\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 228, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/228\hfil Sequences of small homoclinic solutions]
{Sequences of small homoclinic solutions for difference equations
on integers}

\author[R. Stegli\'nski \hfil EJDE-2017/228\hfilneg]
{Robert Stegli\'nski}

\address{Robert Stegli\'nski \newline
Institute of Mathematics,
Lodz University of Technology,
Wolczanska 215, 90-924 Lodz, Poland}
\email{robert.steglinski@p.lodz.pl}

\dedicatory{Communicated by Vicentiu D. Radulescu}

\thanks{Submitted July 18, 2017. Published September 22, 2017.}
\subjclass[2010]{39A10, 47J30, 35B38}
\keywords{Difference equations; discrete $p$-Laplacian; variational methods; 
\hfill\break\indent infinitely many solutions}

\begin{abstract}
 In this article, we determine a concrete interval of positive parameters
 $\lambda $, for which we prove the existence of infinitely many homoclinic
 solutions for a discrete problem
 \[
 -\Delta \big( a(k)\phi _{p}(\Delta u(k-1))\big) +b(k)\phi_{p}(u(k))
 =\lambda f(k,u(k)),\quad k\in \mathbb{Z},
 \]
 where the nonlinear term $f:\mathbb{Z}\times \mathbb{R} \to \mathbb{R}$
 has an appropriate oscillatory behavior at zero. We use both the general
 variational principle of Ricceri and the direct method introduced by
 Faraci and  Krist\'aly \cite{FK}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{claim}[theorem]{Claim}
\allowdisplaybreaks

\section{Introduction}

 In this article we study the nonlinear second-order difference equation
\begin{equation}
\begin{gathered}
-\Delta \left( a(k)\phi _{p}(\Delta u(k-1))\right) +b(k)\phi
_{p}(u(k))=\lambda f(k,u(k)) \quad \text{for all $k\in\mathbb{Z}$} \\
u(k)\to 0 \quad \text{as $|k|\to \infty$}.
\end{gathered}  \label{eq}
\end{equation}
Here $p>1$ is a real number, $\lambda $ is a positive real parameter, 
$\phi _{p}(t)=|t|^{p-2}t$ for all $t\in {\mathbb{R}}$, 
$a,b:{\mathbb{Z}} \to \mathbb{(}0,+\infty )$, while 
$f:{\mathbb{Z}}\times {\mathbb{R}} \to {\mathbb{R}}$ is a continuous function.
 Moreover, the forward difference operator is defined as 
 $\Delta u(k-1)=u(k)-u(k-1)$. We say that a
solution $u=\{u(k)\}$ of \eqref{eq} is homoclinic if $\lim_{|
k| \to \infty }u(k)=0$.

Difference equations represent the discrete counterpart of ordinary
differential equations and are usually studies in connection with numerical
analysis. We may regard \eqref{eq} as being a discrete analogue of the
following second order differential equation
\[
-(a(t)\phi _{p}(x'(t)))'+b(t)\phi _{p}(x(t))=f(t,x(t)), \quad
t\in \mathbb{R}.
\]
The case $p=2$ in \eqref{eq} has been motivated in part by searching
standing waves for the nonlinear Schrodinger equation
\[
i\dot{\psi}_{k}+\Delta ^{2}\psi _{k}-\nu _{k}\psi _{k}+f(k,\psi _{k})=0, \quad
k\in \mathbb{Z}.
\]

Boundary value problems for difference equations can be studied in several
ways. It is well known that variational method in such problems is a
powerful tool. Many authors have applied different results of critical point
theory to prove existence and multiplicity results for the solutions of
discrete nonlinear problems. Studying such problems on bounded discrete
intervals allows for the search for solutions in a finite-dimensional Banach
space (see \cite{APR, BC, CIT, CM, BR1,MBR,RR}). The issue of finding solutions on
unbounded intervals is more delicate. To study such problems directly by
variational methods, \cite{IT} and \cite{MG} introduced coercive weight
functions which allow for preservation of certain compactness properties on $
l^{p}$-type spaces. That method was used in the following papers 
\cite{IR,K,St,St1,SM}.



The goal of the present paper is to establish the existence of a sequence of
homoclinic solutions for problem \eqref{eq}, which has been studied
recently in several papers. Infinitely many solutions were obtained in \cite
{SM} by employing Nehari manifold methods, in \cite{K} by applying a variant
of the fountain theorem, in \cite{St} by use of the Ricceri's theorem (see
\cite{BMB, R}) and in \cite{St1} by applying a direct argumentation. In the
two latter papers the nonlinearity $f$ has a suitable oscillatory behavior
at infinity. In this article we will prove that results analogous to \cite
{St} and \cite{St1} can be obtained assuming that the nonlinearity $f$ has a
suitable oscillatory behavior at zero.


A special case of our contributions reads as follows. For $b:{\mathbb{Z}}
\to {\mathbb{R}}$ and the continuous mapping $f:{\mathbb{Z}}\times {
\mathbb{R}}\to {\mathbb{R}}$ define the following conditions:

\begin{itemize}
\item[(A1)] $b(k)\geq \alpha >0$ for all $k\in \mathbb{Z}$, 
$b(k)\to +\infty $ as $| k| \to +\infty $;

\item[(A2)] there is $T_0>0$ such that $\sup_{| t| \leq T_0}| f(\cdot .t)|
\in l_1$;

\item[(A3)] $f(k,0)=0$ for all $k\in\mathbb{Z}$;

\item[(A4)] there are sequences $\{c_{m}\},\{d_{m}\}$
such that $0<d_{m+1}<c_{m}<d_{m}$,\\
 $\lim_{m\to \infty }d_{m}=0$ and $
f(k,t)\leq 0$ for every $k\in\mathbb{Z}$ and 
$t\in \lbrack c_{m},d_{m}],m\in \mathbb{N}$;

