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\markboth{A.\,Ghobadi and S.\,Heidarkhani}{Doubly eigenvalue discrete boundary value problems with $\phi_c$-Laplacian}

\title[Critical point approaches for doubly eigenvalue discrete boundary value problems driven by $\phi_c$-Laplacian operator]
{Critical point approaches for doubly eigenvalue discrete boundary value problems driven by $\phi_c$-Laplacian operator}

\author[A. Ghobadi and S. Heidarkhani]{Ahmad Ghobadi and Shapour Heidarkhani\corrauth}

\address{Department of Mathematics, Faculty of Sciences, Razi University, 67\,149 Kermanshah, Iran}

\emails{{\tt ahmad673.1356@gmail.com}\,\,(A.\,Ghobadi), {\tt sh.heidarkhani@razi.ac.ir}\,\,(S.\,Heidarkhani)}

\begin{abstract}Under appropriate algebraic conditions on the nonlinearity, using
variational methods and critical point theory we discuss the
existence of one, two and three solutions for nonlinear discrete
Dirichlet boundary value problems driven by $\phi_c$-Laplacian
operator involving two parameters $\lambda$ and $\mu$, without
imposing the symmetry or oscillating behavior at infinity on the
the nonlinearity, which has applications in the dynamic model of
combustible gases, the capillarity problem in hydrodynamics, and the
flux-limited diffusion phenomenon. Some applications and examples
illustrate the obtained results.
\end{abstract}

\keywords{multiple solutions, $\phi_c$-Laplacian boundary value
problem, critical point theory, variational methods, critical
point theory}

\ams{39A23, 53A10, 49Q20}

\maketitle

\section{Introduction}
In this paper, we study the following problem:
\begin{equation}\label{e1.1}
\begin{cases}
-\Delta(\phi_{c}(\Delta u(k-1)))=\lambda f(k,u(k))+\mu g(k,u(k)),
 \,\,k\in[1,N]_{\mathbb{Z}},\\
u(0)=u(N+1)=0,
\end{cases}
\end{equation}
where $\phi_{c}(s)={s\over \sqrt{1+s^2}}$,
$[1,N]_{\mathbb{Z}}=\{1,2,...,N\}$ and
$f,g:[1,N]_{\mathbb{Z}}\times\mathbb{R}\rightarrow\mathbb{R}$ are
continuous functions. $\Delta u(k-1)=u(k)-u(k-1)$ and
$\lambda,\mu$ are two positive parameters.

In the last decades, differential equations involving
$\phi_c$-Laplacian operator ($\phi_c(s)$ $=\frac{s}{\sqrt{1+s^2}}$ is
the mean curvature operator) regarded as a variant of the
Liouville-Bratu-Gelfand problem, which is used to study the
dynamic model of combustible gases \cite{new2,new1}, the
capillarity problem in hydrodynamics \cite{new3,new4}, and the
flux-limited diffusion phenomenon \cite{new5},  have been studied
by some researchers \cite{BJM1,BJM,BJM2,BJT,BLM,OO}.

On the other hand, difference equations describe evolution of
certain phenomena over the route of time. For example, if a
certain population has discrete generations, the size of the
$k+1$th generation $u(k+1)$ is a function of the $k$th generation
$u(k)$. Namely, since difference equations give a natural description of
many discrete models in the real world, in various fields of science
and technology such as statistics, computer science, electrical
circuit analysis, biology, neural networks, optimal control,
economics, finance, and so on, it is of practical importance to
study the existence of solutions of difference equations and
discrete boundary value problems. Lately, there have been
many studies on discrete boundary value problems by different
approaches, for example, see
\cite{BoCa2,BoCa1,CaDa,CZ,CLCH,GKW,GKW1,HZXC,HACHM,JZ,LZ,LC,Maw}.
In \cite{JZ}, by using the three critical point theorems proposed
by Bonanno \cite{B3}, Jiang and Zhou have obtained sufficient
conditions for the existence of at least three solutions of
discrete $p$-Laplacian Dirichlet boundary value problems. In
\cite{BoCa1,BoCa2}, Bonanno and Candito have discussed the
existence of multiple solutions for a class of discrete nonlinear
boundary value problems under rather different assumptions via
variational methods and critical point theory.  In \cite{CaDa}, by using critical point theory, the authors have studied the existence of
at least three solutions for a perturbed nonlinear Dirichlet
boundary value problem for difference equations depending on two
positive parameters.

To the best of our knowledge, there are a few works in the
literature studying the existence of solutions for boundary value
problems involving $\phi_c$-Laplacian, and we have just found
\cite{CZ,ZL}, in which the authors have discussed the existence of
  multiple solutions for   problem \eqref{e1.1} in the case
$\mu=0$, and the existence of infinitely many solutions for
problem \eqref{e1.1} in the case $\mu=0$.  We refer to \cite{DZ},
in which a partial discrete Dirichlet boundary value problem
involving the mean curvature operator was studied, and under proper
assumptions on the nonlinear term, some feasible conditions on the
existence of multiple solutions by the method of critical point
theory were obtained. Also, open intervals of the parameter to
attain at least two positive solutions and an unbounded sequence
of positive solutions with the help of the maximum principle were
separately determined.

We refer to \cite{Hal}, where the author has studied the
characterization of entire solutions of some system of Fermat type
functional equations and   also posed an open problem. The
author has provided a nice discussion and presentation of the
mathematical background to underline the relevance of the topic.
The author
 has distinguished various situations and   supported the finding with
illustrative examples. We also refer to   paper \cite{Bohner},
where using critical point theory and variational methods, the
existence of at least three solutions for a class of double
eigenvalue discrete anisotropic Kirchhoff-type problems was
discussed. Further,  the effects of   Kirchhoff weight on the
principal operator were considered. We note the fact that
establishing the Kirchhoff counterpart of existing models is a
very actual topic of research. In \cite{Nas}, the authors have
established the existence and multiplicity of non-zero homoclinic
solutions to a nonlinear Laplacian difference equation without
using Ambrosetti-Rabinowitz type-conditions. Similarly to our
present paper, the main approach is based on the mountain pass theorem
and the Palais-Smale compactness condition involving suitable
functionals.

 Inspired by the above
results, in this the article, we investigate the existence of one,
 two and three solutions for   problem \eqref{e1.1}. In these cases,
we apply suitable conditions on the nonlinear terms and create
openings for two parameters $\lambda$ and $\mu$ in   problem
\eqref{e1.1}, without imposing the symmetry or oscillating
behavior at infinity on the nonlinear terms $f$ and $g$. We also
give examples to show the use of the proven theorems. At the end,
we discuss the existence of the solutions for the problem in the
case when two parameters are the same. Precisely, we discuss the
regularity properties of energy functionals associated to the main
problem.

