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\title{\sc Eigenvalue problems for anisotropic
discrete boundary value problems\thanks{
Correspondence address: Vicen\c{t}iu R\u{a}dulescu, Department of
Mathematics, University of Craiova,  200585 Craiova, Romania. E-mail:
{\tt
vicentiu.radulescu@math.cnrs.fr}}}
\author{\sc Mihai Mih\u ailescu$\,^{a,b}$,  Vicen\c{t}iu R\u{a}dulescu$\,^{a,c}$ and Stepan  Tersian$\,^{d}$\\
\small
$^a\,$Department of Mathematics, University of Craiova,  200585 Craiova,
Romania\\
\small $^b\,$Department of Mathematics, Central European University,  1051 Budapest,
Hungary\\
\small $^c\,$Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,\\
\small P.O. Box 1-764, 014700 Bucharest, Romania\\
\small $^d\,$Department of Mathematical Analysis, University of Rousse, 7017 Rousse, Bulgaria\\
\small
E-mail: {\tt mmihailes@yahoo.com}\quad {\tt
vicentiu.radulescu@math.cnrs.fr}\quad {\tt sterzian@ru.acad.bg}}

\begin{document}

\baselineskip16pt \maketitle\noindent{\small{\sc Abstract}. In
this paper, we prove the existence of a continuous spectrum for a
family of discrete boundary value problems. The main  existence results are obtained
by using critical point theory. The
equations studied in the paper represent a discrete
variant of some recent anisotropic
variable exponent problems  which deserve as models in different fields of mathematical physics.\\
\small{\bf 2000 Mathematics
Subject Classification:}  47A75, 35B38, 35P30, 34L05, 34L30. \\
\small{\bf Key words:}  eigenvalue problem, discrete boundary
value problem, critical point, weak solution, continuous
spectrum.}


\section{Introduction and main results}
This paper is concerned with the study of the existence of solutions for the discrete boundary value problem
\begin{equation}\label{1}
\left\{\begin{array}{lll}
-\Delta(|\Delta u(k-1)|^{p(k-1)-2}\Delta u(k-1))=\lambda|u(k)|^{q(k)-2}u(k), \quad
k\in\ZZ[1,T],\\
u(0)=u(T+1)=0\,,
\end{array}\right.
\end{equation}
where $T\geq 2$ is a positive integer and $\Delta u(k)=u(k+1)-u(k)$ is the
forward difference operator. Here and hereafter, we denote by $\ZZ[a,b]$ the discrete interval
$\{a,a+1,...,b\}$ where $a$ and $b$ are integers and $a<b$.
Moreover, in this paper we assume
that  functions $p:\ZZ[0,T]\rightarrow[2,\infty)$  and
$q:\ZZ[1,T]\rightarrow[2,\infty)$ are bounded while $\lambda$ is a
positive constant.

The study of discrete boundary value problems has captured special
attention in the last years. We just refer to the recent results
of Agarwal {\it et al.} \cite{APR}, Yu and Guo \cite{YG}, Cai and
Yu \cite{CY}, Zhang and Liu \cite{ZL} and the references therein.
The studies regarding such type of problems can be placed at the
interface of certain mathematical fields such as nonlinear partial
differential equations and numerical analysis. On the other hand,
they are strongly motivated by their applicability in mathematical
physics. We note that the problem \eq{1} is the discrete
variant of the variable exponent anisotropic problem
\begin{equation}\label{P}
\left\{\begin{array}{lll}
-\di\sum_{i=1}^N\di\frac{\partial}{\partial x_i}\left(\left|\di\frac{\partial u}{\partial x_i}
\right|^{p_i(x)-2}\di\frac{\partial u}{\partial x_i}\right)=\lambda|u|^{q(x)-2}u, &\mbox{for}&
x\in\Omega\\
u=0, &\mbox{for}& x\in\partial\Omega\,,
\end{array}\right.
\end{equation}
where $\Omega\subset\RR^N$ ($N\geq 3$) is a bounded domain with
smooth boundary, $\lambda>0$ is a real number, and $p_i(x)$,
$q(x)$ are continuous on $\overline\Omega$ such that $N>p_i(x)\geq
2$ and $q(x)>1$ for any $x\in\overline\Omega$ and all
$i\in \ZZ[1,N]$.  Problem \eq{P} was recently analyzed by Mih\u
ailescu-Pucci-R\u adulescu in \cite{mprcras,anisotropic} (see also
the studies in \cite{fraga,rak1,rak2,tro,ven} for the case when
$p_i(x)$ are constant functions). Problems like \eq{P} have been
intensively studied  in the last decades since they can model
various phenomena arising from the study of elastic mechanics
(see, Zhikov \cite{Z1}), electrorheological fluids (see, Acerbi
and Mingione \cite{acerbi2}, Diening \cite{D}, Halsey \cite{hal},
Ruzicka \cite{R}, Mih\u ailescu and R\u adulescu \cite{RoyalSoc,
AMS, manuscripta,mathscan}) or image restoration (see, Chen,
Levine and Rao \cite{chen}).