\item[(A5)] $\liminf_{t\to 0^{+}}~\frac{\sum_{k\in\mathbb{Z}}\max_{| \xi | 
\leq t}F(k,\xi )}{t^{p}}=0$; 

\item[(A6)] $\limsup_{(k,t)\to (+\infty,0^{+})}\frac{F(k,t)}
{[ a(k+1)+a(k)+b(k)] t^{p}}=+\infty $;

\item[(A7)] $\limsup_{(k,t)\to (-\infty
,0^{+})}\frac{F(k,t)}{[ a(k+1)+a(k)+b(k)] t^{p}}=+\infty $;

\item[(A8)] $\sup_{k\in \mathbb{Z}}
\big( \limsup_{t\to 0^{+}}\frac{F(k,t)}{[a(k+1)+a(k)+b(k)] t^{p}}\big) =+\infty $,\\
where $F(k,t)$ is the primitive function of $f(k,t)$, i.\ e.\ 
$F(k,t)=\int_0^{t}f(k,s)\,ds$ for every $t\in \mathbb{R}$ and
$k\in\mathbb{Z}$ .
\end{itemize}

The solutions are found in the normed space $(X,\| \cdot\| )$, where 
\begin{gather*}
X=\big\{ u:{\mathbb{Z}}\to {\mathbb{R}}\ :\
\sum_{k\in {\mathbb{Z}}}[ a(k)| \Delta u(k-1)|
^{p}+b(k)|u(k)|^{p}] <\infty \},\\
\| u\| =\Big(\sum_{k\in {\mathbb{Z}}}[ a(k)| \Delta u(k-1)|
^{p}+b(k)|u(k)|^{p}] \Big) ^{1/p}.
\end{gather*}

\begin{theorem} \label{tw0}
Assume that {\rm (A1)--(A4)} are satisfied. Moreover,
assume that at least one of the conditions 
{\rm (A6)--(A8)}, is satisfied. Then, for any $\lambda >0$, 
 problem \eqref{eq} admits a sequence of non-negative solutions in 
$X$ whose norms tend to zero.
\end{theorem}

\begin{theorem} \label{tw0'}
Assume that {\rm (A1), (A2), (A5)} are satisfied. Moreover,
assume that at least one of the conditions {\rm (A6)--(A8)}
is satisfied. Then, for any $\lambda >0$,  problem \eqref{eq}
admits a sequence of solutions in $X$ whose norms tend to zero.
\end{theorem}

 The issue of multiplicity of solutions can be investigated through
variational methods, which consist in seeking solutions of a difference
equation as critical points of an energy functional defined on a convenient
Banach space. In the proof for the first theorem a direct variational
approach is used, introduced in \cite{FK} and then used in such papers as
\cite{BRV, KMR, KMT, KRV, St1}. In the proof for the second theorem the
general variational principle of Ricceri is used, which was applied in \cite
{BC, BCh, BMB2, BMBR, BMBR1, St}. To obtain the differentiability of the
energy functional associated with problem \eqref{eq}, so far in the
literature the following condition has been used
\[
\lim_{t\to 0} \frac{| f(k,t)| }{|t| ^{p-1}}=0\quad
\text{uniformly for all }k\in \mathbb{Z},
\]
following \cite{IT,MG} and then used in \cite{St,St1,SM}.

 We cannot use the condition, as it contradicts each of the conditions
(A6)--(A8).

We obtain our results due to a suitable oscillatory behavior of the
nonlinearity $f$. Let us observe that to satisfy the condition (A8) it
suffices that a suitable oscillatory behavior is present for just one 
$k\in\mathbb{Z}$, while for satisfying conditions (A6) or (A7) a suitable
behavior of the nonlinearity $f$ needs to be maintained for an infinite
number of $k\in \mathbb{Z}$.

The plan of the paper is as follows: Section 2 is devoted to our abstract
framework, in Section 3 and Section 4 we prove more general versions of
Theorems \ref{tw0} and \ref{tw0'} respectively. In Section 5 we give
examples and we show that Theorem \ref{tw0} and Theorem \ref{tw0'} are
independent.

\section{Abstract framework}

For all $1\leq p<+\infty $, we denote by $\ell ^{p}$ the set of all
functions $u:{\mathbb{Z}}\to {\mathbb{R}}$ such that
\[
\| u\| _{p}^{p}=\sum_{k\in {\mathbb{Z}}}|u(k)|^{p}<+\infty .
\]
Moreover, we denote by $\ell ^{\infty }$ the set of all functions $u:{
\mathbb{Z}}\to {\mathbb{R}}$ such that
\[
\| u\| _{\infty }=\sup_{k\in {\mathbb{Z}}}|u(k)|<+\infty
\]

\begin{lemma} \label{difpsi}
Let a continuous function  $f:\mathbb{Z}\times \mathbb{R}\to {\mathbb{R}}$ satisfies
\begin{equation} \label{eF1h}
\sup_{| t| \leq T}| f(\cdot ,t)| \in l_1\text{ for all  }T>0.
\end{equation}
Then the functional $\Psi :l^{p}\to \mathbb{R}$ defined by
\begin{equation}
\Psi (u):=\sum_{k\in \mathbb{Z} }F(k,u(k))\quad \text{for all } u\in l^{p},  
\label{psi}
\end{equation}
where $F(k,s)=\int_0^{s}f(k,t)dt$ for $s\in \mathbb{R}$ and $k\in\mathbb{Z}$,
is continuously differentiable.
\end{lemma}

\begin{proof}
Let us fix $u,v\in l^{p}$. We will prove that
\begin{equation}
\lim_{\tau \to 0^{+}}\frac{\Psi (u+\tau v)-\Psi (u)}{\tau }
=\sum_{k\in \mathbb{Z}}f(k,u(k))v(k).  \label{poch}
\end{equation}
Put $r=\| u\| _{\infty }+\| v\| _{\infty }$
and $q(k)=\sup_{| t| \leq r}| f(k,t)| $
for all $k\in \mathbb{Z}$. We have $q\in l^{1}$, by \eqref{eF1h}.