\section{Preliminaries and basic notation}
In this section, we introduce the tools that are necessary for our
main results in the next section. Set
\begin{align}\label{x}
X=\{u:[0,N+1]_{\mathbb{Z}}\rightarrow\mathbb{R}| u(0)=u(N+1)=0\}.
\end{align}
$X$ is an $N$-dimensional Banach space with the following  norm:
\begin{align}\label{norm1}
\|u\|:=\Big(\sum_{k=1}^{N+1}|\Delta u(k-1)|^2\Big)^{1\over2}.
\end{align}
Let $\|u\|_{\infty}:=\max\{u(k):k\in [1,N]_{\mathbb{Z}}\}.$ We see
that $\|u\|_{\infty}$ is another norm in $X$. From Lemma 2.2 of
\cite{JZ}, we have
\begin{lemma} For any $u\in X$, the following relation holds:
\begin{align}\label{norm2}
\|u\|_{\infty}\leq {\sqrt{N+1}\over 2}\|u\|.
\end{align}
\end{lemma}
From (2.1) and (2.3) in \cite{BCD}, we have the following lemma.
\begin{lemma} For any $u\in X$, one has
\begin{align*}
{1\over\sqrt{N\lambda_N}}\|u\|\leq\|u\|_{\infty}\leq
{1\over\sqrt{\lambda_1}}\|u\|,
\end{align*}
where $\lambda_1=4 \sin^2{\pi\over 2(N+1)}$ and $\lambda_N=4
\sin^2{N\pi\over 2(N+1)}$.
\end{lemma}
We define the functionals $\Phi,\ \Psi:X\rightarrow\mathbb{R}$ in
the following way:
\begin{eqnarray}\label{p1}
\Phi(u)\!=\!\sum_{k=1}^{N+1}\!\Big(\!\sqrt{1\!+\!(\Delta u(k-1))^2}\!-\!1\Big),\
\Psi(u)=\sum_{k=1}^{N}\!\Big(F(k,u(k))\!+\!\frac{\mu}{\lambda}G(k,u(k))\Big).
\end{eqnarray}
 Corresponding to the functions $f$ and
$g$, we introduce the functions $F, G:
[1,N]_{\mathbb{Z}}\times\mathbb{R}\rightarrow\mathbb{R}$,
respectively, as follows:
\[
F(k,t)=\int^{t}_{0}f(k,\xi)d\xi,t\in \mathbb{R} \ \textrm{and} \
G(k,t)=\int^{t}_{0}g(k,\xi)d\xi, t\in \mathbb{R}.
\]
For $\gamma > 0$ and $\eta> 0$, set
\begin{align*}
G^\gamma:=\sum_{k=1}^{N}\max_{|t|<\gamma} G(k,\xi)\,\ \textrm{and}
\,\
 G_\eta:=\sum_{k=1}^{N}\inf_{\xi\in[0,\eta]} G(k,\xi).
\end{align*}
If $g$ is sign-changing, then $G^{\gamma}\leq 0$ and
$G_{\eta}\geq0$. We say that a function $u\in X$ is a solution of
  problem \eqref{e1.1} if
\begin{align*}
-\sum_{k=1}^{N}\Delta(\phi_{c}(\Delta
u(k-1)))v(k)-\sum_{k=1}^{N}\Big(\lambda f(k,u(k))v(k)+\mu
g(k,u(k))v(k)\Big)=0
\end{align*}
holds for all $v\in X$.
\begin{definition}\label{d21}
Assume that $X$ is a real reflexive Banach space. We say $I$
satisfies the Palais-Smale condition (denoted as PS-condition for
short) if any sequence $\{u_{k}\}\subset X$, for which $\{I
(u_{k})\}$ is bounded and $I' (u_{k}) \rightarrow 0$ as $ k
\rightarrow 0$, possesses a convergent subsequence.
\end{definition}
If $r_{1}, r_{2}\in [-\infty,+\infty]$  such that $r_{1} < r_{2}$
and $I = \Phi+\lambda\Psi$, where $\Phi,\Psi :
X\rightarrow\mathbb{R}$ are two continuously G\^{a}teaux
differentiable functions, so that  each sequence $\{u_{n}\}$
that has the following conditions:
\begin{itemize}
\item[$(i_{1})$] $\{I(u_{n})\}$ is bounded;
\end{itemize}
\begin{itemize}
\item[$(i_{2})$]
$\lim_{n\rightarrow\infty}\|I'(u_{n})\|_{X^{*}}=0$;
\end{itemize}
\begin{itemize}
\item[$(i_{3})$] $r_{1}<\Phi(u_{n})<r_{2}$,
\end{itemize}
possesses a convergent subsequence. At a time, the functional $J$
applies to the Palais-Smale condition cut off lower at $r_{1}$ and
upper at $r_{2}$ ( $^{[r_{1}]}(PS)^{[r_{2}]}$-condition). Likewise
we can make $(PS)^{[r_{2}]}$ in which $r_{1} =-\infty$ and
$r_{2}\in \mathbb{R}$   and $^{[r_{1}]}(PS)$ in which $r_{1}\in
\mathbb{R}$ and $r_{2} =\infty$. By definition \eqref{d21} of the
(PS)-condition,   it is established that in
$^{[r_{1}]}(PS)^{[r_{2}]}$-condition, $r_{1} = -\infty$ and $r_{2}
= \infty$.
 Indeed, let $\Phi$ and $\Psi$ be two continuously G\^{a}teaux differentiable
functionals defined on a real Banach space $X$; fix $r\in
\mathbb{R}$. The functional $I = \Phi-\lambda\Psi$ is said to
verify the Palais-Smale condition cut off upper at $r$ (in short
$(PS)^{[r]}$) if any sequence $\{u_{n}\}_{n\in \mathbb{N}}$ in $X$
such that
\begin{itemize}
\item[$(j_1)$] $\{I(u_{n})\}$ is bounded;
\end{itemize}
\begin{itemize}
\item[$(j_2)$] $\lim_{n\rightarrow\infty}\|I'(u_{n})\|_{X^{*}}=0$;
\end{itemize}
\begin{itemize}
\item[$(j_3)$] $\Phi(u_{n})<r$ for each $n\in \mathbb{N}$,
\end{itemize}
has a convergent subsequence.

The proofs of our theorems are based on the following four theorems.
\begin{theorem}[{\cite[Theorem 2.3]{B1}}]\label{t21}
Let $X$ be a real Banach space and let
$\Phi,\Psi : X\rightarrow \mathbb{R}$ be two continuously
G\^{a}teaux differentiable functions such that $\inf_{u\in X}
\Phi(u) = \Phi(0) = \Psi(0) = 0$. Assume that there exist $r > 0$
and $\bar{u}\in X$, with $0 < \Phi(\bar{u}) < r$, such that:
\begin{itemize}
\item[$(\textrm{k}_1)$]$ {\sup_{\Phi(u)\leq r}\Psi(u)\over
r}<{\Psi(\bar{u})\over\Phi(\bar{u})}$,
\end{itemize}
\begin{itemize}
\item[$(\textrm{k}_2)$]for each $\lambda\in
\Big({\Phi(\bar{u})\over\Psi(\bar{u})},{r\over \sup_{\Phi(u)\leq
r}\Psi(u)}\Big)$, the functional $I_{\lambda} := \Phi-\lambda\Psi$
satisfies the $(PS)^{[r]}$-condition.
\end{itemize}
Then, for each $\lambda\in\Lambda:=
\Big({\Phi(\bar{u})\over\Psi(\bar{u})},{r\over \sup_{\Phi(u)\leq
r}\Psi(u)}\Big)$, there exists $u_{0,\lambda}\in \Phi^{-1}(0, r)$
such that $I_{\lambda}(u_{0,\lambda})\equiv \vartheta_{X^*}$ and
$I_{\lambda}(u_{0,\lambda})\leq I_{\lambda}(u)$ for all
$u\in\Phi^{-1}(0, r)$.
\end{theorem}
\begin{theorem} [{\cite[Theorem 3.2]{B1}}] \label{t22}
Let $X$ be a real
Banach space, $\Phi,\Psi : X\rightarrow \mathbb{R}$   two
continuously G\^{a}teaux differentiable functionals such that
$\Phi$ is bounded from below and $\Phi(0) = \Psi(0) = 0$. Fix $r >
0$ and assume that for each
\begin{align*}
\lambda\in\Big(0,{r\over
\sup_{u\in\Phi^{-1}(-\infty,r)}\Psi(u)}\Big),
\end{align*}
the functional $I_{\lambda}=\Phi-\lambda\Psi$ satisfies
(PS)-condition and is unbounded from below. Then, for each
\begin{align*}
\lambda\in\Big(0,{r\over
\sup_{u\in\Phi^{-1}(-\infty,r)}\Psi(u)}\Big),
\end{align*}
the functional $I_{\lambda}$ admits two distinct critical points.
\end{theorem}
\begin{theorem} [{\cite[Theorem A]{AB}}] \label{t23}
Let $X$ be a reflexive real Banach space,
$\Phi : X \rightarrow\mathbb{R}$   a continuously G\^{a}teaux
differentiable and sequentially weakly lower semicontinuous
functional whose G\^{a}teaux derivative admits a continuous
inverse on $X^*$ and let $\Psi : X \rightarrow\mathbb{R}$ be a
continuously G\^{a}teaux differentiable functional whose
G\^{a}teaux derivative is compact. Assume that:
\begin{itemize}
\item[$(\textrm{m}_1)$]$
\lim_{\|u\|\rightarrow\infty}(\Phi(u)+\lambda\Psi(u))=\infty$, for
all $\lambda\in[0,\infty)$;
\end{itemize}
\begin{itemize}
\item[$(\textrm{m}_2)$]there is $r\in \mathbb{R}$ such that
$\inf_{X}\Phi<r$ and $\varphi_{1}(r)<\varphi_{2}(r)$, where
\begin{align*}
\varphi_{1}(r)=\inf_{u\in \Phi^{-1}]-\infty,
\bar{r}[}{\Psi(u)-\inf_{\overline{\Phi^{-1}(-\infty,
r)}^\omega}\Psi\over r-\Phi(u)},
\end{align*}
\begin{align*}
\varphi_{2}(\bar{r})=\inf_{u\in \Phi^{-1}(-\infty ,r)}\sup_{v\in
\Phi^{-1}[r,\infty)}{\Psi(u)-\Psi(v)\over \Phi(v)-\Phi(u)},
\end{align*}
\end{itemize}
and $\overline{\Phi^{-1}(-\infty, r)}^\omega$ is the closure of
$\Phi^{-1}(-\infty ,r)$ in the weak topology. Then, for each
$\lambda\in({1\over \varphi_{2}(r) },{1\over \varphi_{1}(r) })$
 the functional $\Phi+\lambda\Psi$ has at least three critical points in
$X$.
\end{theorem}

