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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 195, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/195\hfil Multiplicity of solutions]
{Multiple solutions for a discrete anisotropic
$(p_1(k),p_2(k))$-Laplacian equations}

\author[E. M. Hssini \hfil EJDE-2015/195\hfilneg]
{El Miloud Hssini}

\address{EL Miloud Hssini \newline
University Mohamed I, Faculty of Sciences,
Oujda, Morocco}
\email{hssini1975@yahoo.fr}

\thanks{Submitted July 7, 2015. Published July 27, 2015.}
\subjclass[2010]{39A10, 34B18, 58E30}
\keywords{Discrete nonlinear
boundary value problem; $p(k)$-Laplacian; \hfill\break\indent
multiple solutions; critical point theory}

\begin{abstract}
 This article concerns the existence and multiplicity
 solutions for a discrete Dirichlet Laplacian problems. Our technical
 approach is based on variational methods.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

In this work, we study the existence and multiplicity solutions of
the discrete boundary-value problem
\begin{equation}  \label{E11}  
\begin{gathered}
-\Delta(\phi_{p_1(k-1)}(\Delta
u(k-1)))-\Delta(\phi_{p_2(k-1)}(\Delta u(k-1)))=\lambda
f(k,u(k)),\\ 
\forall k\in \mathbb{Z}[1,T],\\
u(0)=u(T+1)=0,
\end{gathered}
\end{equation}
where, $\phi_{p_i(k)}(t) = |t|^{p_i(k)-2}t$  $(i=1,2)$ for all 
$t\in \mathbb{R}$ and for each $k\in \mathbb{Z}[1,T]$, $T\geq2$ is a
positive integer,  $\mathbb{Z}[1,T]$ is a discrete interval 
$\{1, 2, \dots , T\}$, $\lambda$ is a positive parameter, $\Delta
u(k-1):=u(k)-u(k-1)$ is the forward difference operator,
$f:\mathbb{Z}[1,T]\times\mathbb{R}\to\mathbb{R}$ is a continuous
function and
 $p_1,\,p_2:\mathbb{Z}[0, T]\to [2,+\infty)$.

Discrete boundary value problems have been intensively studied in
the last decade. The modeling of certain nonlinear problems from
biological neural networks, economics, optimal control and other
areas of study have led to the rapid development of the theory of
difference equations; see the monograph of
\cite{AgPerOr1,AgPerOr2,Cai,yu} for an overview on this subject.

Equations involving the discrete $p$-Laplacian operator, subjected
to classical or less classical boundary conditions, have been widely
studied by many authors using various techniques. Recently, many
results have been established by applying variational methods. In
this direction we mention the papers
\cite{Af,BiSunZha,JiZh,Molica,TiGe} and the references therein.
However, problems like \eqref{E11} involving anisotropic exponents
have only been started, by Mihailescu, Radulescu and Tersian
\cite{Rad},  Kone and Ouaro \cite{Con}, where known tools from the
critical point theory are applied in order to get the existence of
solutions. Later considered by many methods and authors, see
\cite{Ay,BerJeb,Gal,Gal2,Molica1,Ser} for an extensive survey of
such boundary value problems.

Our aim is to establish the existence and multiplicity results for
problem \eqref{E11} through variational methods.
 First we will exploit a critical point Theorem \ref{teo:bon}
which provides for the existence of a local minima for a
parameterized abstract functional. Next, Theorem  \ref{critical1}
with the classical Ambrosetti-Rabinowitz condition, guarantee that
\eqref{E11} has at least two distinct nontrivial weak solutions
(Theorem \ref{the4.1}). Finally, we will get the existence of at
least three nontrivial solutions of the problem \eqref{E11} where
the nonlinearity $f(x,u)$ does not satisfy Ambrosetti-Rabinowitz
condition (Theorem \ref{theo33}), by employing a local minimum
Theorem \ref{critical3}.


\section{Preliminaries and basic notation}

In this section, we state some basic properties, definitions and
theorems to be used in this article. Let $(X,\|\cdot\|)$ be a finite
dimensional Banach space. A functional $I_{\lambda}$ is said to
verify the Palais-Smale condition (in short $(P.S.)$) whenever one
has that any sequence $\{u_n\}$ such that
\begin{itemize}
\item $\{I_\lambda (u_n)\}$ is bounded;
\item $\{I'_\lambda (u_n)\}$ is convergent at $0$ in $X^*$
\end{itemize}
admits a subsequence which is converging in $X$.