Let us fix arbitrarily $\epsilon >0$. Then, there exists $h\in\mathbb{N}$ 
such that
\[
\sum_{| k| >h}| q(k)| <\frac{\epsilon }{3\| v\| _{\infty }}.
\]
We can find $0<\tau _0<1$ such that for all $0<\tau \leq \tau _0$,
\[
\sum_{| k| \leq h}\big| \frac{F(k,u(k)+\tau
v(k))-F(k,u(k))}{\tau }-f(k,u(k))v(k)\big| <\frac{\epsilon }{3}.
\]
Now fix $0<\tau <\tau _0$. For all $| k| >h$ we can
find $0\leq \tau _{k}\leq \tau $ such that
\[
\frac{F(k,u(k)+\tau v(k))-F(k,u(k))}{\tau }=f(k,u(k)+\tau _{k}v(k))v(k).
\]
We define $w\in l^{p}$ by putting $w(k)=0$ for all
$| k|\leq h$ and $w(k)=u(k)+\tau _{k}v(k)$ for all $| k| >h$.
So $\| w\| _{\infty }\leq r$ and
\begin{align*}
&\big| \frac{\Psi (u+\tau v)-\Psi (u)}{\tau }-\sum_{k\in\mathbb{Z}}f(k,u(k))v(k)
\big| \\
&\leq \frac{\epsilon }{3}+\sum_{|k| >h}| f(k,w(k))v(k)|
 +\sum_{|k| >h}| f(k,u(k))v(k)| \\
&\leq \frac{\epsilon }{3}+2\| v\| _{\infty}\sum_{| k| >h}q(k)
<\epsilon ,
\end{align*}
which proves \eqref{poch}. From \eqref{eF1h} and the continuity of the
embeddings $l^{p}\hookrightarrow l^{\infty }$ and $l^{1}\hookrightarrow
l^{p'}$, the linear operator on the right-hand side of \eqref{poch}
lies in $l^{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$, so $\Psi $
is Gateaux differentiable and
\[
\langle \Psi '(u),v\rangle =\sum_{k\in\mathbb{Z}}f(k,u(k))v(k).
\]
It remains to prove that $\Psi ':l^{p}\to l^{p'} $ is continuous.
 Let $(u_{n})$ be a sequence such that $u_{n}\to
u $ in $l^{p}$. Put $R=\max \{\| u\| _{\infty },
\sup_{n\in\mathbb{N}}\| u_{n}\| _{\infty }\}$ and
$Q(k)=\sup_{| t| \leq R}| f(k,t)| $ for all $k\in\mathbb{Z}$.
We have $Q\in l^{1}$, by \eqref{eF1h}. Fix an $\epsilon >0$
arbitrarily. There exists $h\in \mathbb{N}$ such that
\begin{equation}
\sum_{| k| >h}| Q(k)| <\frac{\epsilon }{3}  \label{Q}
\end{equation}
and there exists $N\in \mathbb{N}$ such that for all $n>N$ we have
\begin{equation}
\sum_{| k| \leq h}|
f(k,u_{n}(k))-f(k,u(k))| <\frac{\epsilon }{3}.  \label{fn-f}
\end{equation}
Applying \eqref{Q} and \eqref{fn-f}, for every $n>N$ and $v\in l^{p}$
one has
\begin{align*}
&| \langle \Psi '(u_{n})-\Psi '(u),v\rangle |  \\
&\leq \| v\| _{\infty}\sum_{k\in \mathbb{Z}}| f(k,u_{n}(k))-f(k,u(k))| \\
&\leq \| v\| _{p}\Big( \sum_{| k| \leq h}| f(k,u_{n}(k))-f(k,u(k))|
 +\sum_{| k| >h}| f(k,u_{n}(k))| +\sum_{|k| >h}| f(k,u(k))| \Big) \\
&\leq \| v\| _{p}\Big( \frac{\epsilon }{3} +2\sum_{| k| >h}Q(k)\Big) \\
&< \epsilon \| v\| _{p},
\end{align*}
thus, $\| \Psi '(u_{n})-\Psi '(u)\| <\epsilon $. So,
$\Psi '$ is continuous and $\Psi \in C^{1}(l^{p})$.
\end{proof}

Now, we set
\[
X=\big\{ u:{\mathbb{Z}}\to {\mathbb{R}}:
 \sum_{k\in {\mathbb{Z}}}[ a(k)| \Delta u(k-1)| ^{p}+b(k)|u(k)|^{p}]
<\infty \big\}
\]
and
\[
\| u\| =\Big( \sum_{k\in {\mathbb{Z}}}[ a(k)| \Delta
u(k-1)| ^{p}+b(k)|u(k)|^{p}] \Big) {1/p}.
\]
Clearly we have
\begin{equation}
\| u\| _{\infty }\leq \| u\| _{p}\leq \alpha ^{-1/p}\| u\| \quad
 \text{for all $u\in X$.}  \label{a}
\end{equation}

 As is shown in \cite[Propositions 3]{IT},  $(X,\| \cdot \| )$
is a reflexive Banach space and the embedding $X\hookrightarrow l^{p}$ is
compact. See also \cite[Lemma 2.2]{K}.

