\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 186, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2013/186\hfil Multiplicity of solutions]
{Multiplicity of solutions for discrete problems with double-well potentials}
\author[J. Otta, P. Stehl\'ik \hfil EJDE-2013/186\hfilneg]
{Josef Otta, Petr Stehl\'ik} % in alphabetical order
\address{Josef Otta \newline
University of West Bohemia in Pilsen \\
Univerzitni 22 \\
312 00 Czech Republic}
\email{josef.otta@gmail.com}
\address{Petr Stehl\'ik \newline
University of West Bohemia in Pilsen \\
Univerzitni 22 \\
312 00 Czech Republic}
\email{pstehlik@kma.zcu.cz}
\thanks{Submitted April 29, 2013. Published August 23, 2013.}
\subjclass[2000]{39A12, 34B15}
\keywords{Discrete problem; boundary value problem; $p$-Laplacian;
\hfill\break\indent partial difference equation; bistable equation;
double-well; saddle-point; variational method;
\hfill\break\indent periodic problem;
Neumann boundary condition}
\begin{abstract}
This article presents some multiplicity results for a general class
of nonlinear discrete problems with double-well potentials.
Variational techniques are used to obtain the existence of
saddle-point type critical points. In addition to simple discrete
boundary-value problems, partial difference equations as well as problems
involving discrete $p$-Laplacian are considered.
Also the boundedness of solutions is studied and possible applications,
e.g. in image processing, are discussed.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
In this article, we present the existence and multiplicity results for
a general class of discrete problems with the so-called double-well
nonlinearities. In the first part of this work we study the problem
\begin{equation} \label{e:matrix}
Ax+F(x)=0, \quad x \in \mathbb{R}^{N},
\end{equation}
in which $A$ is a symmetric positive semidefinite matrix and $F$ is a continuous
nonlinear vector function whose entries have the form $f_i(x_i):=g_i'(x_i)$
with $g_i$'s being the double-well potentials (one could list $g(s)=(1-s^2)^2$
as a typical example). Later, we generalize
\eqref{e:matrix} and replace $A$ by discrete $p$-Laplacian.
Problems of this type are known as Allen-Cahn or bi-stable equations and have
been extensively analyzed in the continuous settings. Their history reaches
back to the pioneering work by Allen, Cahn \cite{allencahn} which examined
the coarsening of binary alloys. Such a process is represented
by a boundary-value problem
\begin{equation}\label{e:evol-AC}
\begin{gathered}
u_t(x,t)=\varepsilon^2 \Delta u(x,t) - g'(u(x,t)),\quad x \in \Omega,\; t>0,\\
\frac{\partial u(x,t)}{\partial n}=0,\quad\text{for } x \text{ on } \partial \Omega,\ t>0,
\end{gathered}
\end{equation}
in which $u$ is a concentration rate of one of two components in the alloy
and the parameter $\varepsilon$ corresponds to the interfacial energy.
Later, similar problems have appeared in the models of phase changes at
the transition temperatures (see Fusco, Hale \cite{fusco}), in the analysis
of crystals' growth (e.g. Wheeler et al. \cite{krystaly}) and
most recently in the image processing (e.g. Choi et al. \cite{choi}).
The image processing application mentioned above serves as important motivation for the study of
discrete counterparts of \eqref{e:evol-AC}. In this area, similar models are studied and applied for
the object identification, the so-called level set segmentation. The function $u$ corresponds to
the grayscale values (rescaled to the interval $[-1, 1]$) in an $N\times N$-pixel image with
discrete coordinates $(x,y)$ (see e.g. Bene\v{s} et al. \cite{benes} for a nice description of this
process). Apparently, stationary points of such discrete evolutionary equation are solutions of
\eqref{e:matrix}.
To authors' best knowledge, discrete problems with double-well potentials have
not been analyzed so far. Recently, several papers have studied the
solvability of nonlinear discrete problems \eqref{e:matrix},
either in this general form or with specific operators $A$
(e.g. Bai, Zhang \cite{bai}, Galewski, Smejda
\cite{galewski}, Mih\u{a}ilescu, R\u{a}dulescu, Tersian \cite{miha}
or Yang, Zhang \cite{yang}). Applying mountain-pass or linking theorems, they get
conditions on the limit behaviour of functions $f_i$'s near the origin and
in the infinity. Other techniques have been used to get interesting results
for general (e.g. partial) difference operators
(e.g. Bereanu, Mawhin \cite{bereanu}, Galewski, Orpel \cite{galewski2}
or Stehl\'{\i}k \cite{stehlik1}). One
could find similar manuscripts in which the linear discrete operator $A$
is replaced by $p$-Laplacian, e.g. Agarwal et al. \cite{agaral} or
Cabada et al. \cite{cabada}.