In this paper our goal is to use the critical point theory in
order to establish the existence of a continuous spectrum of
eigenvalues for problems of type \eq{1}. Our idea is to transfer
the problem of the existence of solutions for problem \eq{1} into
the problem of existence of critical points for some associated
energy functional. On the other hand, we point out that, to our
best knowledge, discrete problems like \eq{1}, involving
anisotropic exponents, have not yet been discussed. Thus, the
present paper can be regarded as a contribution in this
direction.

We are interested in finding week solutions for problems of type
\eq{1}.  For this purpose we define the function space
$$H=\{u:\ZZ[0,T+1]\rightarrow\RR;\;{\rm such}\;{\rm that}\;u(0)=u(T+1)=0\} .$$

Clearly, $H$ is a $T$-dimensional Hilbert space (see \cite{APR})
with the inner product
$$(u,v)=\sum_{k=1}^{T+1}\Delta u(k-1)\Delta v(k-1),\;\;\;\forall\;u,v\in H\,.$$
This associated norm is defined by
$$\|u\|=\left(\sum_{k=1}^{T+1}|\Delta u(k-1)|^2\right)^{1/2}\,.$$
By a {\it weak solution} for problem \eq{1} we understand a function $u\in H$ such that
$$\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)-2}\Delta u(k-1)\Delta
v(k-1)-\lambda\sum_{k=1}^{T}|u(k)|^{q(k)-2}u(k)v(k)=0\,,$$
for any $v\in H$.

Denote for short $\max_{k\in \ZZ[a,b]}p(k)$ by $\max_{\ZZ[a,b]}p$ and
$\min_{k\in \ZZ[a,b]}p(k)$ by $\min_{\ZZ[a,b]}p$.

The main results of this paper are the following.

\begin{teo}\label{t1}
Assume that  functions $p$ and $q$ verify the hypothesis
\begin{equation}\label{2}
\max_{\ZZ[0,T]}p<\min_{\ZZ[1,T]}q\,.
\end{equation}
Then for any $\lambda>0$ problem \eq{1} has a nontrivial weak
solution.
\end{teo}

\begin{teo}\label{t2}
Assume that  functions $p$ and $q$ verify the hypothesis
\begin{equation}\label{3}
\max_{\ZZ[1,T]}q<\min_{\ZZ[0,T]}p\,.
\end{equation}
Then there exists $\lambda^{\star\star}>0$ such that for any
$\lambda>\lambda^{\star\star}$ problem \eq{1} has a nontrivial
weak solution.
\end{teo}

\begin{teo}\label{t3}
Assume that  functions $p$ and $q$ verify the hypothesis
\begin{equation}\label{4}
\min_{\ZZ[1,T]}q<\min_{\ZZ[0,T]}p\,.
\end{equation}
Then there exists $\lambda^{\star}>0$ such that for any
$\lambda\in(0,\lambda^{\star})$ problem \eq{1} has a nontrivial
weak solution.
\end{teo}

