%-----------------------------------------------------------------------
% Beginning of article.tex
%-----------------------------------------------------------------------
%
% AMS-LaTeX 1.2 sample file for book proceedings, based on amsproc.cls.
%
% Replace amsproc by the documentclass for the target series, e.g. pspum-l.
%
\documentclass{amsproc}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\numberwithin{equation}{section}

%    Absolute value notation
\newcommand{\abs}[1]{\lvert#1\rvert}

%    Blank box placeholder for figures (to avoid requiring any
%    particular graphics capabilities for printing this document).
\newcommand{\blankbox}[2]{%
  \parbox{\columnwidth}{\centering
%    Set fboxsep to 0 so that the actual size of the box will match the
%    given measurements more closely.
    \setlength{\fboxsep}{0pt}%
    \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}%
  }%
}

\newcommand{\RR}{\mathbb R}
\newcommand{\NN}{\mathbb N}
\newcommand{\PP}{\mathbb P}
\newcommand{\ZZ}{\mathbb Z}
\newcommand{\di}{\displaystyle}

\begin{document}

\title[A nonlinear difference equation with oscillatory nonlinearity]{Infinitely many solutions for a nonlinear difference equation with oscillatory nonlinearity}

%    Information for first author
\author{Maria M\u alin}
%    Address of record for the research reported here
\address{Department of Mathematics, University of
Craiova, 200585, Romania}
%    Current address

\email{amy.malin@yahoo.com}
%    \thanks will become a 1st page footnote.
%\thanks{The first author was supported in part by NSF Grant \#000000.}

%    Information for second author
\author{Vicen\c{t}iu D. R\u{a}dulescu}
\address{Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia \& Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, 014700 Bucharest, Romania}
\email{vicentiu.radulescu@math.cnrs.fr}
%\thanks{Support information for the second author.}

%    General info
\subjclass{39A14; 47J30.}
%\date{January 1, 1994 and, in revised form, June 22, 1994.}

\dedicatory{Dedicated with esteem to Professor Hugo Beir\~{a}o da Veiga on his 70th anniversary}

\keywords{Difference equations, Discrete Laplacian, Oscillatory nonlinearities, Variational methods}

\begin{abstract}
In this paper, we study a discrete nonlinear  boundary value problem that involves a
 nonlinear term oscillating near the origin and a power-type nonlinearity $u^p$. By using variational methods, we establish the existence of a sequence of non-negative weak solutions that converges to $0$ if $p\geq 1$. In the sublinear case, we prove that for all $n$ positive integer, the problem has at least $n$ weak  solutions if the parameter lies in a certain range.
\end{abstract}

\maketitle

\section{Introduction and preliminary results}\label{sec1}
Let $n\geq 2$ be an integer number and denote $\mathbb{Z}[1,n]:=\{1,\ldots,n\}$.
The discrete Laplace operator is defined by
$$\Delta u(k)=\nabla (\nabla u(k+1)),$$
where $\nabla$ is the
backward difference operator, namely
$$\nabla u(k)=u(k)-u(k-1)\ \mbox{for all}\ k\in \mathbb{Z}[1,n].$$

In this paper, we are interested in the existence of solutions solutions $u=(u(1), \ldots, u(n))\in \mathbb{R}^T_+$  of  the following problem
$$\left\{  \begin{array}{ll}
	-\Delta u(k)=\lambda a(k) u(k)^p+f(u(k))\ \mbox{for all}\ k\in \mathbb{Z}[1,n], \\
   u(0)=u(n+1)=0,
 \end{array}\right.\hspace{2cm}(P_{\lambda})$$
where  $a=\left(a(1), \ldots, a(n)\right)\in \mathbb{R}^n$, $f:[0,+\infty)\to\RR$ is  continuous, $p>0$ and $\lambda\in {\mathbb R}$.
We refer to R\u adulescu \cite{14} and R\u adulescu and Repov\v{s} \cite{radrep} for related nonlinear discrete problems.
%We  prove the existence of solutions for problem $(P_{\lambda})$ using variational methods, by  %considering an auxiliary problem and, under suitable assumptions on the data, we will prove the %existence of solutions for this equation. The method  consists of proving the fact that the %corresponding     energy functional   admits a minimum, by using the direct methods of the %calculus of variations.

Problem $(P_{\lambda})$ is the discrete version of the semilinear elliptic equation studied in \cite{8}.
Moreover, this problem was recently extended by Molica Bisci,  R\u adulescu and Servadei  \cite{15, 15bis} to general classes of quasilinear elliptic equations.

Motivated by the studies in \cite{8} and \cite{15}, we focus in the present paper on the case of nonlinear difference equations. We are concerned in the study of the number of solutions of  problem $(P_{\lambda})$ and of their behavior in the case when $f$ oscillates near the origin.
Usually, equations involving oscillatory nonlinearities give infinitely many distinct solutions (see \cite{16}, \cite{17}), but the presence of an additional term may alter the situation.

Define the vector space
$$H=\{v=(v(0), v(1), \ldots, v(n),v(n+1))\in \RR^{n+2} \mbox{ such that }\ v(0)=v(n+1)=0\}.$$
Then $H$ is a $n$-dimensional Hilbert space (see \cite{2}) with the inner product
$$\langle u, v\rangle=\sum_{k=1}^{n+1}\nabla u(k)\nabla v(k),\ \forall \, u,v\in H.$$
The associated norm is defined by
$\|u\|=\left(\sum_{k=1}^{n+1}|\nabla u(k)|^2\right)^{1/2}.$

For all $u\in H$ we set
\begin{eqnarray}\label{eqmax}
\|v\|_{\infty}=\max_{k\in\mathbb{Z}[1,n]}|v(k)|.
\end{eqnarray}
Since  $H$ is finite-dimensional,  the norms $\|\cdot\|$ and $\|\cdot\|_{\infty}$ are equivalent on $H$.

%Taking into account that our paper studies the existence of weak solutions of the problem %$(P_{\lambda})$ we will present the variational characterization of
%the corresponding weak solutions.

\begin{definition}\label{MA}
We say that  $u\in H$ is a weak solution for the problem $(P_{\lambda})$ if
\begin{equation}\label{RV}
\sum_{k=1}^{n+1}\nabla u(k)\nabla v(k)-\lambda\sum_{k=1}^{n}a(k)u(k)^pv(k)-\sum_{k=1}^{n}f(u(k))v(k)=0,
\end{equation}
$\mbox{for all}\ v\in H.$
\end{definition}

\begin{remark}
Note that \eqref{RV} can be obtained by multiplying $(P_{\lambda})$  with $v(k)$ for all $k\in \mathbb{Z}[1,n]$ and summing up from $k=0$ to $k=n+1$. By taking into account that
$v(0)=v(n+1)=0$ and using some  simple computations we deduce the variational characterization of weak solutions from \eqref{RV}.
\end{remark}



\section{Main results}\label{sec2}

Throughout this paper, we assume that $f:[0,+\infty)\to\RR$ is a continuous function and we denote for all $s\in (0,+\infty)$,
$F(s):=\int\limits_0^s f(t)dt.$

We assume that $f$ oscillates near the origin, namely the following conditions are fulfilled:
\begin{itemize}
\item[$(f_1^0)$]  $-\infty<\liminf\limits_{s\to 0^{+}}\frac{F(s)}{s^2};\ \limsup\limits_{s\to 0^{+}}\frac{F(s)}{s^2}>\frac{1}{n};$
\item[$(f_2^0)$] $l_0:=\liminf\limits_{s\to 0^{+}}\frac{f(s)}{s}<0.$
\end{itemize}

\begin{example}
Let $\alpha>1, \, \beta\in \mathbb{R}$ and $\gamma>0$.
Define $f_0:[0,+\infty)\to\RR$ by
\begin{eqnarray*}\label{eq4}
f_0(s)=\left\{  \begin{array}{ll}
  0\ &\mbox{if}\ s=0, \\
   s(1+\alpha\sin (\beta s^{-\gamma}))\ &\mbox{if}\ s>0,
    \end{array}\right.
\end{eqnarray*}
 Then $f_0$ satisfies assumptions $(f_1^0)$ and $(f_2^0)$.
\end{example}

\begin{remark}
  Hypotheses $(f_1^0)$ and $(f_2^0)$ imply that
\begin{eqnarray}\label{eq3}
f(0)=0.
\end{eqnarray}
\end{remark}

We point out that condition $(f_1^0)$ allows us to deduce some information about the number of solutions for problem $(P_{\lambda})$, while $(f_2^0)$ yields the existence of the solutions.