Working with a special (and thus narrower) class of problems we are able to get finer
assumptions on the nonlinearities. At the same time, we are trying to preserve the generality by
considering a wide class of discrete operators. Our main tools include the Saddle point theorem and
coercivity variational results.
First, we briefly introduce basic notation and the necessary functional-analytic support (Sections
\ref{s:pre} and \ref{s:not}). Then, in Section \ref{s:3sol}, we prove the existence of at least
three solutions of the problem \eqref{e:matrix} with general double-well potentials. Consequently,
we extend this result to the existence of at least five solutions for a class of double-wells with
special behaviour at the origin (Section \ref{s:5sol}). Finally, we generalize these results
for the case in which the operator $A$ is replaced by $p$-Laplacian (Section \ref{s:plap}). Several
examples are included to illustrate the existence results.
\section{Preliminaries}\label{s:pre}
We study variational formulations of discrete problems and our main tool is
the Saddle point theorem. Thus, we restrict our attention to functionals
satisfying the Palais-Smale condition.
\begin{definition}[{\cite[Definition 1.16]{Willem}}] \rm
Let $X$ be Banach space, $\mathcal{J}\in C^1(X,\mathbb{R})$ and $c\in \mathbb{R}$.
Then the functional $\mathcal{J}$ satisfies the \emph{Palais-Smale condition} if any
sequence $\{u_n\}\subset X $ such that
\begin{align} \label{e:PS} %\tag{PS1}
\mathcal{J}(u_n)\to c,\quad \mathcal{J}'(u_n)\to 0
\end{align}
has a convergent subsequence. We call this sequence PS-sequence.
\end{definition}
We use original version of the Saddle point theorem proven by Paul
Henry Rabinowitz in 1978.
\begin{theorem}\label{thm:saddlePt}
Let $X=Y\oplus Z$ be real Banach space with $\mathrm{dim}(Y)<\infty$
equipped with the norm $\|\cdot\|$.
For some $\rho>0$, we define
$$
M=\{u\in Y:\|u\|\leq\rho\},\quad M_0=\{u\in Y:\|u\|=\rho\}.
$$
Let $\mathcal{J}$ be a real functional, $\mathcal{J}\in C^1(X,\mathbb{R})$, such that
\begin{equation}
\label{e:saddle} % \tag{G}
\inf_{u\in Z}\mathcal{J}(u)>\max_{u\in M_0} \mathcal{J}(u).
\end{equation}
If $\mathcal{J}$ satisfies \eqref{e:PS} with
\[
c:=\inf_{\gamma\in \Gamma}\,\max_{u\in M} \mathcal{J}(\gamma(u)),\quad
\Gamma:=\big\{ \gamma\in C(M,X):\gamma|_{M_0}=I \big\}
\]
then $c$ is the critical value of $\mathcal{J}$.
\end{theorem}
The proof of the above lemma can be found in \cite[Theorem 2.11]{Willem}.
We also use the following statement about weakly coercive functionals in
$\mathbb{R}^N$ whose proof could be found in
\cite[Thoerem 1.9]{rockafellar}.
\begin{theorem} \label{cor:min}
Let $F:\mathbb{R}^N\mapsto\mathbb{R}$ be a continuous and weakly coercive
functional. Then $F$ is bounded from below on $H$ and there exists a
minimizer $u_0\in H$ such that
$F(u_0) =\min_{u\in H} F(u)$. Moreover, if the Fr\'{e}chet
derivative $F'(u_0)$ exists then
$F'(u_0)=o$.
\end{theorem}
\section{Notation and assumptions - survey}\label{s:not}
Let $x=(x_1,x_2,\ldots, x_N)^T$ be a vector in $\mathbb{R}^N$, equipped with the
standard Euclidean norm
$$
\|x\|=\Big(\sum_{i=1}^{N}|x_i|^2\Big)^{1/2}.
$$
Vector $o$ denotes the trivial solution; i.e.,
$o=(0,0,\ldots, 0)^T$.
We study problem \eqref{e:matrix}. Throughout the paper we assume that
an $N\times N$ matrix $A$ satisfies assumptions:
\begin{itemize}
\item[(A1)] $A$ is symmetric and positive semidefinite,
\item[(A2)] multiplicity of eigenvalue $\lambda_1=0$ is one,
\item[(A3)] the corresponding first eigenvector is
$e_1 = (1,1,\dots,1)^\mathrm{T}$.
