|\leq K_1,\quad k\in\mathbb{N}. \end{equation} So passing to a subsequence if necessary, it can be assumed that $u^{(k)}\rightharpoonup u^{(0)}$ in $E$. For any given number $\varepsilon>0$, by (A3), we can choose $\zeta>0$ such that \begin{equation} \label{e2.9} |f_n(u_{n+T},u_n,u_{n-T})|\leq\varepsilon (u_{n+T}^{\delta+1}+u_n^{\delta+1}+u_{n-T}^{\delta+1})^{\frac{\delta}{\delta+1}}, \quad \forall n\in\mathbb{Z}\setminus M,\; u\in\mathbb{R}, \end{equation} where $(u_{n+T}^{\delta+1}+u_n^{\delta+1}+u_{n-T}^{\delta+1}) ^{\frac{1}{\delta+1}}\leq\zeta$. By (A2), we can also choose a positive integer $D>\max\{\max\{|n|:n\in M\}, T\}$ such that \begin{equation} \label{e2.10} q_n\geq\frac{K_1^{\delta+1}}{\zeta^{\delta+1}},\ |n|\geq D. \end{equation} By \eqref{e2.8} and \eqref{e2.10}, we obtain \begin{equation} \label{e2.11} (u^{(k)}_n)^{\delta+1}= \frac{1}{q_n}q_n(u^{(k)}_n)^{\delta+1} \leq\frac{\zeta^{\delta+1}}{K_1^{\delta+1}}\|u^{(k)}\|^{\delta+1} \leq\zeta^{\delta+1},\quad |n|\geq D. \end{equation} Since $u^{(k)}\rightharpoonup u^{(0)}$ in $E$, it is easy to verify that $u^{(k)}_n$ converges to $u^{(0)}_n$ pointwise for all $n\in\mathbb{Z}$; that is, \begin{equation} \label{e2.12} \lim_{k\to \infty} u^{(k)}_n=u^{(0)}_n,\quad \forall n\in\mathbb{Z}. \end{equation} Combining with \eqref{e2.11}, we have \begin{equation} \label{e2.13} (u^{(0)}_n)^{\delta+1}\leq\zeta^{\delta+1},\quad |n|\geq D. \end{equation} It follows from \eqref{e2.12} and the continuity of $f_n(v_1,v_2,v_3)$ on $v_1,v_2,v_3$ that there exists $k_0\in\mathbb{N}$ such that \begin{equation} \label{e2.14} \sum^{D}_{n=-D}\big|f_n\big(u_{n+T}^{(k)},u_n^{(k)},u_{n-T}^{(k)}\big) -f_n\big(u_{n+T}^{(0)},u_n^{(0)},u_{n-T}^{(0)}\big)\big|<\varepsilon,\quad k\geq k_0. \end{equation} On the other hand, it follows from (A3), \eqref{e2.7}, \eqref{e2.8}, \eqref{e2.9}, \eqref{e2.11} and \eqref{e2.13} that \begin{equation} \label{e2.15} \begin{aligned} &\sum_{|n|\geq D}\big| f_n\big(u_{n+T}^{(k)},u_n^{(k)},u_{n-T}^{(k)}\big) -f_n\big(u_{n+T}^{(0)},u_n^{(0)},u_{n-T}^{(0)}\big)\big||u^{(k)}_n-u^{(0)}_n|\\ &\leq\sum_{|n|\geq D}\big[\big|f_n\big(u_{n+T}^{(k)},u_n^{(k)},u_{n-T}^{(k)}\big)\big| +\big|f_n\big(u_{n+T}^{(0)},u_n^{(0)},u_{n-T}^{(0)}\Big)\big|\big] (|u^{(k)}_n|+|u^{(0)}_n|)\\ &\leq\varepsilon\sum_{|n|\geq D} \Big\{\big[(u^{(k)}_{n+T})^{\delta+1}+(u^{(k)}_n)^{\delta+1} +(u^{(k)}_{n-T})^{\delta+1}\big]^{\frac{\delta}{\delta+1}} \\ &\quad +\big[(u^{(0)}_{n+T})^{\delta+1}+(u^{(0)}_n)^{\delta+1} +(u^{(0)}_{n-T})^{\delta+1}\big]^{\frac{\delta}{\delta+1}}\Big\} (|u^{(k)}_n|+|u^{(0)}_n|) \\ &\leq3\varepsilon\sum_{n=-\infty}^{+\infty} \big[|u^{(k)}_n|^\delta+|u^{(0)}_n|^\delta\big] (|u^{(k)}_n|+|u^{(0)}_n|) \\ &\leq 