\begin{theorem} [{\cite[Theorem 1.1]{B}}] \label{t24}
Let $X$ be a reflexive real Banach space,
and let $\Phi,\ \Psi: X \rightarrow\mathbb{R}$  be two sequentially
weakly lower semicontinuous and G\^{a}teaux differentiable
functions. Assume that $\Phi$ is (strongly) continuous and
satisfies $\lim_{\|u\|\rightarrow\infty}$ $\Phi(u)=\infty.$ Also,
suppose there exist two constants $r_{1}$ and $r_{2}$ such that
\begin{itemize}
\item[$(\textrm{n}_1)$] $\inf_{X}\Phi<r_{1}<r_{2}$;
\end{itemize}
\begin{itemize}
\item[$(\textrm{n}_2)$]
$\varphi_{1}(r_{1})<\varphi^{*}_{2}(r_{1},r_{2})$;
\end{itemize}
\begin{itemize}
\item[$(\textrm{n}_3)$]
$\varphi_{1}(r_{2})<\varphi^{*}_{2}(r_{1},r_{2})$, where
$\varphi_{1}$ is defined as in Theorem \ref{t23} and
\[
\varphi^{*}_{2}(r_{1},r_{2})=\inf_{u\in \Phi^{-1}(-\infty,r)}\sup_{v\in\Phi^{-1}[r_{1},r_{2}]}{\Psi(v)-\Psi(u)\over
\Phi(v)-\Phi(u)}.
\]
\end{itemize}
Then, for each $\lambda\in({1\over
\varphi^{*}_{2}(r_{1},r_{2})},\min\{{1\over\varphi_{1}(r_{1})},{1\over\varphi_{1}(r_{2})}\})$,
the functional $\Phi+\lambda\Psi$ admits at least two critical
points which lie in $\Phi^{-1}(-\infty, r_{1}]$ and
$\Phi^{-1}[r_{1}, r_{2})$, respectively.
\end{theorem}
For situations of successful employment of the results such as
Theorems \ref{t21}-\ref{t24} in order to prove the existence of
solutions for various boundary value problems, we refer the reader
to \cite{BC,BO,CHA}.
\section{Main results}
In this section, we present the existence results.
 \begin{theorem}\label{t1}
 Assume that there exist two positive constants $\gamma>\frac{(N+1)\sqrt{N+1}}{2}$ and $\eta$ with the property
\begin{align*}%% \label{gamaa1}
 2\sqrt{1+\eta^2}-1<{2\over
\sqrt{N+1}}\gamma-N,
\end{align*} and assume that
\begin{itemize}
\item[$(A_{1})$] ${\sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)\over
{2\over \sqrt{N+1}}\gamma-N-1}<{\sum_{k=1}^{N}F(k,\eta)\over
2(\sqrt{1+\eta^2}-1)}$;
\end{itemize}
\begin{itemize}
\item[$(A_{2})$] $\limsup_{|\xi|\rightarrow \infty}{\sup_{k\in
[1,N]_{\mathbb{Z}}}F(k,\xi)\over |\xi|}< \infty$.
\end{itemize}
 Then, for every
\begin{align*}
\lambda\in\Lambda:=\Big({2(\sqrt{1+\eta^2}-1) \over
\sum_{k=1}^{N}F(k,\eta)},{{2\over \sqrt{N+1}}\gamma-N-1\over
\sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)}\Big)
\end{align*}
and for every continuous function
$g:[1,N]_{\mathbb{Z}}\times\mathbb{R}\rightarrow\mathbb{R}$
satisfying the following condition:
\begin{align}\label{G2}
\limsup_{|t|\rightarrow \infty}{\sup_{k\in
[1,N]_{\mathbb{Z}}}G(k,t)\over |t|}<\infty,
\end{align}
there exists $\delta_\lambda$ given by
\begin{eqnarray}\label{de}
\min\!
\Big\{\!{2(\sqrt{1\!+\!\eta^2}\!-\!1)\!-\!\lambda\sum_{k=1}^{N}F(k,\eta)\over
G_{\eta}},{{2\over\sqrt{N\!+\!1} }\gamma\!-\!N\!-\!1\!-\!\lambda
\sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)\over G^{\gamma}}\!\Big\}
\end{eqnarray}
 such that for each $\mu\in [0,\delta_\lambda)$,   problem
\eqref{e1.1} admits at least one solution $u_{\lambda}$ in $X$
such that $\max_{ t\in[0,1]} |u_{\lambda}(t)| <\gamma.$
\end{theorem}
\begin{proof}
We want to apply Theorem \ref{t21} with regard to the space $X$
with the norm defined in \eqref{norm1}, and the functionals $\Phi$
and $\Psi$ defined as in \eqref{p1}. According to the definition
of $\Phi$, we realize that $\Phi$ is G\^{a}teaux differentiable and
sequentially weakly lower semicontinuous, its G\^{a}teaux
derivative is the functional $\Phi'(u)\in X^*$, that is as follows:
\[
\Phi'(u)(v)=-\sum_{k=1}^{N}\Delta(\phi_{c}(\Delta u(k-1)))v(k)
\]
and
\[
\Psi'(u)(v)=\sum_{k=1}^{N}\Big(f(k,u(k))v(k)+\frac{\mu}{\lambda }
g(k,u(k))v(k)\Big),
\]
for every $v\in X$. An easy computation
ensures the regularity assumptions required on $\Phi$. Moreover,
the functional $\Psi$ is in $C^{1}(X,\mathbb{R})$ and $\Psi$ has a
compact derivative. By the definition of $\Phi$, one has
\begin{align}\label{ph1}
\Phi(u)&= \sum_{k=1}^{N+1}\sqrt{1+(\Delta
u(k-1))^2}-\sum_{k=1}^{N+1}1\nonumber\\& \geq
\Big(\sum_{k=1}^{N+1}(1+(\Delta
u(k-1))^2)\Big)^{\frac{1}{2}}-\sum_{k=1}^{N+1}1\nonumber\\&\geq
\|u\|-N-1.
\end{align}
 Put $r={2\over
\sqrt{N+1}}\gamma-N-1$ and
\begin{equation*}
\omega(k)=
\begin{cases}
\eta,
 \,\,k\in[1,N]_{\mathbb{Z}},\\
0,\,\ k=0=N+1.
\end{cases}
\end{equation*}
Clearly, $w\in X$.
 Hence, we have
\begin{align*}%% \label{p}
\Phi(\omega)=2(\sqrt{1+\eta^2}-1).
\end{align*}
Thus, by   assumption \eqref{gamaa1}, we get $0<\Phi(\omega)<r$.
Moreover, by \eqref{norm2} and \eqref{ph1}, we have
\begin{align*} |u(t)|\leq \|u\|_{\infty}\leq {\sqrt{N+1}\over
2}\|u\|&\leq {\sqrt{N+1}\over 2}(\Phi(u)+N+1)\\&\leq
{\sqrt{N+1}\over 2}(r+N+1)=\gamma,\,\ \forall\,\
k\in[1,N]_{\mathbb{Z}}.
\end{align*}
 Consequently,
\begin{align*}
\Phi^{-1}(-\infty,r]=\{u\in X; \Phi(u)\leq r\}\subseteq \{u\in X;
|u(t)|\leq\gamma\}.
\end{align*}
Therefore, one has
\begin{align*}
\sup_{u\in \Phi^{-1}(-\infty,r)}\sum_{k=1}^{N}F(k,u(k))\leq
\sup_{|t|\leq\gamma}\sum_{k=1}^{N}F(k,t),
\end{align*}
and this in conjunction with the second inequality in \eqref{G2}
ensures
\begin{align*}
 \sup_{u\in\Phi^{-1}(-\infty,r)}\sum_{k=1}^{N}\Big(F(k,u(k))+{\mu\over
 \lambda}G(k,u(k))\Big)
 &\leq\sup_{u\in\Phi^{-1}(-\infty,r)}\sum_{k=1}^{N}F(k,u(k))+{\mu\over
 \lambda}G^{\gamma}
\\&\leq \sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)+{\mu\over
 \lambda}G^{\gamma},
\end{align*}
for every $u\in X$ such that $\Phi(u) < r$. Thus,
\begin{align*}
\sup_{ \Phi(u)\leq
r}\Psi(u)\leq\sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)+{\mu\over
 \lambda}G^{\gamma}.
\end{align*}
On the other hand, we have
\begin{align*}
\Psi(\omega)=\sum_{k=1}^{N}\big(F(k,\omega(k))+{\mu\over
\lambda}G(k,\omega(k))\Big)\geq
\sum_{k=1}^{N}F(k,\omega(k))+{\mu\over \lambda}G_{\eta}.
\end{align*}
Therefore,
\begin{align}\label{L2}
{\sup_{ u\in\Phi^{-1} (-\infty,r]}\Psi(u)\over
r}&={\sup_{u\in\Phi^{-1}(-\infty,r]}\sum_{k=1}^{N}\Big(F(k,u(k))+{\mu\over
 \lambda}G(k,u(k))\Big)\over r}\nonumber\\&\leq {\sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)+{\mu\over
\lambda}G^{\gamma}\over {2\over \sqrt{N+1}}\gamma-N-1},
\end{align}
and
\begin{align*} %%\label{L3}
&{\Psi(\omega)\over
\Phi(\omega)}\geq{\sum_{k=1}^{N}\Big(F(k,\eta)+{\mu\over
\lambda}G(k,\eta)\Big)\over2(\sqrt{1+\eta^2}-1) }\geq
{\sum_{k=1}^{N}F(k,\eta)+{\mu\over \lambda}G_{\eta} \over
2(\sqrt{1+\eta^2}-1)}.