The main results in this paper distinguish between the superlinear case $p\geq 1$ and the sublinear setting that corresponds to $p\in (0,1)$.

%In this case our main results are the following.
\begin{theorem}\label{th1}
Let $a=\left(a(1), \ldots, a(n)\right)\in \mathbb{R}^n$, $\lambda\in\RR$ and $p\geq 1$. Assume that $f\in C([0,+\infty);\RR)$ satisfies conditions $(f_1^0)$ and $(f_2^0)$. If either
\begin{itemize}
\item[(i)] $p=1$, $l_0\in (-\infty,0)$ and $\lambda a(k)<\lambda_0$ for all $k\in \mathbb{Z}[1,n]$ and some $\lambda_0\in (0,-l_0)$ or
\item[(ii)] $p=1$, $l_0=-\infty$ and $\lambda\in\RR$ is arbitrary or
\item[(iii)] $p>1$ and $\lambda\in\RR$ is arbitrary,
\end{itemize}
then there exists a sequence $\{u_i\}_i$ in $H$ of non-negative, distinct weak solutions of problem $(P_{\lambda})$ such that
\begin{eqnarray}\label{eq5}
\lim\limits_{i\to+\infty}\|u_i\|=\lim\limits_{i\to+\infty}\|u_i\|_{\infty}=0.
\end{eqnarray}
\end{theorem}
\begin{theorem}\label{th2}
Let $a=\left(a(1), \ldots, a(n)\right)\in \mathbb{R}^n$, $\lambda\in\RR$ and $0<p< 1$. Assume that $f\in C([0,+\infty);\RR)$ satisfies conditions $(f_1^0)$ and $(f_2^0)$. Then, for every $n\in\NN$, there exists $\Lambda_n>0$ such that problem $(P_{\lambda})$ has at least $n$ distinct weak solutions $u_{1,\lambda},\ldots,u_{n,\lambda}\in H$ such that
\begin{eqnarray}\label{eq6}
\|u_{i,\lambda}\|<\frac{1}{i}\ \mbox{and}\ \|u_{i,\lambda}\|_{\infty}<\frac{1}{i},\ \mbox{for any}\ i=1,\ldots,n,
\end{eqnarray}
provided $\lambda\in[-\Lambda_n,\Lambda_n]$.
\end{theorem}

\

\section{An auxiliary problem}\label{sec3}
Consider the problem
$$\left\{  \begin{array}{ll}
  -\Delta u(k)+c(k)u(k)=g(k,u(k)),\ k\in\mathbb{Z}[1,n], \\
   u(0)=u(n+1)=0\ .
    \end{array}\right.\hspace{4cm}(P_g^c)$$
Here, we assume that $c=\left(c(1), \ldots, c(n)\right)\in \mathbb{R}^n$ is such that
\begin{eqnarray}\label{eq10}
\min\limits_{k\in\mathbb{Z}[1,n]}c(k)>0,
\end{eqnarray}
while $g:\mathbb{Z}[1,n]\times [0,+\infty)\to\RR$ is a Carath\'{e}odory function satisfying the following conditions
\begin{eqnarray}\label{eq11}
g(k,0)=0\ \mbox{for every}\ k\in \mathbb{Z}[1,n];
\end{eqnarray}
$\mbox{there exists}\ M_g>0\ \mbox{such that}$
\begin{eqnarray}\label{eq12}
|g(k,s)|\leq M_g\ \mbox{for every}\ k\in \mathbb{Z}[1,n]\ \mbox{and all}\ s\geq 0;
\end{eqnarray}
$\mbox{there exist}\ \delta\ \mbox{and}\ \eta,\ \mbox{with}\ 0<\delta<\eta\ \mbox{such that}$
\begin{eqnarray}\label{eq13}
 g(k,s)\leq 0\ \mbox{for every}\ k\in \mathbb{Z}[1,n]\ \mbox{and all}\ s\in [\delta,\eta].
\end{eqnarray}
We extend the function $g$ by taking $g(k,s)=0$ for every $k\in \mathbb{Z}[1,n]$ and $s\leq0$.

%Firstly, in a similar way as in Definition \ref{MA} we give the variational characterization of %a weak solution for $(P_g^c)$.
\begin{definition}\label{def1}
By a weak solution for problem $(P_g^c)$ we understand a vector $u\in H$ such that for all $v\in H$
\begin{eqnarray*}
&&\sum_{k=1}^{n+1}\nabla u(k)\nabla v(k)+ \sum_{k=1}^{n}c(k)u(k)v(k)-\sum_{k=1}^{n}g(k,u(k))v(k)=0.
\end{eqnarray*}
\end{definition}
Let $E_{c,g}:H\to\RR$ be the energy functional associated to problem $(P_g^c)$, namely
\begin{eqnarray}\label{eq14}
E_{c,g}(u)=\frac{1}{2}\|u\|^2+\frac{1}{2}\sum_{k=1}^{n}c(k)u(k)^2-\sum_{k=1}^{n}G(k,u(k)),\ u\in H,
\end{eqnarray}
where $G(k,s):=\int\limits_0^s g(k,t)dt$ for any $s\in\RR$ and $k\in \mathbb{Z}[1,n]$.

Then $E_{c,g}$ is well-defined,  of class $C^1(H;\RR)$ and
$$
\langle E_{c,g}'(u),v\rangle = \langle u,v\rangle+\sum_{k=1}^{n}c(k)u(k)v(k)-\sum_{k=1}^{n}g(k,u(k))v(k),\ \forall u,v\in H.
$$
Thus, the weak solutions of $(P_g^c)$ coincide with the critical points of $E_{c,g}$.

Finally, we introduce the set $W^{\eta}$ defined as follows
$$W^{\eta}:=\{u\in H:\ \|u\|_{\infty}\leq\eta\},$$
where $\eta$ is a positive parameter given in (\ref{eq13}).

Since $g(k,0)=0$ for every $k\in \mathbb{Z}[1,n]$ by (\ref{eq11}), then $u\equiv0$ is clearly a weak solution of problem $(P_g^c)$.
%In the sequel, under some general assumptions, we prove the existence of a non-negative weak %solution for problem $(P_g^c)$.