Let $J_{\lambda }:X\to \mathbb{R}$ be the functional associated with
problem \eqref{eq1} defined by
\[
J_{\lambda }(u)=\Phi (u)-\lambda \Psi (u),
\]
where
\[
\Phi (u):=\frac{1}{p}\sum_{k\in \mathbb{Z}}[ a(k)| \Delta u(k-1)| ^{p}+b(k)|
u(k)| ^{p}] \quad  \text{for all }u\in X
\]
and $\Psi $ is given by \eqref{psi}.

\begin{proposition} \label{propIT}
 Assume that {\rm (A1)} and \eqref{eF1h} are satisfied.
Then 
\begin{itemize}
\item[(a)] $\Psi \in C^{1}(l^{p})$ and  $\Psi \in C^{1}(X)$;

\item[(b)] $\Phi \in C^{1}(X)$;

\item[(c)] $J_{\lambda }\in C^{1}(X)$ and every critical point $u\in X$ of
$J_{\lambda }$ is a homoclinic solution of problem \eqref{eq};

\item[(d)] $J_{\lambda }$ is sequentially weakly lower semicontinuous
functional on $X$.
\end{itemize}
\end{proposition}

\begin{proof}
 Part (a) follows from Lemma \ref{difpsi}. Parts (b) and (c)
can be proved essentially by the same way as 
\cite[Propositions 5 and 7]{IT}, where $a(k)\equiv 1$ on 
$\mathbb{Z}$ and the norm on $X$ is slightly different.
 See also \cite[Lemmas 2.4 and 2.6]{K}. 
The proof of part (d) is based on the following facts: 
$\Phi =\frac{1}{p}\| \cdot \| ^{p}$, $\Psi \in C(l^{p})$ and
the compactness of $X\hookrightarrow l^{p}$ and it is standard.
\end{proof}


\section{Proof of Theorem \ref{tw0}}

 Now we will formulate and prove a stronger form of Theorem \ref{tw0}. Let
\begin{gather}
B_{\pm }:=\limsup_{(k,t)\to (\pm \infty ,0^{+})}\frac{F(k,t)}{
[ a(k+1)+a(k)+b(k)] t^{p}},  \label{B+-} \\
B_0:=\sup_{k\in\mathbb{Z}}\Big( 
\limsup_{t\to 0^{+}}\frac{F(k,t)}{[a(k+1)+a(k)+b(k)] t^{p}}\Big) .  \label{B0}
\end{gather}
Set $B=\max \{B_{\pm },B_0\}$. For convenience we put $1/+\infty=0$.

\begin{theorem} \label{tw1}
Assume that {\rm (A1)--(A4)} are satisfied and assume
that $B>0$. Then, for any $\lambda >\frac{1}{Bp}$,  problem \eqref{eq}
admits a nonzero sequence of non-negative solutions in $X$ whose norms tend
to zero.
\end{theorem}

\begin{proof}
To apply Proposition \ref{propIT}, we need to have a nonlinearity which
satisfies condition \eqref{eF1h}. Let $T_0>0$ be a number satisfying
(A2). Define the truncation function 
\[
\tilde{f}(k,s)=\begin{cases}
0, & s\leq 0\text{ and }k\in \mathbb{Z}, \\
f(x,s), & 0\leq s\leq T_0\text{ and }k\in \mathbb{Z}, \\
f(x,T_0), & s\geq T_0\text{ and }k\in \mathbb{Z}.
\end{cases}
\]
and consider the problem
\begin{equation}
\begin{gathered}
-\Delta \left( a(k)\phi _{p}(\Delta u(k-1))\right) +b(k)\phi
_{p}(u(k))=\lambda \tilde{f}(k,u(k)) \\
u(k)\to 0.
\end{gathered}  \label{eq1}
\end{equation}
 Clearly, if $u$ is a non-negative solution of  problem \eqref{eq1} with
$\| u\| _{\infty }\leq T_0$, then it is also a
non-negative solution of  problem \eqref{eq}, so it is enough to show
that  problem \eqref{eq1} admits a nonzero sequence of non-negative
solutions in $X$ whose norms tend to zero.

Put $\lambda >\frac{1}{Bp}$ and put $\Phi ,\Psi $ and $J_{\lambda }$ as in
the previous section. By Proposition \ref{propIT} we need to find a
nontrivial sequence $\{u_{n}\}$ of critical points of $J_{\lambda }$ with
non-negative terms whose norms tend to zero.

Let $\{c_{n}\},\{d_{n}\}$ be sequences\ satisfying conditions (A4). Up
to subsequence, we may assume that $d_1<T_0$. 
For every $n\in\mathbb{N}$ define the set
\[
W_{n}=\{ u\in X:\| u\| _{\infty }\leq d_{n}\text{ for every }k\in\mathbb{Z}\} .
\]

\begin{claim}  \label{c1}
For every $n\in\mathbb{N}$, the functional $J_{\lambda }$ is bounded from 
below on $W_{n}$ and its infimum on $W_{n}$ is attained.
\end{claim}

The proof of this Claim is essentially the same as the proof of
\cite[Claim 3.2]{St1}.

\begin{claim}  \label{c3}
For every $n\in\mathbb{N}$, let $u_{n}\in W_{n}$ be such that 
$J_{\lambda}(u_{n})=\inf_{W_{n}}J_{\lambda }$. Then, $u_{n}$ is a solution 
of problem \eqref{eq1} with $0\leq u_{n}(k)\leq c_{n}$ for all 
$k\in\mathbb{Z}$.
\end{claim}

Firstly, arguing as in the proof of \cite[Claim 3.3]{St1}, we obtain that
if $u_{n}\in W_{n}$ is such that $J_{\lambda }(u_{n})=\inf_{W_{n}}J_{\lambda
}$, then  $0\leq u_{n}(k)\leq c_{n}$ for all $k\in \mathbb{Z}$. 
Secondly, arguing as in the proof of \cite[Claim 3.4]{St1}, we obtain
that $u_{n}$ is a critical point of $J_{\lambda }$ in $X$, and so is a
solution of problem \eqref{eq1}. This proves Claim \ref{c3}.

\begin{claim}  \label{inf} 
For every $n\in\mathbb{N}$, we have 
$J_{\lambda }(u_{n})<0$ and $\lim_{n\to +\infty}J_{\lambda }(u_{n})=0$.
\end{claim}