\end{align*}
Since
\begin{align*}
\mu<{{2\over\sqrt{N+1} }\gamma-N-1-\lambda
\sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)\over G^{\gamma}},
\end{align*}
this means
\begin{align*}
{\sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)+{\mu\over
\lambda}G^{\gamma}\over {2\over \sqrt{N+1}}\gamma-N-1}<{1\over
\lambda}.
\end{align*}
Furthermore,
\begin{align*}
\mu< {2(\sqrt{1+\eta^2}-1)-\lambda\sum_{k=1}^{N}F(k,\eta)\over
G_{\eta}},
\end{align*}
this means
\begin{align*}
{\sum_{k=1}^{N}F(k,\eta)+{\mu\over \lambda}G_{\eta} \over
2(\sqrt{1+\eta^2}-1)}>{1\over \lambda}.
\end{align*}
Then,
\begin{align}\label{L4}
{\sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)+{\mu\over
\lambda}G^{\gamma}\over {2\over \sqrt{N+1}}\gamma-N-1}<{1\over
\lambda}<{\sum_{k=1}^{N}F(k,\eta)+{\mu\over \lambda}G_{\eta} \over
2(\sqrt{1+\eta^2}-1)}.
\end{align}
Hence, from \eqref{L2} to \eqref{L4}, the condition $(k_{1})$ of
Theorem \ref{t21} is fulfilled. Finally, for $\lambda > 0$, we
will show that the functional $I_{\lambda}=\Phi-\lambda\Psi$ is
coercive. Since $\mu<\delta_{k}$ and by \eqref{G2}, we can fix
$\alpha>0$ such that $\alpha\mu<{2\over N\sqrt{N+1}}$, and there
exists $\rho\in \mathbb{R}$ such that
$
G(k,t)\leq \alpha |t|+\rho,
$
for every $(k,t)\in [1,N]_{\mathbb{Z}}\times\mathbb{R}$.
 Now, we fix $\varepsilon<{2\over \lambda N\sqrt{N+1}}-{\alpha\mu \over \lambda}.$
 From the assumption $(A_{2})$ there exists
$h\in \mathbb{R}$ such that $F(k,t)\leq \varepsilon |t|+h $ for
every $(k,t)\in [1,N]_{\mathbb{Z}}\times\mathbb{R}$. It follows
that for each $u\in X$,
\begin{align*}
\Phi(u)-\lambda\Psi(u)&=\sum_{k=1}^{N+1}\Big(\sqrt{1+(\Delta
u(k-1))^2}-1\Big)- \lambda\sum_{k=1}^{N}\Big(F(k,u(k))+{\mu\over
\lambda}G(k,u(k))\Big)\\&\geq \Big(\sum_{k=1}^{N+1}[1+(\Delta
u(k-1))^2]\Big)^{1\over2}-\sum_{k=1}^{N+1}1-\lambda \varepsilon
\sum_{k=1}^{N}|u(k)|-\lambda h\\&\quad-\alpha\mu
 \sum_{k=1}^{N}|u(k)|-\mu\rho\\&\geq
\|u\|-\lambda{\varepsilon N\sqrt{N+1}\over2}\|u\|-\mu{\alpha
N\sqrt{N+1}\over2}\|u\|-N(1+\lambda h+\mu\rho)-1\\&\geq
\Big(1-\lambda {\varepsilon N\sqrt{N+1}\over2}-\mu {\alpha
N\sqrt{N+1}\over2}\Big)\|u\|-N(1+\lambda h+\mu\rho)-1,
\end{align*}
 so
\begin{align*}
\lim_{\|u\|_{X}\rightarrow\infty}(\Phi(u)-\lambda\Psi(u))=\infty,
\end{align*}
which means the functional $I_{\lambda}=\Phi(u)-\lambda\Psi(u)$ is
coercive. Therefore, by \cite[ Proposition 2.1]{B2}, the functional
$I_{\lambda}=\Phi(u)-\lambda\Psi(u)$ verifies the
$(PS)^{[r]}$-condition for each $r > 0$, so the condition
$(k_{2})$ of Theorem \ref{t21} is fulfilled. From
\eqref{L2}-\eqref{L4}, one also has
\begin{align*}
\lambda\in\Big({\Phi(\omega)\over \Psi(\omega)},{r\over
\sup_{\Phi(u)\leq r}\Psi(u)}\Big).
\end{align*}
Theorem \ref{t21} with $\overline{u} = \omega$ guarantees the
existence of a local minimum point $u_\lambda$ for the functional
$I_{\lambda}$ such that $0 < \Phi(u_\lambda) < r$ and so
$u_\lambda$ is a nontrivial weak solution of   problem
\eqref{e1.1} such that $\max_{k\in[1,N]_{\mathbb{Z}}}
|u_\lambda(k)| < \gamma $.
\end{proof}
Now we give an example to show the use of Theorem \ref{t1}.
\begin{example}\label{ex1}
Consider the problem
\begin{equation}\label{ex11}
\begin{cases}
-\Delta\Big({\Delta u(k-1)\over \sqrt{1+(\Delta
u(k-1))^2}}\Big)=\lambda f(k,u(k))+\mu g(k,u(k)),
 \,\,k\in[1,3]_{\mathbb{Z}},\\
u(0)=u(N+1)=0,
\end{cases}
\end{equation}
where $g(k,t)=2k$ for all $(k,t)\in
 [1,3]_{\mathbb{Z}}\times\mathbb{R}$, we have
 $G(k,t)=2kt+1$ for all $(k,t)\in
 [1,3]_{\mathbb{Z}}\times\mathbb{R}$.
 We see that $\limsup_{|\xi|\rightarrow\infty}{\sup_{k\in
[1,3]_{\mathbb{Z}}} G(k,\xi)\over |\xi|}<\infty$, namely,
\eqref{G2} holds. Letting
\begin{equation*}
f(k,t)=
\begin{cases}
3k,& t\leq 1,\\
{3k\over t}, & t>1,
\end{cases}
\end{equation*}
for every $k\in [1,3]_{\mathbb{Z}}$, we have
\begin{equation*}
F(k,t)=
\begin{cases}
3kt,& t\leq1,
 \\
3k (\ln(t)+1),
 & t>1,
\end{cases}
\end{equation*}
for every $k\in [1,3]_{\mathbb{Z}}$. Hence,
$\lim\sup_{|\xi|\rightarrow\infty} {\sup_{k\in
[1,3]_{\mathbb{Z}}}F(k,\xi)\over |\xi|}<\infty$, the condition
$(A_{2})$ holds. Taking $\gamma=5$ and $\eta= 10^{-6}$, then
$2\sqrt{1+10^{-12}}-1=2\sqrt{1+\eta^2}-1<{2\over
\sqrt{N+1}}\gamma-N=2$, and also
\[
{\sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)\over
{2\over \sqrt{N+1}}\gamma-N-1}=18\times \ln
6<{18\times10^{-6}\over 2(\sqrt{1+10^{-12}}-1)}={\sum_{k=1}^{N}
F(k,\eta)\over 2(\sqrt{1+\eta^2}-1)}.
\]
Therefore, the condition
$(A_{1})$ holds. So, all conditions of Theorem \eqref{t1} are
satisfied. Consequently, it follows that for each
$\lambda\in({2(\sqrt{1+10^{-12}}-1)\over18\times10^{-6}},{1\over
18\ln6})$ and for every
\[
0\leq\mu<\min\Big\{{2\times (\sqrt{1+10^{-12}}-1)-18\times10^{-6}\lambda\over 12\times10^{-6}+3},{1-18 \ln6\times\lambda\over
 63}\Big\}
\]
  problem \eqref{ex11} admits at least one weak
solution in $X$.
\end{example}
 Now, we want to discuss  the
existence of at least two solutions for   problem~\eqref{e1.1}.
\begin{theorem}\label{t2}
Assume that there exist two positive constants
$\gamma>\frac{(N+1)\sqrt{N+1}}{2}$ and $\eta$ with the property
\begin{align}\label{gamaa1}
 2\sqrt{1+\eta^2}-1<{1\over
\sqrt{N+1}}\gamma-N,
\end{align} and  assume that
\begin{itemize}
\item[$(A_{3})$]  there exist $\nu > 1$ and $T > 0$ such that
\[
0<\nu F(k,\xi)<\xi f(k,\xi)
\]
for all $|\xi|>T$ and $k\in [1,N]_{\mathbb{Z}}$.
\end{itemize}
Then, for each
\begin{align*}
\lambda \in \Big(0,{{2\over \sqrt{N+1}}\gamma-N-1\over
\sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)}\Big),
\end{align*}
and for every continuous function
$g:[1,N]_{\mathbb{Z}}\times\mathbb{R}\rightarrow\mathbb{R}$
satisfying the following condition:
\begin{itemize}
\item[$(A_{4})$]  there exist $\nu\geq\nu' > 1$ and $T' > 0$ such
that
\[
0<\nu' G(k,\xi)<\xi g(k,\xi)
\]
for all $|\xi|>T'$ and $k\in [1,N]_{\mathbb{Z}}$,
\end{itemize}
 and  for each $\mu$ in \eqref{de},   problem \eqref{e1.1} admits at least two
 solutions $u_{1}$ and $u_{2}$ in $X$ such that $\max_{k\in
[1,N]_{\mathbb{Z}}}|u_{1}(k)|<\gamma$.
\end{theorem}
\begin{proof}
We want to apply Theorem \ref{t22}.  Recalling the functionals
$\Phi$ and $\Psi$ defined in Theorem \ref{t22}, we first examine
the (PS)-condition for the functional $I_{\lambda}$. To prove
this, let $\{u_{n}\}$ be a sequence in $X$ such that