\noindent {\bf Remark 1.} We point out that if   relation \eq{4}
is verified then  relation \eq{3} is fulfilled, too. Consequently,
the result of Theorem \ref{t2} can be completed with the
conclusion of Theorem \ref{t3}. More exactly, we deduce the
following corollary.

\begin{cor}\label{c1}
Assume that  functions $p$ and $q$ verify the hypothesis
$$
\min_{\ZZ[1,T]}q<\min_{\ZZ[0,T]}p\,.
$$
Then there exist $\lambda^{\star}>0$ and $\lambda^{\star\star}>0$ such that for any
$\lambda\in(0,\lambda^\star)\cup(\lambda^{\star\star},\infty)$  problem \eq{1} possesses a nontrivial weak solution.
\end{cor}

\noindent {\bf Remark 2.} On the other hand, we point out that the result of Theorem \ref{t3} holds true
in situations that extend relation \eq{3} since in relation \eq{4} we could have
$$\min_{\ZZ[1,T]}q<\min_{\ZZ[0,T]}p<\max_{\ZZ[1,T]}q\,.$$

\section{Auxiliary results}
>From now on we will use the following notations:
$$p^-=\min_{\ZZ[0,T]}p,\;\;\;p^+=\max_{\ZZ[0,T]}p,\;\;\;q^-=\min_{\ZZ[1,T]}q,\;\;\;q^+=\max_{\ZZ[1,T]}q\,.$$
On the other hand, it is useful to introduce  other norms on $H$,
namely
$$|u|_m=\left(\sum_{k=1}^{T}|u(k)|^m\right)^{1/m},\;\;\;\forall\;u\in H\;{\rm and}\;m\geq 2\,.$$
It can be  verified (see \cite{CY}) that
\begin{equation}\label{echiv}
T^{(2-m)/(2m)}\cdot|u|_2\leq|u|_m\leq T^{1/m}\cdot|u|_2,\;\;\;\forall\;u\in H\;{\rm and}\;m\geq 2\,.
\end{equation}
We start with the following auxiliary result.
\begin{lemma}\label{LL}
a) There exist two positive constants $C_1$ and $C_2$ such that
$$\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)}\geq C_1\cdot\|u\|^{p^-}-C_2,\;\;\;\forall\;u\in H\;\mbox{
with}\;\|u\|>1\,.$$
\smallskip

\noindent b) There exists a positive constant $C_3$ such that
$$\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)}\geq C_3\cdot\|u\|^{p^+},\;\;\;\forall\;u\in H\;\mbox{ with}\;\|u\|<1\,.$$
\smallskip

\noindent c) For any $m\geq 2$ there exists a positive constant $c_m$ such that
$$\sum_{k=1}^{T}|u(k)|^m\leq c_{m}\cdot\sum_{k=1}^{T+1}|\Delta u(k-1)|^m,\;\;\;\forall\;u\in H\,.$$
\end{lemma}
\proof a) Fix $u\in H$ with $\|u\|>1$. We define

$$\alpha_{k}=\left\{\begin{array}{lll}
p^+, &\mbox{if}& |\Delta u(k)|<1\\
p^-, &\mbox{if}& |\Delta u(k)|>1\,,
\end{array}\right.$$
for each $k\in\ZZ[0,T]$.

We deduce that
\begin{eqnarray*}
\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)}&\geq&\sum_{k=1}^{T+1}|\Delta u(k-1)|^{\alpha_{k-1}}\\
&\geq&\sum_{k=1}^{T+1}|\Delta
u(k-1)|^{p^-}-\sum_{\{k\in\ZZ[0,T];\;\alpha_{k-1}=p^+\}}(|\Delta
u(k-1)|^{p^-}-|\Delta
u(k-1)|^{p^+})\\
&\geq&\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p^-}-T\,.
\end{eqnarray*}
The above inequality and relation \eq{echiv} imply
$$\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)}\geq T^{(2-p^-)/2}\cdot\|u\|^{p^-}-T,\;\;\;
\forall\; u\in H\;{\rm with}\;\|u\|>1\,.$$
Thus, we proved that a) holds true.