\end{itemize}
The vector function $F:\mathbb{R}^N\mapsto \mathbb{R}^N$ has components
$f_i(x_i)$
\[
F(x)=\begin{pmatrix}
f_1(x_1)\\
f_2(x_2)\\
\dots \\
f_{N}(x_N)
\end{pmatrix}
=\begin{pmatrix}
f(1,x_1) \\
f(2,x_2) \\
\dots \\
f(N,x_N)
\end{pmatrix},
\]
with integrable $f_i$, whose potentials are denoted by $g_i$,
\[
g_i'(s) = f_i(s).
\]
Consequently, the potential $G: \mathbb{R}^N \mapsto \mathbb{R}$
corresponding to $F$ is
\[
\nabla G(x) = F(x),\quad G(x)=\sum_{i=1}^{N} g_i(x_i).
\]
As mentioned above, we work with the double-well potentials; i.e.,
we assume that:
\begin{itemize}
\item[(G1)] (evenness) for all $i s\in\mathbb{R}$: $g_i(s)=g_i(-s)$,
\item[(G2)] for all $i$: $g_i(0)>0$, $g_i(\pm 1) = 0$,
\item[(G3)] (non-negativity) for all $i, s\in\mathbb{R}$: $g_i(s) \geq 0$,
\item[(G4)] (weak coercivity) for all $i$:
$\lim_{s\to\infty} g_i(s) = \infty$,
\item[(G5)] for all $i$: $g_i\in C^1(\mathbb{R})$.
\end{itemize}
Under these assumptions solutions of the problem \eqref{e:matrix}
are critical points of the functional $\mathcal{J}:\mathbb{R}^n\mapsto \mathbb{R}$,
\begin{equation}\label{e:energy}
\mathcal{J}(u):=\frac{1}{2}\langle Au,u \rangle+\sum_{i=1}^N g_i(u_i)
= \frac{1}{2} u^{\mathrm{T}} A u + G(u).
\end{equation}
Thus, we study the existence and multiplicity of the critical points of $\mathcal{J}$
in Sections \ref{s:3sol}--\ref{s:5sol}.
To illustrate the content of the set of matrices which satisfy (A1)--(A3)
let us include a couple of examples at this stage.
\begin{example}\label{ex:periodic} \rm
Let us consider a second-order periodic discrete problem
\begin{equation}\label{e:ex:periodic}
\begin{gathered}
-\Delta^2 x_{i-1} + f_i(x_i) = 0, \quad i=1,\ldots,N \\
x_0=x_N,\quad \Delta x_0=\Delta x_N.
\end{gathered}
\end{equation}
One could rewrite this problem as an equation in $\mathbb{R}^N: Ax+F(x)=0$
with
\begin{equation}\label{e:ex:periodic:matrix}
A= \begin{bmatrix}
2 & -1 &&&-1\\
-1 & 2 & -1 &&&\\
&\ddots&\ddots&\ddots&\\
&&-1&2& -1 \\
-1&&&-1&2
\end{bmatrix}, \quad
F(x) = \begin{pmatrix} f_1(x_1) \\ f_2(x_2) \\ \ldots \\ f_N(x_N)
\end{pmatrix}
\end{equation}
Matrix $A$ satisfies assumptions (A1)--(A3) (see e.g. \cite{agarwal}).
\end{example}
The second, slightly more complicated, example considers a partial difference
equation on a square.
\begin{example}\label{ex:pde}
Let us study a two-dimensional nonlinear Poisson equation coupled with
Neumann boundary conditions
\begin{equation}\label{e:ex:pde}
\begin{gathered}
-\Delta^2_1 y_{k-1,l} - \Delta^2_2 y_{k,l-1} + f(y_{k,l}) = 0, \quad
k,l=1,\ldots,N \\
0 = \Delta_1 y_{0,l} = \Delta_1 y_{N,l}
= \Delta_2 y_{k,0} = \Delta_2 y_{k,N},
\end{gathered}
\end{equation}
where $\Delta_i$ denotes the standard difference operator with respect to
the $i$-th variable. Rearranging $y_{k,l}$ to the vector
$x=\left( y_{1,1},\ldots,y_{1,N},y_{2,1},\ldots,y_{N-1,N},y_{N,1},
\ldots,y_{N,N} \right)^T$ one could rewrite the boundary problem
\eqref{e:ex:pde} to the matrix equation \eqref{e:matrix} with A being
a block tridiagonal matrix having dimension $N^2\times N^2$,
\[
A= \begin{bmatrix}
B_1 & -I &&&\\
-I & B_2 & -I &&&\\
&\ddots&\ddots&\ddots&\\
&&-I&B_{N-1}& -I \\
&&&-I&B_N
\end{bmatrix},
\]
where $I$ is the $N\times N$ identity matrix and $N\times N$ matrices
$B_i$ have the form
\begin{gather*}
B_1 = B_N= \begin{bmatrix}
2 & -1 &&&&\\
-1 & 3 & -1 &&&&\\
& -1 & 3 & -1 &&&\\
&&\ddots&\ddots&\ddots&\\
&&&-1&3& -1 \\
&&&&-1&2
\end{bmatrix},
\\
B_k = \begin{bmatrix}
3 & -1 &&&&\\
-1 & 4 & -1 &&&&\\
& -1 & 4 & -1 &&&\\
&&\ddots&\ddots&\ddots&\\
&&&-1&4& -1 \\
&&&&-1&3
\end{bmatrix},
\end{gather*}
for $k=2,\ldots,N-1$. Matrix $A$ satisfies (A1)--(A3) (see e.g. \cite{cheng}).