$\{I_{\lambda}(u_{n})\}$ is bounded and
$I'_{\lambda}(u_{n})\rightarrow 0$ as $n\rightarrow +\infty.$
Then, there exists a positive constant $c_{0}$ such that
$|I_{\lambda}(u_{n})|\leq c_{0}$ and $|I'_{\lambda}(u_{n})|\leq
c_{0}$ for all $n\in \mathbb{N}$. Therefore, by   assumptions
$(A_{3})$, $(A_{4})$, and the definition of $I'_{\lambda}$, we have
\begin{align*}%%\label{ps5}
c_{0}+c_{1}\| u_{n}\|&\geq \nu
I_{\lambda}(u_{n})-I'_{\lambda}(u_{n})(u_{n})\nonumber\\&\geq
(\nu-1)\|u_{n}\|-N-1+\lambda\sum_{k=1}^{N}\Big(f(k,u_{n}(k))u_{n}(k)-\nu
F(k,u_{n}(k))\Big)\nonumber\\&\quad+\mu\sum_{k=1}^{N} \Big(
g(k,u_{n}(k))u_{n}(k)-\nu G(k,u_{n}(k))\Big) \nonumber\\&\geq
(\nu-1)\|u_{n}\|-N-1,
\end{align*}
 for some $c_1>0$. Since $\nu\geq \nu' > 1$, this implies that $\{u_{n}\}$ is
 bounded.
 Next, we prove that there exists $u\in X$ such that $u_{n}\rightarrow
 u$ in $X$, as $n\rightarrow \infty$. Since $X$ is a Banach space, there
 exist a subsequence, still denoted by $\{u_{n}\}$, and a function
 $u$ in $X$ such that
$u_{n}\rightharpoonup u,\,\ in\,\ X. $ We have
\begin{align}\label{ps1}
&\sum_{k=1}^{N}\Big(\sqrt{1+(\Delta
u_{n}(k-1))^2}-1-(\sqrt{1+(\Delta u(k-1))^2}-1)\Big)dt\rightarrow
0,\,\ n\rightarrow \infty.
\end{align}
From the continuity of $f$ and $g$ we get
\begin{align}\label{ps2}
\lambda\sum_{k=1}^{N}(f(k,u_{n}(k))-f(k,u(k)))(u_{n}(k)-u(k))\rightarrow
0,\,\ n\rightarrow \infty,
\end{align}
and
\begin{align}\label{ps3}
\mu
\sum_{k=1}^{N}(g(k,u_{n}(k))-g(k,u(k)))(u_{n}(k)-u(k))\rightarrow
0,\,\ n\rightarrow \infty.
\end{align}
On the other hand, we have
\begin{align*}
<I'_{\lambda}(u_{n})-I'_{\lambda}(u),u_{n}-
u>&=-\sum_{k=1}^{N+1}\Delta (\phi_{c}(\Delta
u_{n}(k-1)))(u_{n}(k)-u(k))
\\&\quad-\lambda\sum_{k=1}^{N}\Big(
f(k,u_{n}(k))(u_{n}(k)-u(k))\Big)
\\&\quad-\mu\sum_{k=1}^{N}\Big(g(k,u_{n}(k))(u_{n}(k)-u(k))\Big)
\end{align*}
\begin{align}\label{ps6}
&\quad+\sum_{k=1}^{N+1}\Delta
(\phi_{c}(\Delta u(k-1)))(u_{n}(k)-u(k))
\nonumber\\&\quad+\lambda\sum_{k=1}^{N}\Big(\lambda
f(k,u(k))(u_{n}(k)-u(k))\Big)\nonumber\\&\quad+\mu\sum_{k=1}^{N}\Big(
g(k,u(k))(u_{n}(k)-u(k))\Big)\nonumber\\&\geq
\|u_{n}-u\|\nonumber\\&\quad-\sum_{k=1}^{N}\Big(\lambda(f(k,u_{n}(k))-f(k,u(k)))(u_{n}(k)-u(k))\nonumber\\&\quad+\mu
(g(k,u_{n}(k))-g(k,u(k)))(u_{n}(k)-u(k))\Big).
\end{align}
It is easy to see that
\begin{align*}
<I'_{\lambda}(u_{n})-I'_{\lambda}(u),u_{n}- u>\rightarrow 0.
\end{align*}
From \eqref{ps1}-\eqref{ps6}, we have
\begin{align*}
\|u_{n}- u\|\rightarrow 0.
\end{align*}
Consequently, the sequence ${u_{n}}$ converges strongly to $u$ in
$X$. Therefore, $I_{\lambda}$ satisfies the $(PS)$-condition. From
the conditions $A_{3}$ and $A_{4}$, there exist constants $a_{1},
a_{2},b_{1},b_2 > 0$ such that
\begin{align}\label{F1}
F(k,t)\geq a_{1}|t|^{\nu}-a_{2}
\end{align}
for all $k\in [1,N]_{\mathbb{Z}}$ and $t\in \mathbb{R}$.
\begin{align}\label{G3}
G(k,t)\geq b_{1}|t|^{\nu'}-b_{2}
\end{align}
for all $k\in [1,N]_{\mathbb{Z}}$ and $t\in \mathbb{R}$. Now,
choosing any $u \in X \backslash \{0\}$, using \eqref{F1} and
\eqref{G3} for each $\tau> 0$, one has
\begin{align*}
I_{\lambda}(\tau u)&\leq \|\tau
u\|-N-1-\lambda\sum_{k=1}^{N}F(k,\tau u(k))-\mu\sum_{k=1}^{N}
G(k,\tau u(k))\\&\leq \tau\|u\|-\lambda \tau^{\nu}\sum
_{k=1}^{N}a_{1}|u(k)|^{\nu}+\lambda a_{2}-\mu \tau^{\nu'}\sum
_{k=1}^{N}b_{1}|u(k)|^{\nu'}+\mu b_{2}-N-1.
\end{align*}
Since $\nu > 2$, this indicates that the functional $I_{\lambda}$
is unbounded from below. Thus, all hypotheses of Theorem \ref{t22}
are verified. Therefore, for each
$
\lambda \in \Big(0,{{2\over \sqrt{N+1}}\gamma-N-1\over
\sum_{k=1}^{N}\sup_{|t|\leq\gamma}F(k,t)}\Big)
$
the functional $I_{\lambda}$ admits two critical points that are
weak solutions of   problem~\eqref{e1.1}.
\end{proof}
\begin{remark} In Theorem \ref{t21} we observe that if $f(k, 0)\neq 0$,
then Theorem \ref{t3} ensures the existence of two nontrivial
solutions for   problem \eqref{e1.1}. If the condition $f(k, 0)
\neq 0$ for all $k\in [1,N]_{\mathbb{Z}}$ does not hold, the
second solution $u_{2}$ of   problem \eqref{e1.1} may be
trivial, but the problem has at least a nontrivial solution.
\end{remark}
Now, we want to present an example to illustrate the application of
Theorem \ref{t2}.
\begin{example}\label{ex2}
 Let $N=20$. Put
 $g(k,t)=3t^2+t+k$ for all
$(k,t)\in [1,20]_{\mathbb{Z}}\times\mathbb{R}$, so
$G(k,t)=t^3+t^2+kt$ for all $(k,t)\in
[1,20]_{\mathbb{Z}}\times\mathbb{R}$, hence
$\lim_{\xi\rightarrow+\infty}{\xi g(k,\xi)\over G(k,\xi)}=3$, by
choosing $\nu'=3$ and $T'=1$, condition $(A_{4})$ holds.
Letting
\begin{equation*}
f(k,t)=
\begin{cases}
t^{3}+\cos \pi t+4,
 & t\leq1,\\
5t^{4}-1,
 & t>1,
\end{cases}
\end{equation*}
for all $k\in [1,20]_{\mathbb{Z}}$, we have
\begin{equation*}
F(k,t)=
\begin{cases}
{1\over4}t^{4}+{1\over \pi}\sin \pi t+4t,
 & t\leq1,\\
t^{5}+{13\over4},
 & t>1.
\end{cases}
\end{equation*}
Hence, $\lim_{\xi\rightarrow+\infty}{\xi f(k,\xi)\over
F(k,\xi)}=5<\infty$ and $\lim_{\xi\rightarrow-\infty}{\xi
f(k,\xi)\over F(k,\xi)}=4<\infty$, thus by choosing $\nu=5>2$ and
$T=1$, condition $(A_{3})$ is satisfied, also $\nu>\nu'$.
Taking $\gamma=100$ and $\eta=1$ such that
$2\sqrt{1+\eta^2}-1=2\sqrt{2}-1<{2\over \sqrt{21}}\times
100-21={2\over \sqrt{N+1}}\gamma-N-1$, we clearly see that all
assumptions of Theorem \ref{t2} are fulfilled. Therefore, it
follows that for each $\lambda\in\Big(0,{{2\over
\sqrt{21}}\times100-21\over21(100^5+{13\over4} )}\Big)$ the
following problem:
\begin{equation*}
\begin{cases}
-\Delta\Big({\Delta u(k-1)\over \sqrt{1+(\Delta
u(k-1))^2}}\Big)=\lambda f(k,u(k))+\mu(3t^2+t+k),
 \,\,k\in[1,20]_{\mathbb{Z}}\\
u(0)=u(21)=0,
\end{cases}
\end{equation*}
 has at least two nontrivial solutions.
\end{example}
Now, we consider $\lambda = \mu$, and in this case we want to
discuss the existence of at least two and three solutions for
problem \eqref{e1.1}.
 \begin{theorem}\label{t3}
 Assume that there exist two positive constants $\bar{\gamma}>\frac{(N+1)\sqrt{N+1}}{2}$ and $\bar{\eta}$ with the
 property
\begin{align}\label{t3-1}
\frac{\sqrt{N+1}}{2} (2\sqrt{1+\bar{\eta}^2}+N-1)>\bar{\gamma},
\end{align}
 and let the assumption $(A_{2})$ in Theorem
\ref{t1} hold. Moreover, assume that
\begin{itemize}
\item[$(A_{5})$] $\sum_{k=1}^{N} F(k,\eta)\geq0$;
\end{itemize}