b) Assume $u\in H$ with $\|u\|<1$. It follows that $|\Delta
u(k)|<1$ for each $k\in\ZZ[0,T]$. So, by \eq{echiv} we deduce that
\begin{eqnarray*}
\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)}&\geq&\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p^+}\\
%&\geq&1/c_{p^+}\cdot\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p^+}\\
&\geq&1/T^{(2-p^+)/2}\cdot\|u\|^{p^+}\,.
\end{eqnarray*}
Thus, we proved that b) holds true.

c) Since
$$|u(k)|\leq\sum_{i=0}^{k-1}|\Delta u(i)|,\;\;\;\forall\;u\in H\;{\rm and}\;k\in\ZZ[0,T]\,,$$
we deduce that for any positive real number $m\geq 2$ there exists a positive constant $c_{m,k}$ such that
$$|u(k)|^m\leq c_{m,k}\cdot\sum_{i=0}^{k-1}|\Delta u(i)|^m,\;\;\;\forall\;u\in H\;{\rm and}\;k\in\ZZ[0,T]\,.$$
The above information implies that there exists a positive constant $c_m$ such that
\begin{equation}\label{5}
\sum_{k=1}^{T}|u(k)|^m\leq c_{m}\cdot\sum_{k=1}^{T+1}|\Delta u(k-1)|^m,\;\;\;\forall\;u\in H\,.
\end{equation}

The proof of Lemma \ref{LL} is complete.  \endproof

\section{Proof of the main results}
For any $\lambda>0$ the energy functional corresponding to problem \eq{1} is defined as
$J_\lambda:H\rightarrow\RR$,
$$J_\lambda(u)=\sum_{k=1}^{T+1}\frac{1}{p(k-1)}|\Delta
u(k-1)|^{p(k-1)}-\lambda\cdot\sum_{k=1}^{T}\frac{1}{q(k)}|u(k)|^{q(k)}\,.$$
Standard arguments assure that $J_\lambda\in C^1(H,\RR)$ and
$$\langle J_\lambda^{'}(u),v\rangle=\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)-2}\Delta u(k-1)\Delta
v(k-1)-\lambda\sum_{k=1}^{T}
|u(k)|^{q(k)-2}u(k)v(k)\,,$$
for all $u,\;v\in H$. Thus the weak solutions of \eq{1} coincide with the critical
points of $J_\lambda$.

\subsection{Proof of Theorem \ref{t1}}
In order to prove that $J_\lambda$ has a nontrivial critical point our idea is to show that
actually $J_\lambda$ possesses a mountain-pass geometry. With that end in view we start by
proving two  auxiliary results.

\begin{lemma}\label{l1}
There exist $\eta>0$ and $\alpha>0$ such that
$J_\lambda(u)\geq\alpha>0$ for any $u\in H$ with $\|u\|=\eta$.
\end{lemma}
\proof
First, we point out that
\begin{equation}\label{l11}
|u(k)|^{q^-}+|u(k)|^{q^+}\geq|u(k)|^{q(k)},\;\;\;\forall\; k\in\ZZ[1,T]\;{\rm and}\;u\in H\,.
\end{equation}
Using the above inequality we find
\begin{equation}\label{l12}
J_\lambda(u)\geq\frac{1}{p^+}\sum_{k=1}^{T+1}\frac{1}{p(k-1)}|\Delta
u(k-1)|^{p(k-1)}-\frac{\lambda}{q^-}(|u|_{q^-}^{q^-}+
|u|_{q^+}^{q^+}),\;\;\;\forall\;u\in H\,.
\end{equation}
Next, we focus  on the case when $u\in H$ with $\|u\|<1$. Thus,
$|\Delta u(k)|<1$ for any $k\in\ZZ[0,T+1]$. Then using Lemma
\ref{LL} c) and relation \eq{echiv} we infer
\begin{equation}\label{clar}
|u|_{q^-}^{q^-}+|u|_{q^+}^{q^+}\leq c_{q^-}\sum_{k=1}^{T+1}|\Delta u(k-1)|^{q^-}+
c_{q^+}\sum_{k=1}^{T+1}|\Delta u(k-1)|^{q^+}\leq c_{q^-} T\|u\|^{q^-}+c_{q^+} T
\|u\|^{q^+}\,.
\end{equation}
For  $u\in H$ with $\|u\|<1$ the above inequalities combined with
relation \eq{l12}, Lemma \ref{LL} b) and relation \eq{echiv} imply
\begin{eqnarray*}
J_\lambda(u)&\geq&\frac{C_3}{p^+}\|u\|^{p^+}-\frac{\lambda}{q^-}(c_{q^-} T\cdot\|u\|^{q^-}+c_{q^+} T\cdot
\|u\|^{q^+})\\
&=&(d_1-d_2\cdot\|u\|^{q^--p^+}-d_3\cdot\|u\|^{q^+-p^+})\cdot\|u\|^{p^+}\,,
\end{eqnarray*}
where $d_1$, $d_2$ and
$d_3$ are positive constants.