Our main tool will be the following three abstract critical point
theorems, which are a simple extension of the Ricceri's Variational
Principle \cite{Ricceri} recalled here on the finite dimensional
Banach spaces.

\begin{theorem}\label{teo:bon}
Let $X$ be a finite dimensional Banach space and let $\Phi$,
$\Psi:X\to \mathbb{R}$  two functions of class $C^1$ on $X$ with
$\Phi$ is coercive. In addition, suppose that there exist $r\in
\mathbb{R}$ and $w\in X$, with $0<\Phi(w)<r$, such that
\begin{equation}\label{condizionealgebrica}
   {\frac{{\sup_{\Phi^{-1}([0, r])}\Psi}}{r}< {\frac{\Psi(w)}{\Phi(w)}}}.
\end{equation}
Then, for each
\[
\lambda \in \Lambda_w:=\big]{\frac{\Phi(w)}{\Psi(w)}},
\frac{r}{{\sup_{\Phi^{-1}([0, r])}\Psi}}\big[,
\]
 the function $I_\lambda=\Phi-\lambda\Psi$ admits at least one local
minimum $\overline{u} \in X$ such that $\overline{u} \neq 0$,
$\Phi(\overline{u})<r$, $I_{\lambda}(\overline{u})\leq
I_{\lambda}(u)$ for all $u\in \Phi^{-1}([0,r])$ and
$I'_\lambda(\overline{u})=0$.
\end{theorem}

\begin{theorem}\label{critical1}
Let $X$ be a finite dimensional Banach space and let $\Phi$,
$\Psi:X\to \mathbb{R}$  two functions of class $C^1$ on $X$ with
$\Phi$ is coercive. Fix $r>0$.
 Assume that for each
\[
\lambda \in \Lambda:=\big]0, \frac{r}{{\sup_{\Phi^{-1}([0,
r])}\Psi}}\big[,
\]
 the function $I_\lambda=\Phi-\lambda\Psi$
satisfies the (PS)-condition and  is unbounded from below. Then, for
each $\lambda \in \Lambda$, the function $I_\lambda$ admits at least
two distinct critical points.
\end{theorem}

\begin{theorem}\label{critical3}
Let $X$ be a reflexive real Banach space, $\Phi:X\to \mathbb{R}$ be
a continuously G\^ateaux differentiable, coercive and sequentially
weakly lower semicontinuous functional whose G\^ateaux derivative
admits a continuous inverse on $X^*$, $\Psi: X\to \mathbb{R}$ be a
continuously G\^ateaux differentiable functional whose G\^ateaux
derivative is compact, moreover
$$
\Phi(0)=\Psi(0)=0.
$$
Assume that there exist $r\in \mathbb{R}$ and $ \bar{u}\in X$, with
$0<r<\Phi(\bar{u}) $, such that
\begin{itemize}
\item[(i)] 
\[
\frac{\sup_{u\in\Phi^{-1}(]-\infty,r])}\Psi(u)}{r}
 <\frac{\Psi(\bar{u})}{\Phi(\bar{u})}
 \]
\item[(ii)] for each $\lambda\in \Lambda$,
\[
\Lambda:=\big]\frac{\Phi(\bar{u})}{\Psi(\bar{u})},
\frac{r}{\sup_{u\in\Phi^{-1}(]-\infty,r])}
\Psi(u)}\big[,
\]
the functional $\Phi-\lambda \Psi$ is coercive.
\end{itemize}
Then, for each $\lambda\in \Lambda$, the functional
$I_{\lambda}=\Phi-\lambda \Psi$ has at least three distinct critical
points in $X$.
\end{theorem}

\begin{remark}\label{rem2.3}\rm
It is worth noticing that whenever $X$ is a finite dimensional
Banach space, for the  Theorem \ref{critical3} shows that regarding
to the regularity of the derivative of $\Phi$ and $\Psi$, it is
enough to require only that $\Phi'$ and $\Psi'$ are two continuous
functionals on $X^\ast$.
\end{remark}