\begin{itemize}
\item[$(A_{6})$]
\begin{align*}
&{\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}F(k,t)\over
 {2\over \sqrt{N+1}}\bar{\gamma}-N-1}<{\sum_{k=1}^{N}F(k,\eta)-\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}
F(k,t)\over {2( \sqrt{1+\bar{\eta}^2}-1)}}.
 \end{align*}
\end{itemize}
Then, for every
\begin{align*}
\lambda\in&\Big({2( {\sqrt{1+\bar{\eta}^2}-1)}\over
 \sum_{k=1}^{N}F(k,\eta)-\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}
F(k,t)},{{2\over \sqrt{N+1}}\bar{\gamma}-N-1 \over
\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}F(k,t)}\Big)
\end{align*}
and for every sign-changing function
$g:[1,N]_{\mathbb{Z}}\times\mathbb{R}\rightarrow\mathbb{R}$
satisfying in \eqref{G2}   problem \eqref{e1.1}, in the case
$\lambda = \mu$, admits at least three solutions in $X$.
\end{theorem}
\begin{proof}
Put $I_{\lambda}=\Phi(u)+\lambda \Psi(u)$, where
\begin{eqnarray}\label{t3-4}
\Phi(u)\!=\!\!\!\sum_{k=1}^{N+1}\!\!\Big(\!\sqrt{1+(\Delta u(k\!-\!1))^2}\!-\!1\!\Big)\
\textrm{and}\ \Psi(u)\!=\!-\!\!\sum_{k=1}^{N}\!\!\Big(\!F(k,u(k))\!+\!G(k,u(k))\!\Big)
\end{eqnarray}
for all $u\in X$. Standard arguments show that $\Phi$ and $\Psi$
are G\^{a}teaux differentiable functionals whose G\^{a}teaux
derivatives at the point $u\in X$ are given by
\begin{align*}
\Phi'(u)(v)=-\sum_{k=1}^{N}\Delta(\phi_{c}(\Delta u(k-1)))v(k)
\end{align*}
and
\begin{align*}
\Psi'(u)v=-\sum_{k=1}^{N}\Big(f(k,u(k))v(k)+g(k,u(k))v(k)\Big)
\end{align*}
for all $u, v\in X$, respectively.  We know that a critical point
for function $\Phi(u)+\lambda \Psi(u)$ represents a solution of
  problem \eqref{e1.1} in the case $\lambda=\mu$. Our objective
is to implement   Theorem \ref{t23} for $\Phi$ and $\Psi$. By
sequentially weakly lower semicontinuity of the norm, the
functional $\Phi$ is sequentially weakly lower semicontinuous.
Moreover, $\Phi$ is continuously G\^{a}teaux differentiable and
its G\^{a}teaux derivative admits a continuous inverse on $X^*$.
The functional $\Psi : X\rightarrow \mathbb{R}$ is well-defined
and   continuously G\^{a}teaux differentiable and its G\^{a}teaux
derivative is compact. Thus it is enough to show that $\Phi$ and
$\Psi$ satisfy $(m_{1})$ and $(m_{2})$ in Theorem \ref{t23}. By
\eqref{G2}, we can fix $\alpha>0$ such that $\alpha\lambda<{2\over
N\sqrt{N+1}}$, and there exists $\rho\in \mathbb{R}$ such that
$G(k,t)\leq \alpha |t|+\rho. $
 Now, we fix $\varepsilon<{2\over \lambda N\sqrt{N+1}}-\alpha.$
 From the assumption $(A_{2})$ there is
$h\in \mathbb{R}$ such that $F(k,t)\leq \varepsilon |t|+h $ for
every $(k,t)\in [1,N]_{\mathbb{Z}}\times\mathbb{R}$. It follows
that for each $u\in X$,
\begin{align*}
\Phi(u)+\lambda\Psi(u)=&\sum_{k=1}^{N+1}\Big(\sqrt{1+(\Delta
u(k-1))^2}-1\Big)-
\lambda\sum_{k=1}^{N}\Big(F(k,u(k))+G(k,u(k))\Big)\\
&\geq
\Big(\sum_{k=1}^{N+1}[1+(\Delta
u(k-1))^2]\Big)^{1\over2}-\sum_{k=1}^{N+1}1-\lambda \varepsilon
\sum_{1}^{N}|u(k)|-\lambda h\\&\quad-\alpha\lambda
 \sum_{1}^{N}|u(k)|-\lambda\rho
\end{align*}
\begin{align*}
&\geq
\|u\|-N-1-\lambda{\varepsilon N\sqrt{N+1}\over2}\|u\|-\lambda N
h-\lambda{\alpha N\sqrt{N+1}\over2}\|u\|-\lambda N \rho\\&\geq
\Big(1-\lambda {\varepsilon N\sqrt{N+1}\over2}-\lambda{\alpha
N\sqrt{N+1}\over2}\Big )\|u\|-\lambda N h-\lambda N \rho-N-1,
\end{align*}
thus
$\lim_{\|u\|_{X}\rightarrow\infty}(\Phi(u)+\lambda\Psi(u))=\infty,$
which means the functional $I_{\lambda}=\Phi(u)+\lambda\Psi(u)$ is
coercive. Now it remains to show that $(m_2)$ of Theorem \ref{t23}
is satisfied. Put $\bar{r}={2\over \sqrt{N+1}}\bar{\gamma}-N-1$
and
\begin{equation*}
\omega(k)=
\begin{cases}
\bar{\eta},
 \,\,k\in[1,N]_{\mathbb{Z}},\\
0,\,\, \textrm{otherwise}.
\end{cases}
\end{equation*}
Clearly, $w\in X$ and
\begin{align}\label{p}
\Phi(\omega)=2(\sqrt{1+\bar{\eta}^2}-1).
\end{align}
Thus, by \eqref{t3-1} we see that $ \Phi(\omega)> \bar{r}$.
Moreover, since $g$ is sign-changing and using the condition
$(A_5)$, we have
\begin{align*}
\Psi(\omega)=-\sum_{k=1}^{N}\Big(F(k,w(k))+G(k,w(k))\Big)\leq
-\sum_{k=1}^{N}F(k,\eta).
\end{align*}
Taking \eqref{norm2} into account, for every $u\in X$ such that
$\Phi(u) < \bar{r}$, we get
\begin{align*} %%\label{q}
\sup_{k\in[1,N]_{\mathbb{Z}}}|u(k)|\leq\bar{\gamma}.
\end{align*}
Thus
\begin{align}\label{t3-3}
 \sup_{u\in\Phi^{-1}(-\infty,\bar{r})}\sum_{k=1}^{N}\Big(F(k,u(k))+G(k,u(k))\Big)
\nonumber&\leq\sup_{u\in\Phi^{-1}(-\infty,\bar{r})}\sum_{k=1}^{N}F(k,u(k))
\nonumber\\&\leq \sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}F(k,t),
\end{align}
for every $u\in X$ with $\Phi(u) < \bar{r}$. So
\begin{align*}
\sup_{ \Phi(u)\leq
\bar{r}}\Psi(u)\leq\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}F(k,t).
\end{align*}
By simple calculations and from the definition of
$\varphi_{1}(\bar{r})$, since $\Phi(0) = \Psi(0) = 0$ and
$\overline{\Phi^{-1}(-\infty, \bar{r})}^\omega =
\Phi^{-1}(-\infty, a\bar{r})$, we have
\begin{align*}
\varphi_{1}(\bar{r})&=\inf_{u\in \Phi^{-1}]-\infty,
\bar{r}[}{\Psi(u)-\inf_{\overline{\Phi^{-1}(-\infty,
\bar{r})}^\omega}\Psi(u)\over \bar{r}-\Phi(u)}\\&\leq
{-\inf_{\overline{\Phi^{-1}(-\infty,
\bar{r})}^\omega}\Psi(u)\over
\bar{r}}\\&\leq{\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}F(k,t)\over
 {2\over \sqrt{N+1}}\bar{\gamma}-N-1}.
\end{align*}
On the other hand, by \eqref{t3-3}, one has
\begin{align*}
\varphi_{2}(\bar{r})&=\inf_{u\in \Phi^{-1}(-\infty,
\bar{r})}\sup_{v\in \Phi^{-1}[
\bar{r},\infty)}{\Psi(u)-\Psi(v)\over \Phi(v)-\Phi(u)}\\&\geq
\inf_{u\in \Phi^{-1}(-\infty, \bar{r})}{\Psi(u)-\Psi(\omega)\over
\Phi(\omega)-\Phi(u)}\\&\geq {\inf_{u\in \Phi^{-1}(-\infty,
\bar{r})}\Psi(u)-\Psi(\omega)\over \Phi(\omega)-\Phi(u)}\\&\geq
{-\sum_{k=1}^{N}\sup_{|k|\leq\bar{\gamma}}F(k,t)+\sum_{k=1}^{N}F(k,\eta)\over
\Phi(\omega)-\Phi(u)}\\&\geq{\sum_{k=1}^{N}F(k,\eta)-\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}
F(k,t)\over {2( \sqrt{1+\bar{\eta}^2}-1)}}.
\end{align*}
 From $(A_{6})$ one has
\begin{align*}
\varphi_{1}(\bar{r})<\varphi_{2}(\bar{r}).
\end{align*}
Therefore, from Theorem \ref{t23}, taking into account that
\begin{align*}
{1\over\varphi_{2}(\bar{r})}\leq {
   2({\sqrt{1+\bar{\eta}^2}-1)}\over
\sum_{k=1}^{N}F(k,\eta)-\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}
F(k,t)}
\end{align*}
and
\begin{align*}
{1\over\varphi_{1}(\bar{r})}\geq{{2\over