We remark that the function $g:[0,1]\rightarrow\RR$ defined by
$$g(t)=d_1-d_2\cdot t^{q^+-p^+}-d_3\cdot t^{q^--p^+}$$
is positive in a neighborhood of the origin by continuity argument. We conclude that Lemma
\ref{l1} holds true.  \endproof

\begin{lemma}\label{l2}
There exists $e\in H$  with $\|e\|>\eta$ (where $\eta$ is given in Lemma \ref{l1}) such that
$J_\lambda(e)<0$.
\end{lemma}
\proof Consider the function $\psi:\ZZ[0,T+1]\rightarrow\RR$ such
that there exists $k_0$ an integer satisfying $0<k_0<T+1$ for
which $\psi(k_0)=1$ and $\psi(k)=0$ for any
$k\in\ZZ[0,T+1]\setminus\{k_0\}$. Thus, we deduce that $\psi\in
H$. For each $t>1$ we have
$$J_\lambda(t\psi)=\frac{t^{p(k_0)}}{p(k_0)}+\frac{t^{p(k_0-1)}}{p(k_0-1)}-\lambda\cdot\frac{t^{q(k_0)}}{q(k_0)}\leq
\frac{2\cdot t^{p^+}}{p^-}-\lambda\cdot\frac{t^{q^-}}{q^+}\,.$$
Since $q^->p^+$ it is clear that $\lim_{t\rightarrow\infty}J_\lambda(t\psi)=-\infty$. Then, for $t>1$ large enough
we can take $e=t\psi$ such that $\|e\|>\eta$ and $J_\lambda(e)<0$.

The proof of Lemma \ref{l2} is complete. \endproof

{\sc Proof of Theorem \ref{t1}.} By Lemmatas \ref{l1} and \ref{l2}
and the mountain-pass theorem of Ambrosetti and Rabinowitz
\cite{AR} we deduce the existence of a sequence $\{u_n\}\subset H$
such that
\begin{equation}\label{mp}
J_\lambda(u_n)\rightarrow\overline c>0\;\;\;\;{\rm and}\;\;\;\;J_\lambda^{'}(u_n)\rightarrow 0\;\;\;{\rm as}\;
n\rightarrow\infty.
\end{equation}
We prove that $\{u_n\}$ is bounded in $H$. Arguing by
contradiction, we assume that passing eventually to a subsequence,
still denoted by $\{u_n\}$, we have $\|u_n\|\rightarrow\infty$.
Thus, we may assume that for $n$ large enough we have $\|u_n\|>1$.