For the rest of this article, we use the following notation:
\begin{gather*}
p_{\rm min}(k):= \min_{i=1,2} p_i(k),\quad
 p_{\rm max}(k) := \max_{i=1,2}p_i(k),\quad  \text{for all } k\in \mathbb{Z}[0,T];\\
p_{\rm min}^-=\min_{k\in[0,T]}p_{\rm min}(k),\quad 
p_{\rm max}^+=\max_{k\in[0,T]}p_{\rm max}(k);\\
p_i^-=\min_{k\in[0,T]}p_i(k),\quad
p_i^+=\max_{k\in[0,T]}p_i(k),\quad \text{for }i=1,2.
\end{gather*}
Define the function space,
$$
H:=\{u:[0,T+1]\to\mathbb{R} : u(0)=u(T+1)=0\}.
$$
Clearly, $H$ is a $T$-dimensional Hilbert space (see
\cite{AgPerOr1}) with the inner product
$$
\langle u,v\rangle:=\sum_{k=1}^{T+1}\Delta u(k-1)\Delta
v(k-1),\quad \forall\,u,v\in H.
$$
The associated norm is defined by
$$
\|u\|:=\Big(\sum_{k=1}^{T+1}|\Delta u(k-1)|^2\Big)^{1/2}.
$$
On the other hand, it is useful to introduce other norms on $H$,
namely
$$
|u|_m=\Big(\sum_{k=1}^{T}|u(k)|^m\Big)^{1/m},\quad \forall u\in
H\text{ and }m\geq2\,.
$$
It can be verified \cite{Cai} that
\begin{equation}\label{e2.3}
T^{\frac{2-m}{2m}}|u|_2\leq|u|_m\leq
T^{\frac{1}{m}}|u|_2,\quad \forall u\in H\text{ and }m\geq2.
\end{equation}
We start with the following auxiliary result. For (a), (b) and
(c) see \cite{Rad} and for (d) see \cite{TiGe}.

\begin{lemma}\label{lem}
We have the following assertions:
\begin{itemize}
\item[(a)] For every $u\in H$ with $\|u\|\leq1$ one has
$$
\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)}\geq
T^{\frac{p^+-2}{2}}\|u\|^{p^+}.
$$

\item[(b)] For every $u\in H$ with $\|u\|\geq1$ one has
$$
\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)}\geq
T^{\frac{2-p^-}{2}}\|u\|^{p^-}-T.
$$

\item [(c)] For any $m\geq2$ there exists a positive constant $c_m$ such that
$$
\sum_{k=1}^{T}|u(k)|^{m}\leq c_m\sum_{k=1}^{T+1}|\Delta
u(k-1)|^{m},\;\;\forall\,u\in H.
$$

\item [(d)] For every $u\in H$ and for any $p,\,q > 1$ such that $\frac1
{p}+\frac1{q}=1$, we have
$$
\max_{k\in
\mathbb{Z}[1,T]}|u(k)|<\Big(T+1\Big)^{1/q}\Big(\sum_{k=1}^{T+1}|\Delta
u(k-1)|^{p}\Big)^{1/p}.
$$
\end{itemize}
\end{lemma}

\begin{definition} \rm
We say that $u\in H$ is a weak solution of problem \eqref{E11} if
\begin{align*}
&\sum_{k=1}^{T+1}\left(\phi_{p_1(k-1)}(\Delta
u(k-1))+\phi_{p_2(k-1)}(\Delta u(k-1))\right)\Delta v(k-1)\\
&-\lambda\sum_{k=1}^{T} f(k,u(k))v(k)=0,
\end{align*}
for all $v\in H$.
\end{definition}

To treat the Dirichlet problem \eqref{E11}, we define the following
two functions:
\begin{equation}\label{E23}
\begin{gathered}
\Phi(u)=\sum_{k=1}^{T+1}\Big(\frac{|\Delta
u(k-1)|^{p_1(k-1)}}{p_1(k-1)}+\frac{|\Delta
u(k-1)|^{p_2(k-1)}}{p_2(k-1)}\Big),\\
 \Psi(u)=\sum_{k=1}^{T} F(k,u(k)),
\end{gathered}
\end{equation}
where $F(k,t)=\int_0^tf(k,s)ds$ for all $(k,t)\in\mathbb{Z}[1,T]\times\mathbb{R}$.
 Further, let us denote
$$
I_\lambda(u):=\Phi(u)-\lambda \Psi(u),\quad \text{for every }u\in H.
$$
The functional $I_\lambda$ is of class $ C^1(H,\mathbb{R})$, and
\begin{align*}
\langle I'_\lambda(u),v\rangle
&=\sum_{k=1}^{T+1}\left(\phi_{p_1(k-1)}(\Delta
u(k-1))+\phi_{p_2(k-1)}(\Delta u(k-1))\right)\Delta v(k-1)\\
&-\lambda\sum_{k=1}^{T} f(k,u(k))v(k),
\end{align*}
for any $u,v\in H$. Thus, critical points of $I_\lambda$ are weak
solutions of \eqref{E11}.