%Thus, the main result of this section is the following.
\begin{theorem}\label{th5}
Assume that $c=\left(c(1), \ldots, c(n)\right)\in \mathbb{R}^n$ satisfies (\ref{eq10}) and that $g:\mathbb{Z}[1,n]\times [0,+\infty)\to\RR$ is a Carath\'{e}odory function satisfying (\ref{eq11}), (\ref{eq12}) and (\ref{eq13}). Then
\begin{itemize}
\item[(a)] the functional $E_{c,g}$ is bounded from below on $W^{\eta}$ attaining  its infimum at some $\tilde{u}\in W^{\eta}$;
\item[(b)] $\tilde{u}(k)\in [0,\delta]$ for every $k\in \mathbb{Z}[1,n]$, where $\delta$ is the positive parameter given in (\ref{eq13});
\item[(c)] $\tilde{u}$ is a non-negative weak solution of problem $(P_g^c)$.
\end{itemize}
\end{theorem}
\begin{proof}
$(a)$ Since the norms $\|\cdot\|_{\infty}$ and $\|\cdot\|$ are equivalent in the finite-dimensional space $H$, the set $W^{\eta}$ is compact in $H$. Combining this fact with the continuity of $E_{c.g}$, we infer that $E_{c,g}\Big|_{W^{\eta}}$ attains its infimum at $\tilde{u}\in W^{\eta}$.

$(b)$ Let $\delta$ be as in assumption (\ref{eq13}) and let
$M:=\{k\in \mathbb{Z}[1,n]:\ \tilde{u}(k)\notin [0,\delta]\}.$
Hence, arguing by contradiction, we suppose that $M\neq\emptyset$.
Define the truncation function $\gamma:\RR\to\RR$ by
$\gamma(s):=\min\{s_{+},\delta\},$
where $s_{+}=\max\{s,0\}$ and set $w:=\gamma\circ\tilde{u}$.
Since $\gamma(0)=0$, we have $w(0)=w(n+1)=0$, so $w\in H$. Besides, $0\leq w(k)\leq\delta$ for every $k\in \mathbb{Z}[1,n]$. By assumption (\ref{eq13}) we know that $\delta<\eta$, and so $w\in W^{\eta}$ .
We introduce the sets
$M_{-}:=\{k\in M:\ \tilde{u}(k)<0\}\ \mbox{and}\  M_{+}:=\{k\in M:\ \tilde{u}(k)>\delta\}.$
Thus, $M=M_{-}\cup M_{+}$ and we have that
$$w(k)=\left\{  \begin{array}{ll}
 \tilde{u}(k)\ &\mbox{for all}\ k\in \mathbb{Z}[1,n]\setminus M, \\
   0\ &\mbox{for all}\ k\in M_{-},\\
   \delta\ &\mbox{for all}\ k\in M_{+}.
    \end{array}\right.$$
Moreover, we have
\begin{equation}\label{eq15}\begin{array}{ll}
\di E_{c,g}(w)-E_{c,g}(\tilde{u})&=\di\frac{1}{2}\left(\|w\|^2-\|\tilde{u}\|^2\right)+\frac{1}{2}\sum_{k=1}^{n}c(k)[(w(k))^2-(\tilde{u}(k))^2]\\
& \di- \sum_{k=1}^{n}[G(k,w(k))-G(k,\tilde{u}(k))]
=:\frac{1}{2}J_1+\frac{1}{2}J_2-J_3.
\end{array}
\end{equation}
Since $\gamma$ is a Lipschitz function with Lipschitz constant $1$, and $w=\gamma\circ\tilde{u}$, we have
\begin{equation}\label{eq16}\begin{array}{ll}
J_1&=\di\|w\|^2-\|\tilde{u}\|^2=\sum_{k=1}^{n+1}[|\nabla w(k)|^2-|\nabla\tilde{u}(k)|^2]\\
&=\di\sum_{k=1}^{n+1}\left[|w(k)-w(k-1)|^2-| \tilde{u}(k)-\tilde{u}(k-1)|^2\right]\leq 0.
\end{array}
\end{equation}
Since $\min\limits_{k\in\mathbb{Z}[1,n]}c(k)>0$ by (\ref{eq10}), we have
\begin{equation}\label{eq17}\begin{array}{ll}
J_2&=\di\sum_{k=1}^{n}c(k)[(w(k))^2-(\tilde{u}(k))^2]=\sum_{k\in M}c(k)[(w(k))^2-(\tilde{u}(k))^2]\\
&=\di -\sum_{k\in M_{-}}c(k)(\tilde{u}(k))^2+\sum_{k\in M_{+}}c(k)[\delta^2-(\tilde{u}(k))^2]\leq 0.
\end{array}
\end{equation}

Next, we estimate $J_3$. Due to the fact that $g(k,s)=0$ for all $s\leq 0$ and for every $k\in \mathbb{Z}[1,n]$, we have
\begin{eqnarray}\label{eq18}
\sum_{k\in M_{-}}\left[G(k,w(k))-G(k,\tilde{u}(k))\right]=0.
\end{eqnarray}
Moreover, by the mean value theorem, for every $k\in M_{+}$, there exists $\theta(k)\in [\delta,\tilde{u}(k)]\subset[\delta,\eta]$ such that
$$G(k,w(k))-G(k,\tilde{u}(k))=G(k,\delta)-G(k,\tilde{u}(k))=g(k,\theta(k))(\delta-\tilde{u}(k)).$$
Thus, taking into account hypothesis (\ref{eq13}) and definition of $M_{+}$, we have
\begin{eqnarray}\label{eq19}
\sum_{k\in M_{+}}\left[G(k,w(k))-G(k,\tilde{u}(k))\right]\geq 0.
\end{eqnarray}
Hence, by (\ref{eq18}) and (\ref{eq19}), we obtain
\begin{eqnarray}\label{eq20}
J_3=\sum_{k\in M_{+}}\left[G(k,w(k))-G(k,\tilde{u}(k))\right]\geq 0.
\end{eqnarray}
Combining relations (\ref{eq16}), (\ref{eq17}), (\ref{eq20}) with (\ref{eq15}), we get
\begin{eqnarray}\label{eq21}
E_{c,g}(w)-E_{c,g}(\tilde{u})\leq 0.
\end{eqnarray}
On the other hand, since $w\in W^{\eta}$, it is easy to see that
$E_{c,g}(w)\geq E_{c,g}(\tilde{u})=\inf\limits_{u\in W^{\eta}}E_{c,g}(u).$
By this and (\ref{eq21}) we get that every term in $E_{c,g}(w)-E_{c,g}(\tilde{u})$ should be zero.
In particular, from $J_2$ and due to (\ref{eq10}), we have
$$\sum_{k\in M_{-}}c(k)(\tilde{u}(k))^2=\sum_{k\in M_{+}}c(k)[\delta^2-(\tilde{u}(k))^2]=0,$$
which implies that $$\tilde{u}(k)=\left\{  \begin{array}{ll}
 0\ \mbox{for every}\ k\in M_{-} \\
   \delta\ \mbox{for every}\ k\in M_{+}.
    \end{array}\right.$$

In view of the definition of the sets $M_{-}$ and $M_{+}$, we deduce that $M_{-}=M_{+}=\emptyset$, which contradicts $M_{-}\cup M_{+}=M\neq \emptyset$.