Firstly, we assume that $B=B_{\pm }$. Without loss of generality we can
assume that $B=B_{+}$.  We begin with $B=+\infty $. Then there exists a
number $\sigma >\frac{1}{\lambda p}$, a sequence of positive integers 
$\{k_{n}\}$ and a sequence of positive numbers $\{t_{n}\}$ which tends to $0$,
such that
\begin{equation}
F(k_{n},t_{n})>\sigma (a(k_{n}+1)+a(k_{n})+b(k_{n}))t_{n}^{p}  \label{kn}
\end{equation}
for all $n\in \mathbb{N}$. Up to extracting a subsequence, we may assume 
that $t_{n}\leq d_{n}$ for all $n\in\mathbb{N}$. Define in $X$ a sequence 
$\{w_{n}\}$ such that, for every $n\in \mathbb{N}$, 
$w_{n}(k_{n})=t_{n}$ and $w_{n}(k)=0$ for every 
$k\in \mathbb{Z}\backslash \{k_{n}\}$. It is clear that $w_{n}\in W_{n}$. 
One then has
\begin{align*}
&J_{\lambda }(w_{n}) \\ 
&= \frac{1}{p}\sum_{k\in\mathbb{Z}}
\left( a(k)| \Delta w_{n}(k-1)| ^{p}+b(k)|
w_{n}(k)| ^{p}\right) -\lambda \sum_{k\in \mathbb{Z}}F(k,w_{n}(k)) \\
&< \frac{1}{p}\left( a(k_{n}+1)+a(k_{n})\right) t_{n}^{p}+\frac{1}{p}
b(k_{n})t_{n}^{p}-\lambda \sigma (a(k_{n}+1)+a(k_{n})+b(k_{n}))t_{n}^{p} \\
&= \big( \frac{1}{p}-\lambda \sigma \big)
(a(k_{n}+1)+a(k_{n})+b(k_{n}))t_{n}^{p}<0
\end{align*}
which gives $J_{\lambda }(u_{n})\leq J_{\lambda }(w_{n})<0$. Next, assume
that $B<+\infty $. Since $\lambda >\frac{1}{Bp}$, we can fix 
$\varepsilon <B-\frac{1}{\lambda p}$. Therefore, also taking $\{k_{n}\}$
 a sequence of positive integers and $\{t_{n}\}$ a sequence of positive 
numbers with $\lim_{n\to +\infty }t_{n}=0$ and $t_{n}\leq d_{n}$ for all 
$n\in\mathbb{N}$ such that
\begin{equation}
F(k_{n},t_{n})>(B-\varepsilon )(a(k_{n}+1)+a(k_{n})+b(k_{n}))t_{n}^{p}
\label{kn'}
\end{equation}
for all $n\in \mathbb{N}$, choosing $\{w_{n}\}$ in $W_{n}$ as above, 
one has
\[
J_{\lambda }(w_{n})<\Big( \frac{1}{p}-\lambda (B-\varepsilon )\Big)
(a(k_{n}+1)+a(k_{n})+b(k_{n}))t_{n}^{p}.
\]
So, also in this case, $J_{\lambda }(u_{n})<0$.

Now, assume that $B=B_0$. We begin with $B=+\infty $. Then there exists a
number $\sigma >\frac{1}{\lambda p}$ and an index 
$k_0\in \mathbb{Z}$ such that
\[
\limsup_{t\to 0^{+}}\frac{F(k_0,t)}{
(a(k_0+1)+a(k_0)+b(k_0))| t| ^{p}}>\sigma .
\]
 Then, there exists a sequence of positive numbers $\{t_{n}\}$ such that
$\lim_{n\to +\infty }t_{n}=0$ and
\begin{equation}
F(k_0,t_{n})>\sigma (a(k_0+1)+a(k_0)+b(k_0))t_{n}^{p}  \label{k0}
\end{equation}
for all $n\in \mathbb{N}$. Up to considering a subsequence, we may assume that
$t_{n}\leq d_{n}$ for all $n\in\mathbb{N}$. Thus, take in $X$ a sequence
$\{w_{n}\}$ such that, for every $n\in \mathbb{N}$,
 $w_{n}(k_0)=t_{n}$ and $w_{n}(k)=0$ for every $k\in \mathbb{Z}\backslash \{k_0\}$.
Then, one has $w_{n}\in W_{n}$ and
\begin{align*}
&J_{\lambda }(w_{n}) \\
&= \frac{1}{p}\sum_{k\in\mathbb{Z}}\left( a(k)| \Delta w_{n}(k-1)| ^{p}+b(k)|
w_{n}(k)| ^{p}\right) -\lambda \sum_{k\in \mathbb{Z}}F(k,w_{n}(k)) \\
&< \frac{1}{p}\left( a(k_0+1)+a(k_0)\right) t_{n}^{p}+\frac{1}{p}
b(k_0)t_{n}^{p}-\lambda \sigma (a(k_0+1)+a(k_0)+b(k_0))t_{n}^{p} \\
&=\big( \frac{1}{p}-\lambda \sigma \big)
(a(k_0+1)+a(k_0)+b(k_0))t_{n}^{p}<0
\end{align*}
which gives $J_{\lambda }(u_{n})<0$. Next, assume that $B<+\infty $.
Since $\lambda >\frac{1}{Bp}$, we can fix $\varepsilon >0$ such that
$\varepsilon<B-\frac{1}{\lambda p}$. Therefore, there exists an index
$k_0\in\mathbb{Z}$ such that
\[
\limsup_{t\to 0^{+}}\frac{F(k_0,t)}{(a(k_0+1)+a(k_0)+b(k_0))t^{p}}>B-\varepsilon .
\]
and taking $\{t_{n}\}$ a sequence of positive numbers with
$\lim_{n\to +\infty }t_{n}=0$ and $t_{n}\leq d_{n}$ for all
$n\in\mathbb{N}$ and
\begin{equation}
F(k_0,t_{n})>\left( B-\varepsilon \right)
(a(k_0+1)+a(k_0)+b(k_0))t_{n}^{p}  \label{k0'}
\end{equation}
for all $n\in \mathbb{N}$, choosing $\{w_{n}\}$ in $W_{n}$ as above, one has
\[
J_{\lambda }(w_{n})
<\big( \frac{1}{p}-\lambda (B-\varepsilon )\big)
(a(k_0+1)+a(k_0)+b(k_0))t_{n}^{p}<0.
\]
So, also in this case, $J_{\lambda }(u_{n})<0$.