\section{Main results}

To introduce our result, for a nonnegative constant
$\gamma$, put
$$
\sigma(\gamma)
:=\frac{T^{\frac{2-p_{\rm max}^+}{2}}}{p_{\rm max}^+}
\Big(\Big(\frac{\gamma}{\sqrt{T+1}}\Big)^{p_{\rm min}^-}
-2T^{\frac{p_{\rm max}^+}2}\Big).
$$

\begin{theorem}\label{the3.1}
Assume that there exist two real constants $\gamma$ and
$\delta\geq1$, with
\begin{gather} \label{e3.0}
\gamma\geq \sqrt{T+1}\Big(T^{\frac{p_{\rm max}^+
 +p_{\rm min}^--4}{2}}+2T^{\frac{p_{\rm max}^+}2}\Big)^{1/p_{\rm min}^-},\\
\label{e3.1}
4\delta^{p_{\rm max}^+}<p_{\rm min}^-\sigma(\gamma)
\end{gather}
such that
\begin{itemize}
\item[(A1)]
$$
\frac{\sum_{k=1}^T\max_{|t|\leq \gamma}F(k,t)}{\sigma(\gamma)} <
\frac{p_{\rm min}^-\sum_{k=1}^TF(k,\delta)} {4\delta^{p_{\rm max}^+}}.
$$

\item[(A2)] $F(k,\delta)\geq 0$ for each  $k\in
\mathbb{Z}[1,T]$.
\end{itemize}
Then, for each
\begin{equation}\label{ww}
 \lambda\in\Lambda_w:=
\Big]\frac{4\delta^{p_{\rm max}^+}}{p_{\rm min}^-\sum_{k=1}^TF(k,\delta)},
\frac{\sigma(\gamma)}{\sum_{k=1}^T\max_{|t|\leq
\gamma}F(k,t)}\Big[ ,
\end{equation}
problem \eqref{E11} admits at least one nontrivial
solution $\overline{u}\in H$, such that $ |\overline{u}|<\gamma$.
\end{theorem}