\sqrt{N+1}}\bar{\gamma}-N-1\over
\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}F(k,t)},
\end{align*}
we obtain the desired result.
\end{proof}
\begin{remark}\label{r3}
In Theorem \ref{t3}, if we replace the condition
\begin{itemize}
\item[$(A_{7})$]${\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}F(k,t)\over
{2\over \sqrt{N+1}}\bar{\gamma}-N-1}<{\sum_{k=1}^{N}F(k,\eta)\over
2({\sqrt{1+\bar{\eta}^2}-1)}}$
\end{itemize}
with the condition $(A_{6})$,   the assumptions  \eqref{t3-1},
$(A_{5})$ and $(A_{7})$ hold. Then, for each
\begin{align*}
\lambda\in\Big({{2(\sqrt{1+\bar{\eta}^2}-1)}\over
\sum_{k=1}^{N}F(k,\eta)},{ {2\over
\sqrt{N+1}}\bar{\gamma}-N-1\over
\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}}F(k,t)}\Big),
\end{align*}
and for every sign-changing function
$g:[1,N]_{\mathbb{Z}}\times\mathbb{R}\rightarrow\mathbb{R}$
  satisfying   condition \eqref{G2},  in the case $\lambda =\mu$ problem
\eqref{e1.1} admits at least three
solutions.
\end{remark}
We now want to show the existence and multiplicity of  solutions
for   problem \eqref{e1.1} by using Theorem \ref{t24} in the case
$\lambda=\mu$.
 \begin{theorem}\label{t4}
 Assume that there exist three positive constants $\bar{\gamma}_{1}>\frac{(N+1)\sqrt{N+1}}{2}$, $\bar{\gamma}_{2}>\frac{(N+1)\sqrt{N+1}}{2}$
  and $\bar{\eta}$ with the property
\begin{align}\label{t4-1}
 {2\over\sqrt{N+1}} \bar{\gamma}_{1}-N<2 \sqrt{1+\bar{\eta}^2}-1<{2\over\sqrt{N+1}}
 \bar{\gamma}_{2}-N
\end{align}
such that the assumption $(A_{5})$ in Theorem \ref{t23} holds, and
\begin{itemize}
\item[$(A_{8})$]
\begin{align*}
&\max\Big\{{\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}_{1}}F(k,t)\over
 {2\over\sqrt{N+1}} \bar{\gamma}_{1}-N-1},{\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}_{2}}F(k,t)\over
 {2\over\sqrt{N+1}} \bar{\gamma}_{2}-N-1}\Big\}<{\sum_{k=1}^{N}F(k,\eta) \over
2({\sqrt{1+\bar{\eta}^2}-1)}}.
 \end{align*}
\end{itemize}
Then, for each
\begin{align*}
\lambda\in\Lambda=\Big({2({\sqrt{1+\bar{\eta}^2}-1)}\over
\sum_{k=1}^{N}F(k,\eta) },\min\Big\{{{2\over
\sqrt{N+1}}\bar{\gamma}_{1}-N-1 \over
\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}_{1}}F(k,t)},{{2\over
\sqrt{N+1}}\bar{\gamma}_{2} -N-1\over
\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}_{2}}F(k,t)}\Big\}\Big),
\end{align*}
and for every sign-changing function
$g:[1,N]_{\mathbb{Z}}\times\mathbb{R}\rightarrow\mathbb{R}$
  satisfying in \eqref{G2}, in the
case $\lambda = \mu$   problem \eqref{e1.1} admits at least two solutions $u_{1,\lambda}$
and $u_{2,\lambda}$ such that $\max_{k\in[1,N]_{\mathbb{Z}}}
|u_{1,\lambda}(k)| < \bar{\gamma}_{1}$ and
$\max_{k\in[1,N]_{\mathbb{Z}}} |u_{2,\lambda}(k)| <
\bar{\gamma}_{2}$.
\end{theorem}
\begin{proof}
Put
\begin{equation*}
\bar{f}(k,t)=
\begin{cases}
f(k,-\bar{\gamma}_{1}),
& \text{if}\,(k,t)\in[1,N]_{\mathbb{Z}}\times (-\infty,-\bar{\gamma}_{1})\\
f(t,x),
& \text{if}\,(k,t)\in[1,N]_{\mathbb{Z}}\times [-\bar{\gamma}_{1},\bar{\gamma}_{2}]\\
f(k,\bar{\gamma}_{2}), &
\text{if}\,(k,t)\in[1,N]_{\mathbb{Z}}\times
(\bar{\gamma}_{2},\infty).
\end{cases}
\end{equation*}
Clearly,  $\bar{f}:[1,N]_{\mathbb{Z}} \times
\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Now
put $ \bar{F}(k,t)=\int_{0}^{t}\bar{f}(k,\xi)d\xi$ for all $(k,
\xi)\in[1,N]_{\mathbb{Z}}\times\mathbb{R}$ and take $X$ and
$\Phi$ as \eqref{x} and \eqref{t3-4}, respectively, and
\begin{align*}
\Psi(u)=-\sum_{k=1}^{N}\Big(\bar{F}(k,u(k))+G(k,u(k))\Big)
\end{align*}
for all $u\in X$. Our goal is to apply Theorem \ref{t24} to $\Phi$
and $\Psi$. It is well known that
$\lim_{\|u\|_{X}\rightarrow\infty}\Phi(u)=\infty$ and $\Psi$ is a
differentiable functional whose differential at the point $u\in X$
is
\begin{align*}
\Psi'(u)v=-\sum_{k=1}^{N}\Big(\bar{f}(k,u(k))v(k)+g(k,u(k))v(k)\Big)
\end{align*}
for any $v\in X$, and it is also sequentially weakly lower
semicontinuous. Furthermore, $\Psi' : X\rightarrow X^*$ is a
compact operator. Thus, it is enough to show that $\Phi$ and
$\Psi$ satisfy   conditions $(n_1)$, $(n_2)$ and $(n_3)$ in
Theorem \ref{t24}. Let
\begin{align*}
\bar{r}_{1}={2\over
\sqrt{N+1}}\bar{\gamma}_{1}-N-1,\,\,\,\,\bar{r}_{2}={2\over
\sqrt{N+1}}\bar{\gamma}_{2}-N-1
\end{align*}
and let $\omega\in X$ as in the proof of Theorem \ref{t24}. Due to the
assumptions \eqref{t4-1} and \eqref{p}, we have  $\bar{r}_{1} <
\Phi(\omega) <\bar{r}_{2}$ and $\inf _{X} \Phi < \bar{r}_{1}<
\bar{r}_{2}$. Moreover, arguing as in the proof of Theorem
\ref{t3} and taking Remark \ref{r3} into account, we obtain
\begin{align*}
\varphi(\bar{r}_{1})\leq
{\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma}_{1}}F(k,t)\over
 {2\over\sqrt{N+1}}\bar{\gamma}_{1}-N-1},
\end{align*}
\begin{align*}
\varphi(\bar{r}_{2})\leq
{\sum_{k=1}^{N}\sup_{|t|\leq\bar{\gamma_{2}}}F(k,t)\over
 {2\over
\sqrt{N+1}}\bar{\gamma}_{2}-N-1}
\end{align*}
and
\begin{align*}
\varphi^{*}_{2}(\bar{r}_{2},\bar{r}_{2})\geq
{\sum_{k=1}^{N}F(k,\eta)\over 2 ({\sqrt{1+\bar{\eta}^2}-1)}}.
\end{align*}
Hence, from  $(A_{8})$, the conditions $(n_2)$ and $(n_3)$ of
Theorem \ref{t24} hold. Therefore, from Theorem \ref{t24} we
obtain that for each $\lambda\in \Lambda$, the problem
\begin{equation*}
\begin{cases}
-\Delta\Big({\Delta u(k-1)\over \sqrt{1+(\Delta
u(k-1))^2}}\Big)=\lambda\Big(\bar{f}(k,u(k))+ g(k,u(k))\Big),
 \,\,k\in[1,N]_{\mathbb{Z}}\\
u(0)=u(N+1)=0,
\end{cases}
\end{equation*}
admits at least two solutions $u_{1,\lambda}$ and $u_{2,\lambda}$
such that $\max_{k\in[1,N]_{\mathbb{Z}}} |u_{1,\lambda}(k)|
<\bar{\gamma}_{1}^{2}$ and $\max_{k\in[1,N]_{\mathbb{Z}}}
|u_{2,\lambda}(k)| <\bar{\gamma}_{2}^{2}$. Observing that these
solutions are also the solutions for   problem \eqref{e1.1} in
the case $\lambda = \mu$, the conclusion follows.
\end{proof}
We now consider the case where variables of function $f$ are
separated, in which case   problem \eqref{e1.1} is written as
follows:
\begin{equation*}
\begin{cases}
-\Delta\Big({\Delta u(k-1)\over \sqrt{1+(\Delta
u(k-1))^2}}\Big)=\lambda\Big(\theta(k)f(u(k))+ g(k,u(k))\Big),
 \,\,k\in[1,N]_{\mathbb{Z}}\\
u(0)=u(N+1)=0,
\end{cases}
\end{equation*}
where $\theta:[1,N]_{\mathbb{Z}}\rightarrow \mathbb{R}$ is a
non-negative and non-zero function such that $\theta\in
l^{1}([1,N]_{\mathbb{Z}})$ and $f
:\mathbb{R}\rightarrow\mathbb{R}$ is a non-negative and continuous
function.