Moreover, by Claim \ref{c3}, for every \thinspace $k\in \mathbb{N}$ one has
\begin{equation}
| F(k,u_{n}(k))| \leq \int_0^{c_{n}}| \tilde{f}
(k,t)| dt\leq c_{n}\max_{t\in \lbrack 0,c_{n}]}| \tilde{f}
(k,t)| \leq c_{n}\max_{t\in \lbrack 0,T_0]}| \tilde{f}
(k,t)|  \label{F}
\end{equation}
 Then
\[
0>J_{\lambda }(u_{n})\geq -\sum_{k\in \mathbb{Z}}F(k,u_{n}(k))
\geq -c_{n}\| \max_{t\in \lbrack 0,T_0]}|
\tilde{f}(\cdot ,t)| \| _1
\]
Since the sequence $\{c_{n}\}$\ tends to zero, then
$J_{\lambda}(u_{n})\to 0$ as $n\to +\infty$.
 This proves Claim \ref{inf}.

Now we are ready to end the proof of Theorem \ref{tw1}. With Proposition 
\ref{propIT}, Claims \ref{c3}--\ref{inf}, up to a subsequence, we have
infinitely many pairwise distinct non-negative homoclinic solutions $u_{n}$
of \eqref{eq1}. Moreover, due to \eqref{F}, we have
\[
\frac{1}{p}\| u_{n}\| ^{p}=J_{\lambda }(u_{n})
+\sum_{k\in\mathbb{Z}}F(k,u_{n}(k))<c_{n}\|
\max_{t\in \lbrack 0,T_0]}| \tilde{f}(\cdot ,t)| \| _1,
\]
which proves that $\| u_{n}\| ^{p}\to 0$ as
$n\to +\infty $. This concludes our proof.
\end{proof}

We remark that Theorem \ref{tw0} follows now from Theorem \ref{tw1}.

\section{Proof of Theorem \protect\ref{tw0'}}

 Our main tool is a general critical points theorem due to Bonanno
and Molica Bisci (see \cite{BMB}) that is a generalization of a result of
Ricceri \cite{R}. Here we state it in a smooth version for the reader's
convenience.

\begin{theorem}\label{cpt} 
Let $(E,\| \cdot \| )$ be a reflexive
real Banach space, let $\Phi ,\Psi :E\to \mathbb{R}$ be two continuously
differentiable functionals with $\Phi $ coercive, i.e.
$\lim_{\| u\| \to \infty }\Phi (u)=+\infty $, and
a sequentially weakly lower semicontinuous functional\ and $\Psi $ a
sequentially weakly upper semicontinuous functional. 
For every $r>\inf_{E}\Phi $, let us put
\begin{gather*}
\varphi (r):=\inf_{u\in \Phi ^{-1}((-\infty ,r))}
\frac{\big( \sup_{v\in\Phi ^{-1}((-\infty ,r))}\Psi (v)\big) -\Psi (u)}{r-\Phi (u)},
\\
\delta :=\liminf_{r\to (\inf_{E}\Phi )^{+}}~\varphi (r).
\end{gather*}
Let $J_{\lambda }:=\Phi (u)-\lambda \Psi (u)$ for all $u\in E$. 
If $\delta <+\infty $ then, for each $\lambda \in ( 0,1/\delta) $,
the following alternative holds: either
\begin{itemize}
\item[(a)] there is a global minimum of $\Phi $ which is a local minimum of $
J_{\lambda }$, or 

\item[(b)] there is a sequence $\{u_{n}\}$ of pairwise distinct critical points
(local minima) of $J_{\lambda }$, with $\lim_{n\to +\infty }\Phi
(u_{n})=\inf_{E}\Phi $, which weakly converges to a global minimum of 
$\Phi $.
\end{itemize}
\end{theorem}

Now we  formulate and prove a stronger form of Theorem \ref{tw0'}. 
 Let
\[
A:=\liminf_{t\to 0^{+}}~\frac{\sum_{k\in\mathbb{Z}}
\max_{| \xi | \leq t}F(k,\xi )}{t^{p}}.
\]
Set $B:=\max \{B_{\pm },B_0\}$, where $B_{\pm }$ and $B_0$ are given by
\eqref{B+-} and \eqref{B0}, respectively.  For convenience we put
$\frac{1}{0^{+}}=+\infty $ and $\frac{1}{+\infty }=0$.

\begin{theorem} \label{tw1'}
 Assume that {\rm (A1), (A2), (A5)} are satisfied and
assume that the following inequality holds
 $A<\alpha  B$. Then, for
each $\lambda \in ( \frac{1}{Bp},\frac{\alpha }{Ap}) $, problem 
\eqref{eq} admits a sequence of solutions in $X$ whose norms tend to zero.
\end{theorem}

\begin{proof}[Proof]
To apply Proposition \ref{propIT}, we need to have a nonlinearity which
satisfies condition \eqref{eF1h}. Let $T_0>0$ be a number satisfying 
(A2). Define the truncation function
\[
\bar{f}(k,s)=\begin{cases}
f(x,-T_0), & s\leq -T_0\text{ and }k\in \mathbb{Z}, \\
f(x,s), & -T_0\leq s\leq T_0\text{ and }k\in \mathbb{Z}, \\
f(x,T_0), & s\geq T_0\text{ and }k\in \mathbb{Z}.
\end{cases}
\]
and consider the problem
\begin{equation}
\begin{gathered}
-\Delta \left( a(k)\phi _{p}(\Delta u(k-1))\right) +b(k)\phi
_{p}(u(k))=\lambda \bar{f}(k,u(k)) \\
u(k)\to 0.
\end{gathered}  \label{eq2}
\end{equation}
 Clearly, if $u$ is a solution of  problem \eqref{eq2} with
$\|u\| _{\infty }\leq T_0$, then it is also a solution of the
problem \eqref{eq}, so it is enough to show that  problem \eqref{eq2}
admits a nonzero sequence of solutions in $X$ whose norms tend to zero.