\begin{proof}
Take the real Banach space $H$ as defined in Section 2, and put
$\Phi,\Psi$, as in \eqref{E23}. Our aim is to apply Theorem
\ref{teo:bon} to function $I_\lambda$. For each $u\in H$ such that
$\| u\| \geq 1$, from assertion $(b)$ in Lemma \ref{lem}, we have
\begin{align*}
 \Phi(u)
&\geq \frac{1}{p_{\rm max}^+}\sum_{k=1}^{T+1}\Big(|\Delta
u(k-1)|^{p_1(k-1)}+|\Delta u(k-1)|^{p_2(k-1)}\Big)\\
&\geq \frac{1}{p_{\rm max}^+}\Big(T^{\frac{2-p_1^-}{2}}
 \|u\|^{p_1^-}+T^{\frac{2-p_2^-}{2}}\|u\|^{p_2^-}-2T\Big)
\to \infty \quad \text{as } \|u\|\to \infty .
\end{align*}
So, $\Phi$ is a coercive, and we have the regularity assumptions required on
$\Phi$ and $\Psi$. 
Therefore, it remains to verify assumption
\eqref{condizionealgebrica}. To this end, we put 
$r:=\sigma(\gamma)$, and pick $w\in H$, defined as 
\begin{equation}\label{w}
w(k):=\begin{cases}
\delta, & \text{if } k\in \mathbb{Z}[1,T],\\
0, & \text{otherwise }.
\end{cases}
\end{equation}
Clearly, with $\delta\geq1$ one has
\begin{equation}\label{e3.3}
\Phi(w)=\sum_{k=1}^{T+1}\Big(\frac{|\Delta
w(k-1)|^{p_1(k-1)}}{p_1(k-1)}+\frac{|\Delta
w(k-1)|^{p_2(k-1)}}{p_2(k-1)}\Big)\leq
\frac{4\delta^{p_{\rm max}^+}}{p_{\rm min}^-}.
\end{equation}
Hence, it follows from  \eqref{e3.1} that $0<\Phi(w)<r$. Now, let
$u\in H$ such that $u\in \Phi^{-1}([0,r])$, by Lemma \ref{lem}
(a), for any $u\in H$ with $\|u\|<1$ we obtain
\begin{equation}\label{e1}
\begin{aligned}
r\geq\Phi(u)&\geq \frac1{p_{\rm max}^+}\Big(T^{\frac{p_1^+
 -2}{2}}\|u\|^{p_1^+}+T^{\frac{p_2^+-2}{2}}\|u\|^{p_2^+}\Big) \\
&\geq \frac{T^{\frac{p_{\rm min}^--2}{2}}}{p_{\rm max}^+}\|u\|^{p_{\rm max}^+}.
\end{aligned}
\end{equation}
Similarly, from Lemma \ref{lem} (b), for any $u\in H$ with
$\|u\|>1$, we obtain
\begin{equation}\label{e12}
\begin{aligned}
r\geq\Phi(u)&\geq \frac1{p_{\rm max}^+}\Big(T^{\frac{2-p_1^-}{2}}
\|u\|^{p_1^-}+T^{\frac{2-p_2^-}{2}}\|u\|^{p_2^+}-2T\Big) \\