$(c)$ Fix $v\in H$ arbitrarily and let
$\varepsilon_0:=\frac{\eta-\delta}{\|v\|_{\infty}+1}>0,$
where $\delta$ and $\eta$ are given as in (\ref{eq13}). Moreover, let $I:[-\varepsilon_0,\varepsilon_0]\to\RR$ be the function defined as
$I(\varepsilon):=E_{c,g}(\tilde{u}+\varepsilon v).$
First of all, thanks to $(b)$, for any $\varepsilon\in [-\varepsilon_0,\varepsilon_0]$ we have
\begin{eqnarray*}
|\tilde{u}(k)+\varepsilon v(k)|\leq\tilde{u}(k)+\frac{\eta-\delta}{\|v\|_{\infty}+1}\|v\|_{\infty}\leq\eta,
\end{eqnarray*}
for every $k\in \mathbb{Z}[1,n]$. Thus, $\tilde{u}+\varepsilon v\in W^{\eta}$.
Consequently, due to $(a)$, we have $I(\varepsilon)\geq I(0)$ for every $\varepsilon\in [-\varepsilon_0,\varepsilon_0]$, that is, $0$ is an interior minimum point for $I$. Then
$I'(0)=0\ \mbox{and}\ \langle E_{c,g}'(\tilde{u}),v\rangle=0.$
Taking into account that $v\in H$ is arbitrary and using the definition of $E_{c,g}$, we obtain that $\tilde{u}$ is a weak solution of problem $(P_g^c)$. Moreover, due to $(b)$, $\tilde{u}$ is non-negative in $\mathbb{Z}[1,n]$.
\end{proof}

 Theorem \ref{th5} does not guarantee that the solution $\tilde{u}$ of problem $(P_g^c)$ is not the trivial one. In spite of this, by Theorem \ref{th5} we will derive the existence of nontrivial solutions for the original problem $(P_{\lambda})$, provided that the nonlinear term $f$ is chosen appropriately.
Finally, we define the continuous truncation function $\tau_{\eta}:[0,+\infty)\to\RR$ as follows
\begin{eqnarray}\label{eq22}
\tau_{\eta}(s):=\min\{\eta,s\}\ \mbox{for every}\ s\geq 0,
\end{eqnarray}
where $\eta$ is the positive constant given in assumption (\ref{eq13}).

\

\section{Oscillation near the origin}\label{sec4}

In order to prove Theorem \ref{th1} and Theorem \ref{th2}, we consider  problem $(P_g^c)$,
where $c=\left(c(1), \ldots, c(n)\right)\in \mathbb{R}^n$ fulfills (\ref{eq10}) and $g:\mathbb{Z}[1,n]\times [0,+\infty)\to\RR$ is a Carath\'{e}odory function which satisfies the following assumptions
\begin{eqnarray}\label{eq23}
&&g(k,0)=0\ \mbox{for all}\ k\in \mathbb{Z}[1,n],\ \mbox{and there exist}\nonumber\\
 && \overline{s}>0\ \mbox{and}\ M>0\ \mbox{such that}\ \max\limits_{s\in [0,\overline{s}]}|g(k,s)|\leq M,\ \mbox{for all}\ k\in\mathbb{Z}[1,n];
\end{eqnarray}
\begin{eqnarray}\label{eq24}
&&\mbox{there exist two sequences}\ \{\delta_i\}_i\ \mbox{and}\ \{\eta_i\}_i\ \mbox{with}\ 0<\eta_{i+1}<\delta_i<\eta_i\ \mbox{such that}\nonumber\\
&&\lim\limits_{i\to+\infty}\eta_i=0\ \mbox{and}\ g(k,s)\leq 0\ \mbox{for every}\ k\in \mathbb{Z}[1,n]\ \mbox{and all}\ s\in[\delta_i,\eta_i],\ i\in\NN;
\end{eqnarray}
\begin{eqnarray}\label{eq25}
&&-\infty<\liminf\limits_{s\to 0^{+}}\frac{G(k,s)}{s^2}\ \mbox{and}\ \limsup\limits_{s\to 0^{+}}\frac{G(k,s)}{s^2}>\frac{1}{n}\ \mbox{uniformly for all}\ k\in \mathbb{Z}[1,n].
\end{eqnarray}
%In the sequel, we will prove Theorem \ref{th1}. The strategy will consists in applying Theorem %\ref{th5} with a suitable choices of the vector $c$ and the function $g$.



\textbf{Proof of Theorem \ref{th1}}.
We first show that under suitable assumptions, problem $(P_{\lambda})$ has infinitely many distinct weak solutions, provided that $p\geq 1$. We will consider separately the cases  $p=1$ and  $p>1$ and in both situations the strategy will consist in using Theorem \ref{th5}.

 We start by proving assertion $(i)$. In this setting we suppose that $p=1$ and $l_0\in (-\infty,0)$. Let $\lambda\in\RR$ be such that $\lambda a(k)<\lambda_0$ for all $k\in \mathbb{Z}[1,n]$ and some $0<\lambda_0<-l_0$.
Fix $\overline{\lambda}_0\in (\lambda_0,-l_0)$ and let
\begin{eqnarray}\label{eq26}
&&c(k):=\overline{\lambda}_0-\lambda a(k)\ \mbox{and}\ g(k,s):=f(s)+\overline{\lambda}_0 s,
\end{eqnarray}
$\mbox{for all}\ (k,s)\in \mathbb{Z}[1,n]\times [0,+\infty).$
The first step consist in proving that the vector $c$ and the function $g$ given in (\ref{eq26}) satisfy the assumptions (\ref{eq10}), (\ref{eq23}), (\ref{eq24}) and (\ref{eq25}).
Note that $c\in \mathbb{R}^n$  and
$\min_{k\in \mathbb{Z}[1,n]}c(k)>\overline{\lambda}_0-\lambda_0>0,$
which obviously implies (\ref{eq10}). By (\ref{eq3}) we know that $f(0)=0$. Thus, using the regularity of $f$, we obtain that $g$ is a continuous function in $\mathbb{Z}[1,n]\times[0,+\infty)$ and $g(k,0)=0$ for all $k\in \mathbb{Z}[1,n]$. Next, the continuity of $s\mapsto g(\cdot,s)$ and the Weierstrass theorem yield (\ref{eq23}).
Moreover, since for any $k\in\mathbb{Z}[1,n]$ and $s>0$ we have
$G(k,s)/s^2=\overline{\lambda}_0/2+F(s)/s^2,$
hypothesis $(f_1^0)$ immediately implies (\ref{eq25}).