It is clear that $A\geq 0$. Put $\lambda \in \big( \frac{1}{Bp},\frac{
\alpha }{Ap}\big) $ and put $\Phi ,\Psi ,J_{\lambda }$ as above. Our aim
is to apply Theorem \ref{cpt} to function $J_{\lambda }$. By Lemma 
\ref{propIT}, the functional $\Phi $ is the continuously differentiable and
sequentially weakly lower semicontinuous functional\ and $\Psi $ is the
continuously differentiable and sequentially weakly upper semicontinuous
functional. We will show that $\delta <+\infty $. Let $\{c_{m}\}\subset
(0,T_0)$ be a sequence such that $\lim_{m\to \infty }c_{m}=0$ and
\[
\lim_{m\to +\infty } \frac{\sum_{k\in \mathbb{Z}}
\max_{| \xi | \leq c_{m}}F(k,\xi )}{c_{m}^{p}}=A.
\]
Set
\[
r_{m}:=\frac{\alpha }{p}c_{m}^{p}
\]
for every $m\in \mathbb{N}$. Then, if $v\in X$ and $\Phi (v)<r_{m}$, one has
\[
\| v\| _{\infty }\leq \alpha ^{-\frac{1}{p}}\| v\|
 \leq \alpha ^{-\frac{1}{p}}\big( p\Phi (v)\big) {1/p}<c_{m}
\]
which gives
\begin{equation}
\Phi ^{-1}\big( ( -\infty ,r_{m}) \big)
\subset \big\{ v\in X:\| v\| _{\infty }\leq c_{m}\big\} .
\end{equation}
From this and $\Phi (0)=\Psi (0)=0$ we have
\begin{align*}
\varphi (r_{m})
&\leq \frac{\sup_{\Phi (v)<r_{m}}\sum_{k\in\mathbb{Z}}F(k,v(k))}{r_{m}}
\leq \frac{\sum_{k\in\mathbb{Z}}\max_{| t|
\leq c_{m}}F(k,t)}{r_{m}} \\
&=\frac{p}{\alpha }\cdot \frac{\sum_{k\in \mathbb{Z}}
\max_{| t| \leq c_{m}}F(k,t)}{c_{m}^{p}}
\end{align*}
for every $m\in \mathbb{N}$. This gives
\[
\delta \leq \lim_{m\to +\infty } \varphi (r_{m})
\leq \frac{p}{\alpha}\cdot A<\frac{1}{\lambda }<+\infty .
\]

Now, we show that the point (a) in Theorem \ref{cpt} does not hold, i.e. we
show that the global minimum $\theta $ of $\Phi $\ is not a local minimum of
$J_{\lambda }$. Arguing as in the proof of Claim \ref{inf}, we can find a
sequence $\{w_{n}\}$ in $X$\ with $\| w_{n}\| _{\infty
}\to 0$ as $n\to +\infty $, such that $J_{\lambda }(w_{n})<0$
for $n\in \mathbb{N}$. We have to show that $\| w_{n}\| \to 0$. Note
that
\[
\| w_{n}\| =\big((a(k_{n}+1)+a(k_{n})+b(k_{n}))t_{n}^{p}\big) {1/p},
\]
where $\{k_{n}\}$ is a sequence divergent to $+\infty $ or $-\infty $,
as in \eqref{kn} and \eqref{kn'} or $\{k_{n}\}$ is a constant sequence,
as in \eqref{k0} and \eqref{k0'} and $\{t_{n}\}$ is a sequence convergent to
$0^{+}$ from relevant \eqref{kn}, \eqref{kn'}, \eqref{k0} or \eqref{k0'}.
 From this
\[
\| w_{n}\| \leq \gamma F(k_{n},t_{n})
\]
for some positive constant $\gamma $ and all $n\in\mathbb{N}$. Since
\[
\lim_{m\to +\infty } \frac{\sum_{k\in\mathbb{Z}}\max_{| \xi |
\leq c_{m}}F(k,\xi )}{c_{m}^{p}}<+\infty
\]
and $\lim_{m\to +\infty }c_{m}=0$, we have
\[
\lim_{m\to +\infty }\sum_{k\in\mathbb{Z}}\max_{| \xi | \leq c_{m}}F(k,\xi )=0
\]
and, as $\max_{| \xi | \leq c_{m}}F(k,\xi )\geq 0$, we
obtain $\lim_{m\to +\infty }\left( \max_{| \xi |
\leq c_{m}}F(k,\xi )\right) =0$ uniformly for all
$k\in\mathbb{Z}$. This and $F(k_{n},t_{n})>0$ easily gives
$\lim_{n\to +\infty}F(k_{n},t_{n})=0$ and so $\lim_{n\to +\infty }\|w_{n}\| =0$.

From the above it follows that $\theta $ is not a local minimum of 
$J_{\lambda }$ and, by (b), there is a sequence $\{u_{n}\}$ of pairwise
distinct critical points of $J_{\lambda }$ with 
$\lim_{n\to +\infty}\Phi (u_{n})=\inf_{E}\Phi $. 
This  means that 
$0=\inf_{E}\Phi =\lim_{n\to +\infty }\Phi (u_{n})=\frac{1}{p}\|
u_{n}\| ^{p}$, and so $\{u_{n}\}$\ strongly converges to zero. The
proof is complete.
\end{proof}

We remark that Theorem \ref{tw0'} follows now from Theorem \ref{tw1'}.