&\geq \frac{1}{p_{\rm max}^+}\Big(T^{\frac{2-p_{\rm max}^+}{2}}
\|u\|^{p_{\rm min}^-}-T\big).
\end{aligned}
\end{equation}
 Then
$$
\|u\|\leq\max\Big\{\Big(\frac{rp_{\rm max}^+}{T^{\frac{p_{\rm min}^--2}{2}}}\Big)^{1/p_{\rm max}^+},
\Big(\frac{rp_{\rm max}^+}{T^{\frac{2-p_{\rm max}^+}{2}}}
+2T^{\frac{p_{\rm max}^+}2}\Big)^{1/{p_{\rm min}^-}}\Big\}.
$$
Bearing in mind \eqref{e3.0}, we obtain
$$
rp_{\rm max}^+\geq T^{\frac{p_{\rm min}^--2}{2}}.
$$
Then, from \eqref{e1} and \eqref{e12} we have
$$
\|u\|\leq\Big(\frac{rp_{\rm max}^+}{T^{\frac{2-p_{\rm max}^+}{2}}}
+2T^{\frac{p_{\rm max}^+}2}\Big)^{1/p_{\rm min}^-}.
$$
This together with Lemma \ref{lem} $(d)$, yields
 $$
 |u(k)|\leq \sqrt{T+1}\|u\|\leq
\sqrt{T+1}\Big(\frac{rp_{\rm max}^+}{T^{\frac{2-p_{\rm max}^+}{2}}}
+2T^{\frac{p_{\rm max}^+}2}\Big)^{1/p_{\rm min}^-}=\gamma
 $$ for all $k\in\mathbb{Z}[1,T]$.
 Therefore, we have that
\begin{equation}\label{e3.30}
\sup_{u\in \Phi^{-1}([0,r])}\Psi(u) =\sup_{u\in
\Phi^{-1}([0,r])}\sum_{k=1}^T F(k,u(k))
\leq\sum_{k=1}^T\max_{|t|\leq \gamma}F(k,t).
\end{equation}
 In view of \eqref{e3.3} and \eqref{e3.30}, taking into account (A1) and (A2), 
we obtain
\begin{equation}\label{eq}
\begin{aligned}
\frac{\sup_{\Phi^{-1}([0, r])}\Psi(u)}{r}
&\leq \frac{\sum_{k=1}^T\max_{|t|\leq \gamma}F(k,t)}{\sigma(\gamma)} \\
&< \frac{p_{\rm min}^-\sum_{k=1}^TF(k,\delta)}
{4\delta^{p_{\rm max}^+}}
\leq \frac{\Psi(w)}{\Phi(w)}.
\end{aligned}
\end{equation}
Therefore, condition \eqref{condizionealgebrica} of Theorem
\ref{teo:bon} is verified and all the assumptions of Theorem
\ref{teo:bon} are satisfied. So, for each 
$\lambda \in \Lambda_w\subset ] \frac{\Phi(w)}{\Psi(w)},
\frac{r}{\sup_{\Phi^{-1}([0, r])}\Psi(u)} [$, the functional
$I_\lambda$ admits at least one critical point $\overline{u}$ such
that $0<\Phi(\overline{u})< r$, and so $\overline{u}$ is a
nontrivial weak solution of problem \eqref{E11} such that
$|\overline{u}|<\gamma$.
\end{proof}