Relation \eq{mp} and the above considerations imply that for $n$ large enough we have
\begin{eqnarray*}
1+\overline c+\|u_n\|&\geq&J_\lambda(u_n)-\frac{1}{q^-}\langle J_\lambda^{'}(u_n),u_n\rangle\\
&\geq&\left(\frac{1}{p^+}-\frac{1}{q^-}\right)\sum_{k=1}^{T+1}|\Delta u_n(k-1)|^{p(k-1)}\,.
\end{eqnarray*}

By Lemma \ref{LL} a) and the above inequality we deduce that there
exist two positive constants $D_1$ and $D_2$ such that
$$1+\overline c+\|u_n\|\geq D_1\cdot\|u_n\|^{p^-}-D_2\,,$$
for $n$ large enough. Dividing by $\|u_n\|^{p^-}$ in the above inequality and passing to the limit as
$n\rightarrow\infty$ we obtain a contradiction. It follows that $\{u_n\}$ is bounded in $H$.
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, by relation \eq{mp} we have
$$J_\lambda(u_0)=\overline c>0\;\;\;{\rm and}\;\;\;J_\lambda^{'}(u_0)=0.$$
We conclude that $u_0$ is a nontrivial weak solution of problem
\eq{1}.   \endproof

\subsection{Proof of Theorem \ref{t2}}
For any $\lambda>0$ let $J_\lambda$ be defined
as above.

Now we show that $J_\lambda$ possesses a nontrivial global
minimum point in $H$. With that end in view we remark that Lemma
\ref{LL} a) implies that $J_\lambda$ is coercive on $H$. On the
other hand, it is obvious that it is also weakly lower
semicontinuous on the finite dimensional Hilbert space $H$. These
two facts enable us to apply Theorem 1.2 in \cite{S} in order to
find that there exists $u_\lambda\in H$ a global minimizer of
$J_\lambda$ and thus a weak solution of problem \eq{1}.

We show that $u_\lambda$ is not trivial for $\lambda$ large
enough. Indeed, letting $t_0>1$ be a fixed real and defining the
function $v_0:\ZZ[0,T+1]\rightarrow\RR$ such that there exists an
integer $k_0$ with $0<k_0<T+1$ for which $v_0(k_0)=t_0$ and
$v_0(k)=0$ for any $k\in\ZZ[0,T+1]\setminus\{k_0\}$ we deduce that
$v_0\in H$ and
$$J_\lambda(v_0)=\frac{t_0^{p(k_0-1)}}{p(k_0-1)}+\frac{t_0^{p(k_0)}}{p(k_0)}-\frac{\lambda\cdot t_0^{q(k_0)}}{q(k_0)}
\leq L_1-L_2\cdot\lambda\,,$$ where $L_1$ and $L_2$ are two
positive constants. Thus, there exists $\lambda^{\star\star}>0$
such that $J_\lambda(v_0)<0$ for any
$\lambda\in[\lambda^{\star\star},\infty)$. It follows that
$J_\lambda(u_\lambda)<0$ for any $\lambda\geq\lambda^{\star\star}$
and thus $u_\lambda$ is a nontrivial weak solution of problem
\eq{1} for $\lambda$ large enough. The proof of Theorem \ref{t2}
is complete.  \endproof

\subsection{Proof of Theorem \ref{t3}}
For any $\lambda>0$ let $J_\lambda$ be defined
as above.

We show that, by using the hypothesis of Theorem \ref{t3}, the
functional $J_\lambda$ has a nontrivial critical point by applying
Ekeland's variational principle \cite{E}. In order to do that we
first prove two auxiliary results.