\section{Examples}

 Consider the problem
\begin{equation}
\begin{gathered}
-\Delta \left( \phi _{p}(\Delta u(k-1))\right) +| k| \phi
_{p}(u(k))=\lambda f(k,u(k)) \quad
 \text{for all $k\in\mathbb{Z}$} \\
u(k)\to 0 \quad  \text{as $|k|\to \infty$},
\end{gathered}   \label{ex}
\end{equation}
where $p>1$ and $f:\mathbb{Z}\times\mathbb{R}\to\mathbb{R}$ is defined by
\begin{equation}
f(k,s)=\sum_{m\in \mathbb{N}}e_{m}
\Big( d_{m}-c_{m}-2| s-\frac{1}{2}\left( c_{m}+d_{m}\right)
| \Big) \cdot \mathbf{1}_{\{m\}\times \lbrack c_{m},d_{m}]}(k,s)
\label{fun}
\end{equation}
with sequences $\{c_{m}\},\{d_{m}\},\{e_{m}\},\{h_{m}\}$ defined by
\begin{equation}
\begin{gathered}
c_{m}=1/ 2^{2^{2m}} \quad \text{for }m\in \mathbb{N}; \\
d_{m}=1/2^{2^{2m-1}}\quad \text{for }m\in\mathbb{N}; \\
h_{m}=1/2^{(p+1)2^{2m-2}} \quad \text{for }m\in \mathbb{N}; \\
e_{m}= 2h_{m}/(d_{m}-c_{m})^{2} \quad \text{for }m\in\mathbb{N}.
\end{gathered}  \label{seq}
\end{equation}
Here $\mathbf{1}_{A\times B}$ is the indicator of $A\times B$. It is easily
seen that $f$ is continuous and conditions (A2), (A3) are satisfied.
Set $F(k,t):=\int_0^{t}f(k,s)ds$ for every $t\in \mathbb{R}$ and 
$k\in\mathbb{Z}$. Then $F(k,d_{k})=\int_{c_{k}}^{d_{k}}f(k,t)dt=h_{k}$ and
\begin{equation} \label{A}
\begin{aligned}
\liminf_{t\to 0^{+}}~\frac{\sum_{k\in \mathbb{Z}}
\max_{| \xi | \leq t}F(k,\xi )}{t^{p}}
&\leq \lim_{m\to +\infty }\frac{\sum_{k\in \mathbb{Z}}\max_{| \xi |
 \leq c_{m}}F(k,\xi )}{c_{m}^{p}}   \\
&= \lim_{m\to +\infty }\frac{\sum_{k=m+1}^{\infty }F(k,d_{k})}{
c_{m}^{p}}   \\
&=\lim_{m\to +\infty }\frac{\sum_{k=m+1}^{\infty }h_{k}}{c_{m}^{p}}\\
&\leq \lim_{m\to +\infty }\frac{2h_{m+1}}{c_{m}^{p}}=0
\end{aligned}
\end{equation}
and
\begin{equation} \label{BB}
\begin{aligned}
\limsup_{(k,t)\to (+\infty ,0^{+})}\frac{F(k,t)}{(2+k)t^{p}}
&\geq \lim_{m\to +\infty }\frac{F(m,d_{m})}{(2+m)d_{m}^{p}}  \\
&= \lim_{m\to +\infty }\frac{h_{m}}{(2+m)d_{m}^{p}}=+\infty .
\end{aligned}
\end{equation}
So, conditions (A4)--(A6) are satisfied and so for any
$ \lambda >0$, problem \eqref{ex} admits a sequence of non-negative
solutions in $X$ whose norms tend to zero, by Theorem \ref{tw0} or Theorem
\ref{tw0'}. Note also that $f$ does not satisfy (A8).

\begin{remark} \rm
For a fixed $k_0\in\mathbb{Z}$, if we define 
$\tilde{f}:\mathbb{Z}\times\mathbb{R}\to\mathbb{R}$ by
\[
\tilde{f}(k,s)=\sum_{m\in\mathbb{N}}e_{m}\big( d_{m}-c_{m}-2
| s-\frac{1}{2}( c_{m}+d_{m})
| \big) \cdot \mathbf{1}_{\{k_0\}\times [c_{m},d_{m}]}(k,s)
\]
with sequences $\{c_{m}\},\{d_{m}\},\{e_{m}\},\{h_{m}\}$ defined as above,
then $\tilde{f}$ satisfies conditions
(A2)--(A5)  and (A8), but does not satisfy conditions (A6) and (A7).
\end{remark}

\begin{remark} \rm
Theorems \ref{tw0} and \ref{tw0'}
are independent of each other. Indeed, let us replace $h_{m}$ in \eqref{seq} by
\[
h_{m}=1/2^{p2^{2m-2}}\quad \text{for }m\in\mathbb{N}.
\]
Then, the function $f$ given by \eqref{fun} is continuous if $p>2$. It can
be seen that the first inequality in \eqref{A} is in fact equality. Then, an
easy computation shows that
\begin{gather*}
\liminf_{t\to 0^{+}}~\frac{\sum_{k\in\mathbb{Z}
}\max_{| \xi | \leq t}F(k,\xi )}{t^{p}}\geq 1, \\
B_{+}=\limsup_{(k,t)\to (+\infty ,0^{+})}\frac{F(k,t)}{
(2+k)t^{p}}=+\infty .
\end{gather*}
This means that we can not apply Theorem \ref{tw0'}, but
Theorem \ref{tw0} works. On the other hand, it is easy to see that we
can modify $f$ in the way, that for some (or even infinitely many) $k$ we
have $f(k,t)>0$ for all $t>0$ and the limits \eqref{A}, \eqref{BB}
 do not change. Therefore, such
an $f$ does not satisfy (A4)\ and can not be used in Theorem \ref{tw0}.
\end{remark}

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\end{document}