The following result, in which the global Ambrosetti-Rabinowitz
condition is also used, ensures the existence at least two weak
solutions.

\begin{theorem}\label{the4.1}
We suppose that the assumptions \eqref{e3.0} and \eqref{e3.1} of
Theorem \ref{the3.1} be satisfied and $f(k,0)\neq 0$ for every
$k\in\mathbb{Z}[1,T]$. Assume that there are two positive constants
$\mu>p_{\rm max}^+$ and $R>0$ such that,
\begin{equation}\label{r}
0<\mu F(k,t)\leq tf(k,t),
\end{equation}
for all  $k\in \mathbb{Z}[1,T]$ and $|t|\geq R$. Then, for each
$\lambda\in\Lambda:=\Big]0,
\frac{\sigma(\gamma)}{\sum_{k=1}^T\max_{|t|\leq
\gamma}F(k,t)}\Big[$, problem \eqref{E11} admits at least two
nontrivial solutions.
\end{theorem}

\begin{proof}
Let $\Phi,\Psi$ be the functionals defined in \eqref{E23} satisfy
all regularity assumptions requested in Theorem \ref{critical1}.
Arguing as in the proof of Theorem \ref{the3.1}, put $w(k)$ as in
\eqref{w} and $r=\sigma(\gamma)$, for $\lambda\in \Lambda$ we obtain
$$
\frac{\sup_{\Phi^{-1}([0,r])}\Psi(u)}{r}
\leq\frac{\sum_{k=1}^T\max_{|t|\leq
\gamma}F(k,t)}{\sigma(\gamma)}<\frac1{\lambda}.
$$
Now, From condition \eqref{r}, by standard computations, there is a
positive constant $c_1$ such that
\begin{equation}\label{e3.7}
F(k,s)\geq c_1|s|^{\mu}\quad \text{for all } k\in \mathbb{Z}[1,T].
\end{equation}
Hence, for every $\lambda\in\Lambda$, $u\in H\backslash \{0\}$ and
$t>1$, we obtain
\begin{align*}
I_\lambda(tu)
&\leq \sum_{k=1}^{T+1}\Big(\frac{|\Delta
tu(k-1)|^{p_1(k-1)}}{p_1(k-1)}+\frac{|\Delta
tu(k-1)|^{p_2(k-1)}}{p_2(k-1)}\Big)-\lambda c_1
t^\mu\sum_{k=1}^T|u(k)|^\mu \\
&\leq t^{p_{\rm max}^+}\sum_{k=1}^{T+1}\Big(\frac{|\Delta
u(k-1)|^{p_1(k-1)}}{p_1(k-1)}+\frac{|\Delta
u(k-1)|^{p_2(k-1)}}{p_2(k-1)}\Big)-\lambda c_1
t^{\mu}\sum_{k=1}^T|u(k)|^\mu.
\end{align*}
Since $\mu>p_{\rm max}^+$, $I_\lambda(tu)\to -\infty$ as $t\to \infty$.
Then $I_\lambda$ is unbounded from below. Finally, we verify the
$(PS)$-condition, it is sufficient to prove that any Palais-Smale
sequence is bounded. Arguing by contradiction, suppose that there
exists a sequence $\{u_n\}$ such that  $\{I_{\lambda}(u_n)\}$ is
bounded and $\|I'_{\lambda}(u_n)\|_{X^{\ast}}\to 0\;\text{as}\; n\to
+\infty$ and $\lim_{n\to +\infty} \|u_n\|=+\infty$. Using also
\eqref{r}, we deduce that, for all $n\in\mathbb{N}$, it holds
\begin{align*}
&\sum_{k=1}^{T}\Big(\mu F(k,u_n(k))-u_n(k)f(k,u_n(k))\Big)\\
&\leq \sum_{|u_n(k)|\leq R}\Big(\mu F(k,u_n(k))-u_n(k)f(k,u_n(k))\Big)\\
&\leq \sum_{k=1}^{T}\max_{|x|\leq R}|\mu F(k,x)-xf(k,x)|=:c_2.
\end{align*}
To this end, taking into account Lemma \ref{lem} (b) one has
\begin{align*}
M+\|u_n\| 
&\geq I_\lambda(u_n)-\frac{1}{\mu}\langle
I'_\lambda(u_n),u_n\rangle\\
&=\sum_{k=1}^{T+1}\Big(\frac{|\Delta
u_n(k-1)|^{p_1(k-1)}}{p_1(k-1)}+\frac{|\Delta
u_n(k-1)|^{p_2(k-1)}}{p_2(k-1)}\Big)
-\lambda\sum_{k=1}^{T}F(x,u_n(k))\\
&\quad -\frac{1}{\mu}\sum_{k=1}^{T+1}\Big(|\Delta
u_n(k-1)|^{p_1(k-1)}+|\Delta
u_n(k-1)|^{p_2(k-1)}\Big)\\
&\quad+\lambda\sum_{k=1}^{T}\frac1{\mu}f(x,u_n(k))u_n(k)
\\
&\geq\Big(\frac{1}{p_{\rm max}^+}-\frac{1}{\mu}\Big)
\sum_{k=1}^{T+1}\Big(|\Delta u_n(k-1)|^{p_1(k-1)}+|\Delta
u_n(k-1)|^{p_2(k-1)}\Big)\\
&\quad-\frac{\lambda}{\mu} \sum_{k=1}^{T} \left(\mu F(x,u_n(k))-u_n(k)f(x,u_n(k))\right)\\
&\geq \Big(\frac{1}{p_{\rm max}^+}-\frac{1}{\mu}\Big)
\Big(T^{\frac{2-p_1^-}{2}}\|u_n\|^{p_1^-}
 +T^{\frac{2-p_2^-}{2}}\|u_n\|^{p_2^-}-2T\Big)-\frac{\lambda}{\mu}c_2.
\end{align*}
But, this cannot hold true since $p_1^-,\,p_2^->1$ and
$\mu>p_{\rm max}^+$. Hence, $\{u_n\}$ is bounded. That information
combined with the fact that $H$ is a finite dimensional Hilbert
space implies that there exists a subsequence, still denoted by
$\{u_n\}$, and $u_0\in H$ such that $u_n$ converges to $u_0$ in $H$.
Then, for each $\lambda \in \Lambda$, the function $I_\lambda$
admits at least two distinct critical points.
\end{proof}