6\varepsilon\sum_{n=-\infty}^{+\infty}\big[(u^{(k)}_n)^{\delta+1} +(u^{(0)}_n)^{\delta+1}\big] \\ &\leq\frac{6\varepsilon}{\underline{q}}(K_1^{\delta+1}+\|u^{(0)}\|^{\delta+1}). \end{aligned} \end{equation} Since $\varepsilon$ is arbitrary, we obtain \begin{equation} \label{e2.16} \sum^{+\infty}_{n=-\infty} \big|f_n\big(u_{n+T}^{(k)},u_n^{(k)},u_{n-T}^{(k)}\big) -f_n\big(u_{n+T}^{(0)},u_n^{(0)},u_{n-T}^{(0)}\big)\big|\to 0,\quad k\to \infty. \end{equation} It follows from \eqref{e2.2}, \eqref{e2.4} and \eqref{e2.7} that \begin{align*} &\langle J'(u^{(k)})-J'(u^{(0)}),u^{(k)}-u^{(0)}\rangle \\ &=\|u^{(k)}-u^{(0)}\|^{\delta+1} \\ &\quad -\sum^{+\infty}_{n=-\infty}\big[f_n\big(u_{n+T}^{(k)},u_n^{(k)},u_{n-T}^{(k)}\big) -f_n\big(u_{n+T}^{(0)},u_n^{(0)},u_{n-T}^{(0)}\big)\big] (u^{(k)}-u^{(0)}). \end{align*} Therefore, \begin{align*} &\|u^{(k)}-u^{(0)}\|^{\delta+1} \\ &\leq \langle J'(u^{(k)})-J'(u^{(0)}),u^{(k)}-u^{(0)}\rangle \\ &\quad +\sum^{+\infty}_{n=-\infty}\big[f_n\big(u_{n+T}^{(k)},u_n^{(k)},u_{n-T}^{(k)}\big) -f_n\big(u_{n+T}^{(0)},u_n^{(0)},u_{n-T}^{(0)}\big)\big] (u^{(k)}-u^{(0)}). \end{align*} Since $\langle J'(u^{(k)})-J'(u^{(0)}),u^{(k)}-u^{(0)}\rangle \to 0$ as $ k\to \infty$, we have $u^{(k)}\to u^{(0)}$ in $E$. The proof is complete. \end{proof} \section{Proofs of theorems} In this section, we shall obtain the existence of a nontrivial solution of \eqref{e1.5} satisfying \eqref{e1.4} by using the critical point method. \begin{proof}[Proof of Theorem \ref{thm1.1}] We shall prove the existence of a nontrivial solution to \eqref{e1.5} satisfying \eqref{e1.4}. It is clear that $J(0)=0$. We have already known that $J\in C^1(E,\mathbb{R})$ and $J$ satisfies the (PS) condition. Hence, it suffices to prove that $J$ satisfies the conditions for the (PS) condition. By (A3), there exists $\eta\in(0,1)$ such that \begin{equation} \label{e3.1} |F_n(u_{n+T},u_n)|\leq\frac{1}{4(\delta+1)}(u_{n+T}^{\delta+1}+u_n^{\delta+1}),\quad \forall n\in\mathbb{Z}\setminus M,\; (u_{n+T}^{\delta+1}+u_n^{\delta+1})^{\frac{1}{\delta+1}}\leq\eta. \end{equation} Set \begin{equation} \label{e3.2} G=\sup\{W_n(v_2)|v_2\in\mathbb{R},\ v_2^{\delta+1}=1\}, \end{equation} and $$ \theta=\min\big\{[\frac{\underline{q}}{8(\delta+1)(G+1)} ]^{\mu-(\delta+1)},\eta\big\}. $$ If $\|u\|=\underline{q}^{\frac{1}{\delta+1}}\theta:=\rho$, then by Lemma \ref{lem2.3}, $|u_n| \leq\theta\leq\eta<1$ for $n\in\mathbb{Z}$. By (A3), \eqref{e3.1}, \eqref{e3.2} and Lemma \ref{lem2.3}, we have \begin{equation} \label{e3.3} \begin{aligned} \sum_{n\in M}W_n(u_n) &\leq\sum_{n\in M,\ u_n\neq0}W_n(\frac{u_n}{|u_n|})|u_n|^\mu \\ &\leq G\sum_{n\in M}|u_n|^\mu \\ &\leq G\theta^{\mu-(\delta+1)}\sum_{n\in M}u_n^{\delta+1} \\ &\leq \frac{G\theta^{\mu-(\delta+1)}}{\underline{q}} \sum_{n\in M}q_nu_n^{\delta+1}\\ &\leq\frac{1}{8(\delta+1)}\sum_{n\in M}q_nu_n^{\delta+1}. \end{aligned} \end{equation} Set $\alpha=\frac{1}{2(\delta+1)}\theta^{\delta+1}$. Hence, from \eqref{e2.3}, \eqref{e3.1}, \eqref{e3.2}, (A2)--(A4), we have \begin{equation} \label{e3.4} \begin{aligned} J(u) &\geq\frac{1}{\delta+1}\|u\|^{\delta+1} -\sum_{n\in \mathbb{Z}\setminus M}F_n(u_{n+T},u_n) -\sum_{n\in M}F_n(u_{n+T},u_n) \\ &\geq\frac{1}{\delta+1}\|u\|^{\delta+1} -\frac{1}{8(\delta+1)}\sum_{n\in \mathbb{Z}\setminus M} (u_{n+T}^{\delta+1}+u_n^{\delta+1})-\sum_{n\in M}W_n(u_n) \\ &\quad +\sum_{n\in M}H_n(u_{n+T},u_n) \\ &\geq\frac{1}{\delta+1}\|u\|^{\delta+1} -\frac{1}{4(\delta+1)}\sum_{n\in \mathbb{Z}\setminus M}q_nu_n^{\delta+1} -\frac{1}{4(\delta+1)}\sum_{n\in M}q_nu_n^{\delta+1} \\ &\geq\frac{1}{\delta+1}\|u\|^{\delta+1}-\frac{1}{4(\delta+1)}\|u\|^{\delta+1} -\frac{1}{4(\delta+1)}\|u\|^{\delta+1} \\ &=\frac{1}{2(\delta+1)}\|u\|^{\delta+1} =\alpha. \end{aligned} \end{equation} This inequality shows that $\|u\|=\rho$ implies that $J(u)\geq\alpha$, i.e., $J$ satisfies assumption (1) in Lemma \ref{lem2.1}. Next we shall verify the condition (2). Take $\tau\in E$ such that \begin{equation} \label{e3.5} |\tau_n|=\begin{cases} 1,& \text{for } |n|\leq1,\\ 0,& \text{for } |n|\geq2, \end{cases} \end{equation} and $|\tau_n|\leq 1$ for $|n|\in(1,2)$. For any $u\in E$, it follows from \eqref{e2.7} and (A7) that \begin{equation} \label{e3.6} \begin{aligned} &\sum_{n=-2}^2H_n(u_{n+T},u_n)\\ &=\sum_{n\in\mathbb{Z}(-2,2),\ u_{n+T}^{\delta+1}+u_n^{\delta+1}>1}H_n(u_{n+T},u_n) +\sum_{n\in\mathbb{Z}(-2,2),\ u_{n+T}^{\delta+1}+u_n^{\delta+1} \leq1}H_n(u_{n+T},u_n) \\ &\leq c\sum_{n\in\mathbb{Z}(-2,2),\ u_{n+T}^{\delta+1} +u_n^{\delta+1}>1}(u_{n+T}^{\delta+1}+u_n^{\delta+1})^{\frac{\varrho}{\delta+1}}\\ &\quad +\sum_{n\in\mathbb{Z}(-2,2),\ u_{n+T}^{\delta+1}+u_n^{\delta+1}\leq1} H_n(u_{n+T},u_n) \\ &\leq 2c\underline{q}^{-\frac{\varrho}{\delta+1}}\|u\|^\varrho+K_2, \end{aligned} \end{equation} where \[ K_2=\sum_{n\in\mathbb{Z}(-2,2),\ u_{n+T}^{\delta+1}+u_n^{\delta+1}\leq1} H_n(u_{n+T},u_n). \] For $\sigma>1$, by Lemma \ref{lem2.4} and \eqref{e3.5}, we have \begin{equation} \label{e3.7} \sum_{n=-1}^1W_n(\sigma u_n)\geq\sigma^\mu\sum_{n=-1}^1W_n(u_n)=K_3\sigma^\mu, \end{equation} where $K_3=\sum_{n=-1}^1W_n(u_n)>0$. By \eqref{e2.3}, \eqref{e3.5}, \eqref{e3.6} and \eqref{e3.7}, for $\sigma>1$, we have \begin{equation} \label{e3.8} \begin{aligned} J(\sigma\tau) &=\frac{1}{\delta+1}\|\sigma\tau\|^{\delta+1} +\sum_{n=-\infty}^{+\infty} [H_n(\sigma\tau_{n+T},\sigma\tau_n) -W_n(\sigma\tau_n) ] \\ &\leq\frac{\sigma^{\delta+1}}{\delta+1}\|\tau\|^{\delta+1}+\sum_{n=-2}^2 