\begin{lemma}\label{l3}
There exists $\lambda^\star>0$ such that for any $\lambda\in(0,\lambda^\star)$
there exist $\rho$, $a>0$  such that $J_\lambda(u)\geq a>0$
for any $u\in H$ with $\|u\|=\rho$.
\end{lemma}
\proof First, let us remark that for any $u\in H$, Lemma \ref{LL}
c) implies
$$c_2\cdot\|u\|\geq|u|_2\,.$$
Combining that fact and inequality \eq{echiv} we deduce that
\begin{equation}\label{stea1}
c_2\cdot T^{1/q^-}\cdot\|u\|\geq|u|_{q^-},\;\;\;\forall\;u\in H\,.
\end{equation}
We fix $\rho\in(0,1)$ such that $\rho<\min\{1,{1}/({c_2\cdot
T^{1/q^-}})\}$. Thus, for any $u\in H$ with $\|u\|=\rho$ we have
$|u|_{q^-}<1$. It follows that, in this case, $|u(k)|<1$ holds for
any $k\in\ZZ[0,T+1]$. Therefore
\begin{equation}\label{stea2}
\sum_{k=1}^{T}|u(k)|^{q(k)}\leq|u|_{q^-}^{q^-},\;\;\;\forall\;u\in H\;{\rm with}\;\|u\|=\rho\,.
\end{equation}
By relations \eq{stea1} and \eq{stea2} we obtain
$$\sum_{k=1}^{T}|u(k)|^{q(k)}\leq c_2^{q^-}\cdot T\cdot\|u\|^{q^-},\;\;\;\forall\;u\in H\;{\rm with}
\;\|u\|=\rho\,.$$
By Lemma \ref{LL} b) and the above relation we deduce that
for any $u\in H$ with $\|u\|=\rho$ the following inequalities hold true
\begin{eqnarray*}
J_\lambda(u)&\geq&\frac{C_3}{p^+}\cdot\|u\|^{p^+}-\frac{\lambda\cdot c_2^{q^-}\cdot T}{q^-}\cdot\|u\|^{q^-}\\
&=&(C_4\cdot\rho^{p^+-q^-}-\lambda\cdot C_5)\cdot\rho^{q^-}\,,
\end{eqnarray*}
where $C_4$ and $C_5$  are  positive constants. By the above inequality
and the fact that $q^-<p^-\leq p^+$ we remark that if we define
\begin{equation}\label{l33}
\lambda^\star=\frac{C_4\cdot\rho^{p^+-q^-}}{2\cdot C_5}
\end{equation}
then for any $\lambda\in(0,\lambda^\star)$ and any $u\in H$ with $\|u\|=\rho$ there
exists $a=\frac{C_4\cdot\rho^{p^+}}{2}$ such that
$$J_\lambda(u)\geq a>0.$$
The proof of Lemma \ref{l3} is complete.  \endproof

\begin{lemma}\label{l4}
There exists $\phi\in H$ such that $\phi\geq 0$, $\phi\neq 0$, and
$J_\lambda(t\phi)<0$, for $t>0$ small enough.
\end{lemma}
\proof
Since $q^-<p^-$ it follows that there exists an integer $k_0$ such that $0<k_0<T+1$ and $q^-=q(k_0)<p^-\leq p(k_0)$.
We define the function $\phi:\ZZ[0,T+1]\rightarrow\RR$ such that
 $\phi(k_0)=1$ and $\phi(k)=0$ for any $k\in\ZZ[0,T+1]\setminus\{k_0\}$. We
deduce that $\phi\in H$ and for any $t\in(0,1)$ we have
$$J_\lambda(t\cdot\phi)=\frac{t^{p(k_0-1)}}{p(k_0-1)}+\frac{t^{p(k_0)}}{p(k_0)}-\lambda\cdot\frac{t^{q(k_0)}}{q(k_0)}
\leq\frac{2\cdot t^{p^-}}{p^-}-\frac{\lambda\cdot t^{q^-}}{q^+}\,.$$
The above inequality implies
$$J_\lambda(t\cdot\phi)<0$$
for any $t<\delta^{1/(p^--q^-)}$ where
$$0<\delta<\frac{p^-\cdot\lambda}{2\cdot q^+}\,.$$