Finally, we give an application of Theorem \ref{critical3}.

\begin{theorem}\label{theo33}
Suppose that there exist two constants $\gamma$ and $\delta\geq1$
with \eqref{e3.0} and
\begin{equation}\label{e2}
4\delta^{p_{\rm min}^-}>p_{\rm max}^+\sigma(\gamma)
\end{equation}
 such that the assumptions {\rm (A1)} and {\rm (A2)}
 in Theorem \ref{the3.1}
 hold. Assume also
\begin{equation}\label{f}
|f(k,t)|\leq a_0(1+|t|^{\alpha(k)-1}),
\end{equation}
where $a_0>0$ and $2\leq \alpha^-=\min_{k\in
[0,T]}\alpha(k)\leq\alpha^+=\max_{k\in
 [0,T]}\alpha(k)<p_{\rm min}^-$. Then, for each $\lambda\in \Lambda_w$, where $\Lambda_w$ as in \eqref{ww}, problem $\eqref{E11}$
admits at last three weak solutions.
\end{theorem}

\begin{proof}
Our aim is to verify (i) and (ii) of Theorem  \ref{critical3}.
Arguing as in the proof of Theorem \ref{the3.1}, put $w(k)$ as in
\eqref{w} and $r=\sigma(\gamma)$, bearing in mind \eqref{e2} we
obtain
$$
\Phi(w)>r>0.
$$
Therefore, \eqref{eq} holds and the assumption $(i)$ of Theorem
\ref{critical3} is satisfied. Now, we prove that the functional
$I_\lambda$ is coercive. For $u\in H$ such that $\|u\|\to +\infty$,
in fact by using condition \eqref{f}, we have
\begin{align*}
I_{\lambda}(u)&\geq \frac{1}{p_{\rm max}^+}\sum_{k=1}^{T+1}\Big(|\Delta
u(k-1)|^{p_1(k-1)}+|\Delta u(k-1)|^{p_2(k-1)}\Big)\\
&\quad -\lambda a_1
\sum_{k=1}^T\frac{|u(k)|^{\alpha(k)}}{\alpha(k)} -a_2,
\end{align*}
where $a_1,a_2$ are positive constants. Now, for 
$k\in \mathbb{Z}[1,T]$ we point out that
$$
|u(k)|^{\alpha(k)}\leq|u(k)|^{\alpha^-}+|u(k)|^{\alpha^+}.
$$
Thus, using \eqref{e2.3} and Lemma \ref{lem} (c), we obtain
\begin{align*}
|u|^{\alpha^{\pm}}_{\alpha^{\pm}}
&=\sum_{k=1}^{T} |u(k)|^{\alpha^{\pm}}\leq T|u|^{\alpha^{\pm}}_2
=T\Big(\sum_{k=1}^{T} |u(k)|^2\Big)^{\alpha^{\pm}/2}\\
&\leq  T \Big(c_2\sum_{k=1}^{T+1} |\Delta
u(k-1)|^2\Big)^{\alpha^{\pm}/2}
=TC_{\alpha^{\pm}}\|u\|^{\alpha^{\pm}}.
\end{align*}
Then, for every $\lambda\in\Lambda$ we obtain
\begin{align*}
I_{\lambda}(u)
&\geq \frac{1}{p_{\rm max}^+}\Big(T^{\frac{2-p_1^-}{2}}\|u\|^{p_1^-}
 +T^{\frac{2-p_2^-}{2}}\|u\|^{p_2^-}-2T\Big)\\
&\quad -\frac{\lambda a_1}{\alpha^-}\Big(TC_{\alpha^-}\|u\|^{\alpha^-}
 +TC_{\alpha^+}\|u\|^{\alpha^+}\Big)-a_2\\
&\geq \frac{1}{p_{\rm max}^+}\Big(T^{\frac{2-p_{\rm max}^+}{2}}
 \|u\|^{p_{\rm min}^-}-2T\Big)-a_3\|u\|^{\alpha^+}-a_2\to
+\infty,
\end{align*}
since $p_{\rm min}^->\alpha^+$, the functional $I_\lambda$ is coercive,
also condition (ii) holds. So, for each $\lambda\in\Lambda_w$, the
functional $I_\lambda$ has at least three distinct critical points
that are weak solutions of \eqref{E11}.
\end{proof}

\begin{example}\label{ex4.4}\rm
For $T=2$, consider the problem
\begin{equation} \label{ex}  
\begin{gathered}
-\Delta\Big(\Big( |\Delta u(0)|^{p_1(0)-2}+|\Delta
u(0)|^{p_2(0)-2}\Big)\Delta
u(0)\Big)=-2\lambda(u(1)-1)\\
-\Delta\Big(\Big( |\Delta u(1)|^{p_1(1)-2}+|\Delta
u(1)|^{p_2(1)-2}\Big)\Delta u(1)\Big)=-2\lambda(u(2)-2)\\
u(0)=u(3)=0,
\end{gathered}
\end{equation}
where  $f(k,t)=-2(t-k)$ for $k=1,2$ and for 
$$
p_1(k)=\frac12 k+2,\quad 
p_2(k)=-\frac{1}2 k+4\quad\text{for }k=0,1,2.
$$
Then one has
$$
p_1^-=2,\quad p_2^-=3,\quad p_1^+=3,\quad
p_2^+=4,\quad p_{\rm min}^-=2,\quad p_{\rm max}^+=4.
$$
In fact, if we choose, for example $\delta=1$ and $\gamma=6\sqrt{3}$
such that \eqref{e3.0} is verified, we obtain $\sigma(\gamma)=7/2$
and condition \eqref{e3.1} holds. Moreover, one has
$$
\frac{\sum_{k=1}^2\max_{|t|\leq 6\sqrt{3}}F(k,t)}{7/2}=\frac{10}7
<2= \frac{p_{\rm min}^-\sum_{k=1}^2F(k,1)} {4\delta^{p_{\rm max}^+}}.
$$
Then, owing to Theorem \ref{the3.1}, for each
$\lambda\in\big]\frac12,\frac7{10}\big[$, problem \eqref{ex}
admits at least one nontrivial solution $\overline{u}$, such that $
|\overline{u}|<6\sqrt{3}$.
\end{example}

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\end{document}