Next, we show that $g$ satisfies (\ref{eq24}). By $(f_2^0)$,  there exists a sequence $\{s_i\}_i\subset (0,1)$ converging to $0$ such that
$\lim\limits_{i\to+\infty}\frac{f(s_i)}{s_i}=l_0.$
Since $\overline{\lambda}_0<-l_0$ by assumption, there exists $\overline{\varepsilon}>0$ such that $\overline{\lambda}_0+\overline{\varepsilon}<-l_0$. By this and the above relation we get that for  $i\geq i^*\in\NN$,
\begin{eqnarray}\label{eq27}
f(s_i)<-\overline{\lambda}_0 s_i.
\end{eqnarray}
Thus we obtain that
$g(k,s_i)=f(s_i)+\overline{\lambda}_0 s_i<0.$
Consequently, by  the continuity of $f$, there is a neighborhood of $s_i$, say $(\delta_i,\eta_i)$ and there are two sequences $\{\delta_i\}_i$, $\{\eta_i\}_i\subset(0,1)$ such that $0<\eta_{i+1}<\delta_i<s_i<\eta_i$, $\lim\limits_{i\to+\infty}\eta_i=0$ and $g(k,s)=\overline{\lambda}_0 s+f(s)\leq 0$ for any $k\in\mathbb{Z}[1,n]$ and all $s\in[\delta_i,\eta_i]$ and $i\geq i^*$.
In this way, hypothesis (\ref{eq24}) is verified for $g$ on every interval $[\delta_i,\eta_i]$, $i\in\NN$.
In the sequel, since $\eta_i\to 0$ as $i\to+\infty$, by (\ref{eq24}), without any loss of generality, we may assume that
\begin{eqnarray}\label{eq28}
0<\delta_i<\eta_i<\overline{s},
\end{eqnarray}
for $i$ sufficiently large, where $\overline{s}>0$ is given by (\ref{eq23}).
For every $i\in\NN$, let $g_i:\mathbb{Z}[1,n]\times[0,+\infty)\to\RR$ be the truncation function defined by
\begin{eqnarray}\label{eq29}
g_i(k,s):=g(k,\tau_{\eta_i}(s))\ \mbox{and}\ G_i(k,s):=\int_0^s g_i(k,t)dt,
\end{eqnarray}
for every $k\in \mathbb{Z}[1,n]$ and $s\geq 0$, where $\tau_{\eta_i}$ is the function defined in (\ref{eq22}) with $\eta=\eta_i$.
Let $E_i:H\to\RR$ be the energy functional associated with problem $(P_{g_i}^c)$, that is $E_i:=E_{c,g_i}$, where $E_{c,g_i}$ is the functional given in (\ref{eq14}) with $g=g_i$.
We note that the function $g_i$ verifies all the assumptions of Theorem \ref{th5} for $i\in\NN$ large enough with $[\delta_i,\eta_i]$. Indeed, thanks to the regularity of $g$, the continuity of $\tau_{\eta}$ and the fact that $g(k,0)=0$ for all $k\in\mathbb{Z}[1,n]$, the function $g_i$ is Carath\'{e}odory and such that $g_i(k,0)=0$ for every $k\in\mathbb{Z}[1,n]$. Moreover, by (\ref{eq23}), (\ref{eq28}) and (\ref{eq29}), $g_i$ satisfies (\ref{eq11}) and (\ref{eq12}). Finally, condition (\ref{eq13}) is satisfied thanks to (\ref{eq24}).
Hence, as a consequence of Theorem \ref{th5}, for every $i\in\NN$, there exists $u_i\in W^{\eta_i}$ such that
\begin{eqnarray}\label{eq30}
\min\limits_{u\in W^{\eta_i}}E_i(u)=E_i(u_i);
\end{eqnarray}
\begin{eqnarray}\label{eq31}
u_i(k)\in[0,\delta_i]\ \mbox{for every}\ k\in\mathbb{Z}[1,n];
\end{eqnarray}
\begin{eqnarray}\label{eq32}
u_i\ \mbox{is a non-negative weak solution of}\ (P_{g_i}^c).
\end{eqnarray}
Using the definition of $\tau_{\eta}$, relation (\ref{eq29}) and the fact that
$0\leq u_i(k)\leq\delta_i<\eta_i\ \mbox{for every}\ k\in\mathbb{Z}[1,n],$
we have
$g_i(k,u_i(k))=g(k,\tau_{\eta_i}(u_i(k)))=g(k,u_i(k))$ for every $k\in\mathbb{Z}[1,n].$
Thus, by the above relation and (\ref{eq32}), $u_i$ is a non-negative weak solution not only for $(P_{g_i}^c)$ but also for problem $(P_g^c)$.
In the sequel, we prove that there are infinitely many distinct elements in the sequence $\{u_i\}_i$. In order to see this, the first step consists in proving that
\begin{eqnarray}\label{eq33}
E_i(u_i)<0\ \mbox{for}\ i\in\NN\ \mbox{large enough and}
\end{eqnarray}
\begin{eqnarray}\label{eq34}
\lim\limits_{i\to+\infty}E_i(u_i)=0.
\end{eqnarray}
Due to $(f_1^0)$ and (\ref{eq26}), we have that
$\limsup\limits_{s\to 0^{+}}\frac{G(k,s)}{s^2}>\frac{\overline{\lambda}_0}{2}+\frac{1}{n}.$
In particular, there exists a sequence $\{\tilde{s}_i\}_i$, with
\begin{eqnarray}\label{eq35}
0<\tilde{s}_i\leq\delta_i\ \mbox{for all}\ i\in\NN\ \mbox{and}
\end{eqnarray}
\begin{eqnarray}\label{eq36}
G(k,\tilde{s}_i)>\left(\frac{1}{n}+\frac{\overline{\lambda}_0}{2}\right)\tilde{s}_i^2.
\end{eqnarray}
Now, let us fix $i\in\NN$ sufficiently large and let us define the function $w_i\in H$ by
$w_i(k):=\tilde{s}_i\ \mbox{for every}\ k\in\mathbb{Z}[1,n].$
Then $\|w_i\|_{\infty}=\tilde{s}_i\leq\delta_i<\eta_i<1$ by (\ref{eq24}) and (\ref{eq35}). Hence, $w_i\in W^{\eta_i}$.
This yields that for every $k\in\mathbb{Z}[1,n]$, we have
\begin{eqnarray}\label{eq37}
G_i(k,w_i(k))=G_i(k,\tilde{s}_i)=\int_0^{\tilde{s}_i}g_i(k,t)dt=G(k,\tilde{s}_i).
\end{eqnarray}
By this and taking into account (\ref{eq10}), (\ref{eq26}), (\ref{eq36}), (\ref{eq37}), for $i$ sufficiently large we have
\begin{eqnarray*}
E_i(w_i)&=&\frac{1}{2}\sum_{k=1}^{n+1}|\nabla w_i(k-1)|^2+\frac{1}{2}\sum_{k=1}^{n}c(k)(w_i(k))^2-\sum_{k=1}^{n}G_i(k,w_i(k))\\
&<&(\tilde{s}_i)^2+\frac{1}{2}\overline{\lambda}_0T(\tilde{s}_i)^2-n\left(\frac{1}{n}+\frac{\overline{\lambda}_0}{2}\right)(\tilde{s}_i)^2<0.
\end{eqnarray*}
Consequently, using also (\ref{eq30}) for $i$ sufficiently large, the above estimation and $w_i\in W^{\tilde{s}_i}\subset W^{\eta_i}$ show that
\begin{eqnarray}\label{eq38}
E_i(u_i)=\min\limits_{u\in W^{\eta_i}}E_i(u)\leq E_i(w_i)<0,
\end{eqnarray}
which proves in particular (\ref{eq33}).
Next, we prove (\ref{eq34}). For every $i\in\NN$ sufficiently large, by using the definition of $G_i$, the mean value theorem, (\ref{eq23}), (\ref{eq24}), (\ref{eq28}), (\ref{eq29}) and (\ref{eq31}), we have
\begin{eqnarray*}
E_i(u_i)&\geq&-\sum_{k=1}^{n}G_i(k,u_i(k))=-\sum_{k=1}^{n}G(k,u_i(k))\\
&\geq&-\sum_{k=1}^{n}\max\limits_{s\in[0,\overline{s}]}|g(k,s)|u_i(k)\geq -\delta_i TM.
\end{eqnarray*}
Since $\lim\limits_{i\to+\infty}\delta_i=0$, the above estimate and (\ref{eq38}) leads to (\ref{eq34}).

Finally, it is easy to see that relation (\ref{eq5}) is an immediate consequence of (\ref{eq31}) combined with $\lim\limits_{i\to+\infty}\delta_i=0$, and to the fact that norms $\|\cdot\|_{\infty}$ and $\|\cdot\|$ are equivalent.
Thus, we get the existence of infinitely many distinct nontrivial non-negative solutions $\{u_i\}_i$ for problem $(P_g^c)$ satisfying condition (\ref{eq5}). Due to the choice of $c$ and $g$ in (\ref{eq26}) and taking into account that $p=1$, it is easy to see that $u_i$ is a weak solution of problem $(P_{\lambda})$ and this ends the proof of assertion $(i)$ in Theorem \ref{th1} in the case $p=1$.