Put
\begin{align*}
F(\xi)=\int^{\xi}_{0}f(t)dt,\,\ for \,\, all\,\  \xi\in\mathbb{R}.
\end{align*}
We apply the results of Theorems \ref{t3} and \ref{t4} to the
case when $f(k,t)=\theta(k)f(t)$ ( $\forall$ $(k, t)\in
[1,N]_{\mathbb{Z}}\times \mathbb{R}$).
\begin{theorem}\label{t5}
 Assume that there exist two positive constants $\bar{\gamma}>\frac{(N+1)\sqrt{N+1}}{2}$ and $\bar{\eta}$ with the property
\begin{align*}
\frac{\sqrt{N+1}}{2} (2\sqrt{1+\bar{\eta}^2}+N-1)>\bar{\gamma}
\end{align*}
 and let the assumption $(A_{2})$ and
\begin{itemize}
\item[$(A_{8})$]
 $ {F(\bar{\gamma})\over \frac{2}{\sqrt{N+1}}\bar{\gamma}-N-1}<{F(\eta) \over
2({\sqrt{1+\bar{\eta}^2}-1)}}$ hold.
\end{itemize}
Then, for every
\begin{align*}
\lambda\in\Big( {2({\sqrt{1+\bar{\eta}^2}-1)} \over
\|\theta\|_{l^{1}([1,N]_{\mathbb{Z}})}F(\eta)}
,{\frac{2}{\sqrt{N+1}}\bar{\gamma}-N-1\over
\|\theta\|_{l^{1}([1,N]_{\mathbb{Z}})}F(\bar{\gamma})}\Big)
\end{align*}
and for every sign-changing function
$g:[1,N]_{\mathbb{Z}}\times\mathbb{R}\rightarrow\mathbb{R}$
  satisfying   \eqref{G2},
    in the case
$\lambda = \mu$, problem \eqref{e1.1} admits at least three solutions in $X$.
\end{theorem}
 \begin{theorem}\label{t6}
 Assume that there exist three positive constants $\bar{\gamma}_{1}>\frac{(N+1)\sqrt{N+1}}{2}$, $\bar{\eta}$
  and $\bar{\gamma}_{2}>\frac{(N+1)\sqrt{N+1}}{2}$ with the property
\begin{align*}
 {2\over\sqrt{N+1}} \bar{\gamma}_{1}-N< 2\sqrt{1+\bar{\eta}^2}-1<{2\over\sqrt{N+1}}
 \bar{\gamma}_{2}-N
\end{align*}
and
\begin{itemize}
\item[$(A_{9})$]
\begin{align*}
&\max\Big\{{F(\bar{\gamma_1})\over
 {2\over\sqrt{N+1}} \bar{\gamma}_{1}-N-1},{F(\bar{\gamma_2})\over
 {2\over\sqrt{N+1}} \bar{\gamma}_{2}-N-1}\Big\}<{F(\eta) \over
2(\sqrt{1+\bar{\eta}^2}-1)}.
 \end{align*}
\end{itemize}
Then, for each
\begin{align*}
\lambda\in\Lambda&=\Big({{2(\sqrt{1+\bar{\eta}^2}-1)} \over
\|\theta\|_{l^{1}([1,N]_{\mathbb{Z}})}F(\eta)},\min\Big\{{{2\over\sqrt{N+1}}
\bar{\gamma}_{1}-N-1\over
\|\theta\|_{l^{1}([1,N]_{\mathbb{Z}})}F(\bar{\gamma_1})},{{2\over\sqrt{N+1}}
\bar{\gamma}_{2}-N-1 \over
\|\theta\|_{l^{1}([1,N]_{\mathbb{Z}})}F(\bar{\gamma_2})}\Big\}\Big)
\end{align*}
and for every sign-changing function
$g:[1,N]_{\mathbb{Z}}\times\mathbb{R}\rightarrow\mathbb{R}$
  satisfying   \eqref{G2},    in the
case $\lambda = \mu$, problem \eqref{e1.1} admits at least two solutions $u_{1,\lambda}$
and $u_{2,\lambda}$ such that
$\max_{k\in[1,N]_{\mathbb{Z}}}|u_{1,\lambda}(k)| <
\bar{\gamma}_{1}$ and $\max_{k\in[1,N]_{\mathbb{Z}}}
|u_{2,\lambda}(k)| < \bar{\gamma}_{2}$.
\end{theorem}

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\end{document}