H_n(\sigma\tau_{n+T},\sigma\tau_n) -\sum_{n=-1}^1W_n(\sigma\tau_n) \\ &\leq\frac{\sigma^{\delta+1}}{\delta+1}\|\tau\|^{\delta+1} +2c\underline{q}^{-\frac{\varrho}{\delta+1}}\|u\|^\varrho+K_2 -K_3\sigma^\mu. \end{aligned} \end{equation} Since $\mu>\varrho>\delta+1$ and $K_3>0$, \eqref{e3.8} implies that there exists $\sigma_0>1$ such that $\sigma_0\tau>\rho$ and $J(\sigma_0\tau)< 0$. Set $e=\sigma_0\tau$. Then $e\in E$, $\|e\|=\|\sigma_0\tau\|>\rho$ and $J(e)=J(\sigma_0\tau) <0$. By Lemma \ref{lem2.1}, $J$ possesses a critical value $d\geq\alpha$ given by $$ d=\inf_{g\in \Gamma}\max_{s\in[0,1]} J(g(s)), $$ where $$ \Gamma =\{g\in C([0,1],E)|g(0)=0,\ g(1)=e\}. $$ Hence, there exists $u^*\in E$ such that $$ J(u^*)=d,\quad J'(u^*)=0. $$ Then function $u^*$ is a desired solution of \eqref{e1.5} satisfying \eqref{e1.4}. Since $d>0$, $u^*$ is a nontrivial solution. The desired results follow. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.2}] In the proof of Theorem \ref{thm1.1}, the condition that $H_n(v_1,v_2)\geq0$ for $(n,v_1,v_2)\in M\times \mathbb{R}^2$, $\beta_1=(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}$ in (A4) is only used in the proof of hypothesis (1) of Lemma \ref{lem2.1}. Thus, we only prove hypothesis (1) of Lemma \ref{lem2.1} still hold replacing (A4) by (A4'). By (A4'), we have \begin{equation} \label{e3.9} |F_n(u_{n+T},u_n)|\leq\frac{1}{4(\delta+1)}(u_{n+T}^{\delta+1}+u_n^{\delta+1}),\quad \forall n\in\mathbb{Z},\; (u_{n+T}^{\delta+1}+u_n^{\delta+1})^{\frac{1}{\delta+1}}\leq\eta. \end{equation} If $\|u\|=\underline{q}^{\frac{1}{\delta+1}}\eta:=\rho$, then by Lemma \ref{lem2.3}, $|u_n|\leq\eta$ for $n\in\mathbb{Z}$. Set $\alpha=\frac{1}{2(\delta+1)}\eta^{\delta+1}$. Hence, from \eqref{e2.3} and \eqref{e3.9}, we have \begin{equation} \label{e3.10} \begin{aligned} J(u) &\geq\frac{1}{\delta+1}\|u\|^{\delta+1} -\sum_{n=-\infty}^{+\infty}F_n(u_{n+T},u_n) \\ &\geq\frac{1}{\delta+1}\|u\|^{\delta+1} -\frac{1}{4(\delta+1)}\sum_{n=-\infty}^{+\infty}(u_{n+T}^{\delta+1}+u_n^{\delta+1})\\ &\geq\frac{1}{\delta+1}\|u\|^{\delta+1} -\frac{1}{2(\delta+1)}\sum_{n=-\infty}^{+\infty}q_nu_n^{\delta+1} \\ &\geq\frac{1}{\delta+1}\|u\|^{\delta+1}-\frac{1}{2(\delta+1)}\|u\|^{\delta+1} \\ &=\frac{1}{2(\delta+1)}\|u\|^{\delta+1} =\alpha. \end{aligned} \end{equation} This inequality shows that $\|u\|=\rho$ implies that $J(u)\geq\alpha$, i.e., $J$ satisfies assumption (1) of Lemma \ref{lem2.1}. The proof is complete. \end{proof} \begin{thebibliography}{00} \bibitem{CaY} X. C. Cai, J. S. Yu; \emph{Existence theorems of periodic solutions for second-order nonlinear difference equations}, Adv. Difference Equ., 2008 (2008), 1-11. \bibitem{ChT1} Chen, X. H. 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