Now, let us consider assertion $(ii)$. At this purpose, let $p=1$, $l_0=-\infty$ and $\lambda\in\RR$ be arbitrary fixed. In this setting we choose $\overline{\lambda}_0\in(\lambda_0,-l_0)$ and
$$c(k):=\overline{\lambda}_0\ \mbox{and}\ g(k,s)=f(s)+(\lambda a(k)+\overline{\lambda}_0)s\ \mbox{for all}\ (k,s)\in\mathbb{Z}[1,n]\times[0,+\infty).$$
This case can be dealt with in a similar way as $(i)$, using relation
$f(s_i)<-(|\lambda|\cdot\|a\|_{\infty}+\overline{\lambda}_0)s_i,$
instead of $f(s_i)<-\overline{\lambda}_0 s_i$, for $i$ large enough, and taking into account that for every $k\in\mathbb{Z}[1,n]$ and $s\geq 0$ one has
$g(k,s)=f(s)+(\lambda a(k)+\overline{\lambda}_0)s\leq f(s)+(|\lambda|\cdot\|a\|_{\infty}+\overline{\lambda}_0)s.$

Now, let us prove assertion $(iii)$. At this purpose, let $p>1$ and $\lambda\in\RR$ be arbitrary fixed. Let us also fix a number $\overline{\lambda}_0\in (0,-l_0)$ and choose
\begin{eqnarray}\label{eq39}
&&c(k):=\overline{\lambda}_0\ \mbox{and}\ g(k,s):=\lambda a(k)s^p+\overline{\lambda}_0 s+f(s)
\end{eqnarray}
$\mbox{for all}\ (k,s)\in\mathbb{Z}[1,n]\times[0,+\infty).$
Also in this setting our aim is to prove that $c$ and $g$ given in (\ref{eq39}) satisfy the conditions (\ref{eq10}), (\ref{eq23}), (\ref{eq24}) and (\ref{eq25}).
Clearly, (\ref{eq10}) is satisfied and also thanks to $(f_1^0)$, $(f_2^0)$ we have $g(k,0)=0$ for all $k\in\mathbb{Z}[1,n]$. Moreover, since $a\in \mathbb{R}^n$ the continuity of $s\mapsto g(\cdot,s)$ and the Weierstrass theorem yield that (\ref{eq23}) holds true.
Furthermore, since $p>1$ and
$\frac{G(k,s)}{s^2}=\lambda\frac{a(k)}{p+1}s^{p-1}+\frac{\overline{\lambda}_0}{2}+\frac{F(s)}{s^2},\ \mbox{for all}\ k\in\mathbb{Z}[1,n]\ \mbox{and}\ s\in(0,+\infty),$
hypothesis $(f_1^0)$ implies (\ref{eq25}).
In the sequel, note that for all $k\in\mathbb{Z}[1,n]$ and every $s\in[0,+\infty)$, we have
\begin{eqnarray}\label{eq40}
g(k,s)\leq |\lambda|\cdot\|a\|_{\infty}s^p+\overline{\lambda}_0s+f(s).
\end{eqnarray}
As a consequence of this and of $(f_2^0)$ we get
\begin{eqnarray}\label{eq41}
\liminf\limits_{s\to 0^{+}}\frac{g(k,s)}{s}\leq\overline{\lambda}_0+l_0<0,
\end{eqnarray}
for all $k\in\mathbb{Z}[1,n]$, thanks to the choice of $p$.
In particular, there exists a sequence $\{s_i\}_i\subset (0,1)$ converging to $0$ as $i\to+\infty$ such that $g(k,s_i)<0$ for $i\in\NN$ large enough and for all $k\in\mathbb{Z}[1,n]$. Thus, by using the continuity of $s\mapsto g(\cdot,s)$, there exist two sequences $\{\delta_i\}_i$, $\{\eta_i\}_i\subset (0,1)$ such that $0<\eta_{i+1}<\delta_i<s_i<\eta_i$, $\lim\limits_{i\to+\infty}\eta_i=0$ and $g(k,s)\leq 0$, for every $k\in\mathbb{Z}[1,n]$ and all $s\in[\delta_i,\eta_i]$ and $i\in\NN$ large enough.
Summarizing, we deduce that hypothesis (\ref{eq24}) hold true.

Finally, an argument analogous to that used in $(i)$ proves that problem $(P_g^c)$ is equivalent to problem $(P_{\lambda})$ through the choice (\ref{eq39}) and so, we get the existence of infinitely many distinct nontrivial solutions $\{u_i\}_i$ for problem $(P_{\lambda})$  satisfying (\ref{eq5}). This concludes the proof of Theorem \ref{th1}.
\qed