The proof of Lemma \ref{l4} is complete.  \endproof

{\sc Proof of Theorem \ref{t3}.}
Let $\lambda^\star>0$ be defined as in \eq{l33} and $\lambda\in(0,\lambda^\star)$.
By Lemma \ref{l3} it follows that on the boundary of the  ball centered at the origin and of
radius $\rho$ in $H$, denoted by $B_\rho(0)$, we have
\begin{equation}\label{et8}
\inf\limits_{\partial B_\rho(0)}J_\lambda>0.
\end{equation}
On the other hand, by Lemma \ref{l4}, there exists $\phi\in H$ such that
$J_\lambda(t\phi)<0$ for all $t>0$ small enough. Moreover, relation \eq{echiv} and
Lemma \ref{LL} b) imply that for any $u\in B_\rho(0)$ we have
$$J_\lambda(u)\geq\frac{C_3}{p^+}\|u\|^{p^+}-\frac{\lambda}{q^-}(c_{q^-} T\cdot\|u\|^{q^-}+c_{q^+} T\cdot
\|u\|^{q^+})\,.$$
It follows that
$$-\infty<\underline{c}:=\inf\limits_{\overline{B_\rho(0)}}J_\lambda<0.$$
We let now $0<\epsilon<\inf_{\partial
B_\rho(0)}J_\lambda-\inf_{B_\rho(0)}J_\lambda$. Applying Ekeland's
variational principle to the functional
$J_\lambda:\overline{B_\rho(0)}\rightarrow\RR$,  we find
$u_\epsilon\in\overline{B_\rho(0)}$ such that
\begin{eqnarray*}
J_\lambda(u_\epsilon)&<&\inf\limits_{\overline{B_\rho(0)}}J_\lambda+\epsilon\\
J_\lambda(u_\epsilon)&<&J_\lambda(u)+\epsilon\cdot\|u-u_\epsilon\|,\;\;\;u\neq u_\epsilon.
\end{eqnarray*}
Since
$$J_\lambda(u_\epsilon)\leq\inf\limits_{\overline{B_\rho(0)}}J_\lambda+\epsilon\leq
\inf\limits_{B_\rho(0)} J_\lambda+\epsilon<\inf\limits_{\partial B_\rho(0)}J_\lambda\,,$$
we deduce that $u_\epsilon\in B_\rho(0)$. Now, we define $I_\lambda:
\overline{B_\rho(0)}\rightarrow\RR$ by $I_\lambda(u)=J_\lambda(u)+\epsilon\cdot
\|u-u_\epsilon\|$. It is clear that $u_\epsilon$ is a minimum point
of $I_\lambda$ and thus
$$\di\frac{I_\lambda(u_\epsilon+t\cdot v)-{I_\lambda}(u_\epsilon)}
{t}\geq 0$$
for small $t>0$ and any $v\in B_1(0)$. The
above relation yields
$$\di\frac{J_\lambda(u_\epsilon+t\cdot v)-J_\lambda(u_\epsilon)}{t}+
\epsilon\cdot\|v\|\geq 0.$$
Letting $t\rightarrow 0$ it follows that $\langle J_\lambda^{'}
(u_\epsilon),v\rangle+\epsilon\cdot\|v\|>0$ and we infer that
$\|J_\lambda^{'}(u_\epsilon)\|\leq\epsilon$.

We deduce that there exists a sequence
$\{w_n\}\subset B_\rho(0)$ such that
\begin{equation}\label{et9}
J_\lambda(w_n)\rightarrow{\underline c}\;\;\;{\rm and}\;\;\;
J_\lambda^{'}(w_n)\rightarrow 0.
\end{equation}
Since the sequence $\{w_n\}$ is bounded in $H$, there exists $w\in
H$ such that, up  to a subsequence, $\{w_n\}$ converges  to $w$ in
$H$. So, by \eq{et9},
\begin{equation}\label{et11}
J_\lambda(w)=\underline c<0\;\;\;{\rm and}\;\;\;J_\lambda^{'}(w)=0.
\end{equation}
We conclude that $w$ is a nontrivial weak solution for problem \eq{1}.

The proof of Theorem \ref{t3} is complete.  \endproof


\medskip
{\bf Acknowledgements.} M. Mih\u ailescu and V. R\u adulescu have
been supported by Grant CNCSIS PN II--79/2007 {\it ``Procese
Neliniare Degenerate \c si Singulare"}. V.~R\u{a}dulescu also acknowledges the support through Grants 2-CEx06-11-18/2006
and GAR-18/315/2008. S.~Tersian has been supported by Grant VU-MI-02/05 of
the National Research Fund in Bulgaria. First and third authors thank 
Department of Mathematics and Its Applications, CEU, Budapest for the opportunity to meet and
discuss on discrete boundary value problems.  





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