\medskip
\textbf{Proof of Theorem \ref{th2}}.
%First of all, we point out that here we will proceed as in the proof of Theorem \ref{th1} and %the strategy will consist in applying Theorem \ref{th5} to problem $(P_g^c)$ with a suitable %choices of $c$ and $g$.
Let $\overline{\lambda}_0\in (0,-l_0)$, where $l_0<0$ is given in assumption $(f_2^0)$ and let us choose
\begin{eqnarray}\label{eq42}
&&c(k):=\overline{\lambda}_0\ \mbox{and}\ g(k,s,\lambda):=\lambda a(k)s^p+\overline{\lambda}_0 s+f(s),
\end{eqnarray}
for all $(k,s)\in\mathbb{Z}[1,n]\times[0,+\infty),\ \lambda\in\RR$.
Note that for all $k\in\mathbb{Z}[1,n]$ and every $s\in[0,+\infty)$, we have
$ g(k,s,\lambda)\leq|\lambda|\cdot \|a\|_{\infty}s^p+\overline{\lambda}_0 s+f(s).$
Next, on account of $(f_2^0)$, there exists a sequence $\{s_i\}_i\subset(0,1)$ converging to $0$ as $i\to+\infty$ such that
$f(s_i)<-\overline{\lambda}_0 s_i,\ \mbox{for}\ i\in\NN\ \mbox{large enough}.$
Consequently, we have
$g(k,s_i,0)=\overline{\lambda}_0 s_i+f(s_i)<0,$
for $i\in\NN$ large enough and for all $k\in\mathbb{Z}[1,n]$. Thus, due to the continuity of $s\mapsto g(\cdot,s,\cdot)$ we get that there exist three sequences $\{\delta_i\}_i$, $\{\eta_i\}_i$, $\{\lambda_i\}_i\subset(0,1)$ such that,
\begin{eqnarray}\label{eq43}
&&0<\eta_{i+1}<\delta_i<s_i<\eta_i<1,\ \lim\limits_{i\to+\infty}\eta_i=0,
\end{eqnarray}
and for $i\in\NN$ large enough,
\begin{eqnarray}\label{eq44}
&&g(k,s,\lambda)\leq 0,\ \mbox{for all}\ k\in\mathbb{Z}[1,n],\ \lambda\in[-\lambda_i,\lambda_i]\ \mbox{and}\ s\in[\delta_i,\eta_i].
\end{eqnarray}
For any $i\in\NN$ and $\lambda\in[-\lambda_i,\lambda_i]$, let $g_i:\mathbb{Z}[1,n]\times[0,+\infty)\times[-\lambda_i,\lambda_i]\to\RR$ be the function defined by
\begin{eqnarray}\label{eq45}
&&g_i(k,s,\lambda):=g(k,\tau_{\eta_i}(s),\lambda)
\end{eqnarray}
and
$G_i(k,s,\lambda):=\int\limits_0^s g_i(k,t,\lambda)dt,$
for all $k\in\mathbb{Z}[1,n]$ and $s\geq 0$.
In the sequel, let us prove that $c$ given in (\ref{eq42}) and $g_i$ satisfy all the assumptions of Theorem \ref{th5}. Due to relation (\ref{eq3}), it is easy to see that $g_i$ satisfies condition (\ref{eq11}). Also, the assumption (\ref{eq10}) is trivially verified.
Moreover, the regularity of $g$ and the continuity of $\tau_{\eta}$ show that $g_i$ is a Carath\'{e}odory function. Also, thanks to (\ref{eq45}), (\ref{eq22}), the continuity of $s\mapsto g(\cdot,s,\cdot)$ and the Weierstrass Theorem give that $g_i$ satisfies (\ref{eq12}). Finally, (\ref{eq44}) and (\ref{eq45}) yield (\ref{eq13}) for $i$ large enough. Hence, $g_i$ satisfies all the assumptions of Theorem \ref{th5} for $i$ large.
Next, for any $i\in\NN$, let $E_{i,\lambda}:H\to\RR$ be the energy  associated with the problem $(P_{g_i(\cdot,\cdot,\lambda)}^c)$, that is,
\begin{eqnarray}\label{eq46}
E_{i,\lambda}:=E_{c,g_i(\cdot,\cdot,\lambda)},
\end{eqnarray}
where $E_{c,g_i(\cdot,\cdot,\lambda)}$ is the functional given in (\ref{eq14}) with $g=g_i(\cdot,\cdot,\lambda)$. So, Theorem \ref{th5} allows us to deduce that, for $i\in\NN$ sufficiently large and $\lambda\in[-\lambda_i,\lambda_i]$, there exists $u_{i,\lambda}\in W^{\eta_i}$ such that
\begin{eqnarray}\label{eq47}
\min\limits_{u\in W^{\eta_i}}E_{i,\lambda}(u)=E_{i,\lambda}(u_{i,\lambda}),
\end{eqnarray}
\begin{eqnarray}\label{eq48}
u_{i,\lambda}(k)\in [0,\delta_i]\ \mbox{for all}\ k\in\mathbb{Z}[1,n]
\end{eqnarray}
and
\begin{eqnarray}\label{eq49}
u_{i,\lambda}\ \mbox{is a non-negative weak solution of}\ (P_{g_i(\cdot,\cdot,\lambda)}^c).
\end{eqnarray}
Since for $i$ sufficiently large
\begin{eqnarray}\label{eq50}
0\leq u_{i,\lambda}(k)\leq \delta_i<\eta_i,
\end{eqnarray}
for all $k\in\mathbb{Z}[1,n]$ by (\ref{eq43}) and (\ref{eq48}), we get
$g_i(k,u_{i,\lambda}(k),\lambda)=g(k,u_{i,\lambda}(k),\lambda).$
Thus, using (\ref{eq42}) we obviously have that $u_{i,\lambda}$ is a non-negative weak solution of $(P_{\lambda})$, provided $i$ is large and $|\lambda|\leq\lambda_i$.

In the sequel, we  prove that for any $n\in\NN$ problem $(P_{\lambda})$ admits at least $n$ distinct solutions, for suitable values of $\lambda$. We first observe that due to the choice of $c$ and $g_i$ and (\ref{eq50}), the functional $E_{i,\lambda}$ is given by
\begin{eqnarray}\label{eq51}
E_{i,\lambda}(u)=E_{i,0}(u)-\lambda\sum\limits_{k=1}^{n}a(k)\frac{|u(k)|^{p+1}}{p+1},\ \mbox{for any}\ u\in H.
\end{eqnarray}
For $\lambda=0$, the function $g_i(\cdot,\cdot,\lambda)=g_i(\cdot,\cdot,0)$ verifies the hypotheses (\ref{eq10}), (\ref{eq23}), (\ref{eq24}) and (\ref{eq25}). More precisely, $g_i(\cdot,\cdot,0)$ is exactly the function appearing in (\ref{eq29}) and $E_i:=E_{i,0}$ is the energy functional associated with problem $(P_{g_i(\cdot,\cdot,0)}^c)$. Thus by (\ref{eq47})-(\ref{eq49}), the elements $u_i:=u_{i,0}$ also verify
\begin{eqnarray}\label{eq52}
E_i(u_i)=\min\limits_{u\in W^{\eta_i}}E_i(u)\leq E_i(w_i)<0\ \mbox{for all}\ i\in\NN,
\end{eqnarray}
where $w_i\in W^{\eta_i}$ is given in the proof of Theorem \ref{th1}, see for instance (\ref{eq38}).

In the sequel, let $\{\theta_i\}_i$ be an increasing sequence with negative terms such that $\lim\limits_{i\to+\infty}\theta_i=0$. On account of (\ref{eq52}), up to a subsequence, we may assume that
\begin{eqnarray}\label{eq53}
\theta_{i-1}<E_i(u_i)\leq E_i(w_i)<\theta_i,\ \mbox{for}\ i\geq i^*,\ \mbox{with}\ i^*\in\NN.
\end{eqnarray}
Now, for any $i\geq i^*$ let
\begin{eqnarray}\label{eq54}
\lambda_i':=\frac{(p+1)(E_i(u_i)-\theta_{i-1})}{(\|a\|_{\infty}+1)n}\ \mbox{and}\ \lambda_i^{''}:=\frac{(p+1)(\theta_{i}-E_i(w_i))}{(\|a\|_{\infty}+1)n}.
\end{eqnarray}
Note that $\lambda_i'$ and $\lambda_i^{''}$ are strictly positive, due to (\ref{eq53}) and they are independent of $\lambda$.
Now, for any fixed $n\in\NN$, let
$$\Lambda_n:=\min\{\lambda_{i^*+1},\ldots,\lambda_{i^*+n},\lambda_{i^*+1}',\ldots,\lambda_{i^*+n}',\lambda_{i^*+1}^{''},\ldots,\lambda_{i^*+n}^{''}\}.$$
On account of (\ref{eq53}), $\Lambda_n>0$ and it is independent of $\lambda$. Moreover, if $|\lambda|\leq\Lambda_n$, then $|\lambda|\leq\lambda_i$ for any $i=i^*+1,\ldots,i^*+n$. Consequently, for any $\lambda\in\RR$ with $|\lambda|\leq\Lambda_n$, we have that
$u_{i,\lambda}\ \mbox{is a non-negative weak solution of problem}\ (P_{\lambda})$,
for any $i=i^*+1,\ldots,i^*+n$.
In the sequel, we show that these solutions are distinct. For this purpose, note that $u_{i,\lambda}\in W^{\eta_i}$ by (\ref{eq50}) and so for any $\lambda\in\RR$ with $|\lambda|\leq\Lambda_n$ we have
\begin{eqnarray}\label{eq55}
E_i(u_i)=\min\limits_{u\in W^{\eta_i}}E_i(u)\leq E_i(u_{i,\lambda}).
\end{eqnarray}
Thus by (\ref{eq51}) and (\ref{eq55}), for any $\lambda$ with $|\lambda|\leq\Lambda_n$ we obtain
\begin{equation}\label{eq56}
E_{i,\lambda}(u_{i,\lambda})\geq E_i(u_i)-\frac{|\lambda|}{p+1}\|a\|_{\infty}\eta_i^{p+1}n
\geq E_i(u_i)-\frac{\lambda_i'}{p+1}\|a\|_{\infty}n>\theta_{i-1},
\end{equation}
for any $i=i^*+1,\ldots,i^*+n$, due to (\ref{eq43}), (\ref{eq50}), the choice of $\Lambda_n$ and the definition of $\lambda_i'$.
On the other hand, by (\ref{eq51}), (\ref{eq52}) and using the fact that $\|w_i\|_{\infty}=\tilde{s}_i\leq\delta_i<\eta_i<1$ (see the proof of Theorem \ref{th1}), for any $\lambda$ with $|\lambda|\leq\Lambda_n$ we deduce that
\begin{equation}\label{eq57}
E_{i,\lambda}(u_{i,\lambda})\leq E_i(w_i)+\frac{|\lambda|}{p+1}\|a\|_{\infty}n
\leq E_i(w_i)+\frac{\lambda_i^{''}}{p+1}\|a\|_{\infty}n< \theta_{i},
\end{equation}
for any $i=i^*+1,\ldots,i^*+n$, again thanks to the choice of $\Lambda_n$ and the definition of $\lambda_i^{''}$.
In conclusion, by (\ref{eq56}), (\ref{eq57}) and the properties of $\{\theta_i\}_i$, we deduce that for every $i=i^*+1,\ldots,i^*+n$ and $\lambda\in[-\Lambda_n,\Lambda_n]$, we have
\begin{eqnarray}\label{eq58}
\theta_{i-1}< E_{i,\lambda}(u_{i,\lambda})<\theta_i<0,
\end{eqnarray}
which yields that
$E_{1,\lambda}(u_{1,\lambda})<\ldots<E_{n,\lambda}(u_{n,\lambda})<0.$
But $u_{i,\lambda}\in W^{\eta_1}$ for every $i=i^*+1,\ldots,i^*+n$, so $E_{i,\lambda}(u_{i,\lambda})=E_{1,\lambda}(u_{i,\lambda})$, see relation (\ref{eq45}). Therefore, from above, we obtain that for every $\lambda\in[-\Lambda_n,\Lambda_n]$,
$E_{1,\lambda}(u_{1,\lambda})<\ldots<E_{1,\lambda}(u_{n,\lambda})<0=E_{1,\lambda}(0).$
These inequalities show that the elements $u_{1,\lambda},\ldots,u_{n,\lambda}$ are all distinct and non-trivial, provided $\lambda\in[-\Lambda_n,\Lambda_n]$.

Finally, it remains to prove conclusion (\ref{eq6}). For this, by (\ref{eq43}), (\ref{eq50}), (\ref{eq51}), (\ref{eq58}) and the continuity of $f$ we have that
\begin{eqnarray*}
\frac{1}{2}\|u_{i,\lambda}\|^2&<&\theta_i+\frac{|\lambda|}{p+1}\|a\|_{\infty}\delta_i^{p+1}n+\sum\limits_{k=1}^{n}\int\limits_0^{\delta_i}|f(s)|ds\\
&<&\frac{\Lambda_n}{p+1}\|a\|_{\infty}\delta_i n+n\max\limits_{s\in[0,1]}|f(s)|\delta_i,
\end{eqnarray*}
for any $i=i^*+1,\ldots,i^*+n$ and $|\lambda|\leq\Lambda_n$. Hence, we obtain
$\|u_{i,\lambda}\|\leq\tilde{c}\delta_i^{1/2},$
where $\tilde{c}=2^{-1}\left(\frac{\Lambda_n}{p+1}\|a\|_{\infty}n+n\max\limits_{s\in[0,1]}|f(s)|\right)>0$.
Since $\delta_i\to 0$ as $i\to+\infty$, without loss of generality, we may assume that
\begin{eqnarray}\label{eq59}
\delta_i\leq\min\{\tilde{c}^{-2},1\}\frac{1}{i^2},
\end{eqnarray}
which gives that
$\|u_{i,\lambda}\|\leq\frac{1}{i},$
for any $i=i^*+1,\ldots,i^*+n$, provided $|\lambda|\leq\Lambda_n$.
In conclusion, by (\ref{eq50}) and (\ref{eq59}) we obtain that
$\|u_{i,\lambda}\|_{\infty}\leq\frac{1}{i^2}<\frac{1}{i},$
for any $i=i^*+1,\ldots,i^*+n$, with $|\lambda|\leq\Lambda_n$.
This concludes the proof of Theorem \ref{th2}.
\qed


\medskip
\textbf{Acknowledgements.} V. R\u adulescu acknowledges the support through the grant CNCS-UEFISCDI PCCA-23/2014.
V. R\u adulescu also thanks the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme \textit{Free Boundary Problems and Related Topics}, where work on this paper was undertaken.

\begin{thebibliography}{99}

\bibitem{2} {\sc R. P. Agarwal, K. Perera, D. O'Regan}, {\em Multiple positive solutions of singular and nonsingular discrete problems via variational methods}, Nonlinear Anal. \textbf{58} (2004), 69-73.

\bibitem{BERR} {\sc C. Bereanu, J. Mawhin}, {\em Existence and multiplicity results for nonlinear difference equations with Dirichlet boundary conditions}, Math. Bohemica \textbf{131} (2006), 145-160.

\bibitem{MEJ} {\sc M. Galewski, J. Smejda},  {\em On the dependence on parameters for mountain pass solutions
of second order discrete BVP's}, Applied Mathematics and Computation \textbf{219} (2013), 5963-5971.

\bibitem{8} {\sc A. Krist\'{a}ly, Gh. Moro\c sanu}, {\em New competition phenomena in Dirichlet problems}, J. Math. Pures Appl. (9) \textbf{94} (2010), 6, 555-570.

\bibitem{15} {\sc G. Molica Bisci, V. R\u adulescu, R. Servadei}, {\em Low and high energy solutions of nonlinear elliptic oscillatory problems}, C. R. Acad. Sci. Paris, Ser. I \textbf{352} (2014), 117-122.

\bibitem{15bis} {\sc G. Molica Bisci, V. R\u adulescu, R. Servadei}, {\em Competition phenomena for elliptic equations involving a general operator in divergence form}, Analysis and Applications
    (DOI:10.1142/S0219530515500116).

\bibitem{16} {\sc F. Obersnel, P. Omari}, {\em Positive solutions of elliptic problems with locally oscillating nonlinearities}, J. Math. Anal. Appl. \textbf{323} (2) (2006), 913-929.

\bibitem{17} {\sc P. Omari, F. Zanolin}, {\em Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential}, Comm. Partial Differential Equations \textbf{21} (1996), 721-733.

\bibitem{OTTA} {\sc J. Otta, P. Stehlik}, {\em Multiplicity of solutions for discrete problems with double-well potentials},  Electronic Journal of Differential Equations, \textbf{186} Vol. 2013 (2013), pp. 1-14.

\bibitem{14} {\sc V. R\u adulescu}, {\em Nonlinear elliptic equations with variable exponent: old and new}, Nonlinear Analysis, {\bf 121} (2015), 336-369.

\bibitem{radrep} {\sc V. R\u adulescu, D. Repov\v{s}}, {\em  Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis}, CRC Press, Taylor \& Francis Group, Boca Raton FL, 2015.

\end{thebibliography}
\end{document}

%-----------------------------------------------------------------------
% End of article.tex
%-----------------------------------------------------------------------