\end{example}
\section{Saddle-point geometry - three solutions}\label{s:3sol}
To establish the saddle-point geometry from Theorem \ref{thm:saddlePt},
we decompose $\mathbb{R}^N$:
\[
\mathbb{R}^N = Y \oplus Z.
\]
The subspaces $Y$ and $Z$ are generated by eigenvectors $e_i$ of
matrix $A$ in the following way
\begin{equation}\label{e:RN-dec}
Y = \operatorname{span} \{ e_1 \},\quad
Z=\operatorname{span} \{ e_2,e_3\ldots,e_N\}.
\end{equation}
\begin{lemma}\label{lem:3sol}
Let $A$ be a matrix satisfying {\rm (A1)--(A3)},
$G:\mathbb{R}^N\to\mathbb{R}$ be a function satisfying
{\rm (G1)--(G5)}. Then there exist at least three
solutions of equation \eqref{e:matrix}, with at least one being
the saddle-point type critical point of the functional $\mathcal{J}$.
\end{lemma}
\begin{proof}
The functional $\mathcal{J}$ is weakly coercive and non-negative. Indeed,
the positive semidefiniteness
(A1) of $A$ and the limit behavior (G4) of $G$ imply $\mathcal{J}(u)\leq \sum_{i=1}^N G(u_i)\to
\infty$. The non-negativity of $\mathcal{J}$ follows from the semi-definiteness (A1)
of $A$ and the non-negativity (G3) of $G$.
Thanks to the first eigenvector assumption (A3) and the fact that
$g_i(\pm 1) = 0$ (assumption (G2)), one can observe that the constant
functions $e_1=(1,1,\dots,1)^\mathrm{T}$ and $-e_1$ are
global minimizers of $\mathcal{J}$ and consequently trivial solutions
of \eqref{e:matrix}, $\mathcal{J}(e_1)=0$.
Obviously, $e_1, -e_1 \in Y$, we define $M_0:=\{-e_1,e_1\}$.
We prove the saddle-point geometry of functional $\mathcal{J}$ by contradiction.
Due to the continuity and weak coercivity of $\mathcal{J}$ there exists a minimizer
$\tilde{u}\in Z$ such that
$\mathcal{J}(\tilde{u})=\inf_{u\in Z}\mathcal{J}(u)$, cf. Theorem \ref{cor:min}.
Let us assume that $\mathcal{J}(\tilde{u})\leq \mathcal{J}(e_1)$.
But by the definition of $Z$ and (G2),
$\mathcal{J}(\tilde{u})\geq \frac{1}{2} \langle A \tilde{u},\tilde{u} \rangle>0$
if $\tilde{u}\in Z\setminus\{o\}$.
Moreover, $\mathcal{J}(o)=\sum_{i=1}^N g_i(0)>0$ holds true. This implies that
$\mathcal{J}(\tilde{u})> \mathcal{J}(e_1)$.
It remains to show that the Palais-Smale condition \eqref{e:PS} holds.
This is a straightforward consequence of the weak coercivity of $\mathcal{J}$.
Let us assume that $\{u_n\}$ is a PS-sequence. Then
there exists some $K>0$ such that $\|{u_n\|}0$, $K>0$ $\beta\in(1,2)$ such that
for all $s\in \mathbb{R}$ and all $i$: $|s|<\delta$
implies $ g_i(s)\leq g_i(0)-{K}|s|^\beta$.
\end{itemize}
This assumption ensures the maximality of $\mathcal{J}$ at the
origin $o=(0,0,\dots,0)^T$.
\begin{lemma}\label{lem:maximizer}
Let $G$ satisfy {\rm (G6)}. Then $o$ is a local maximizer of
$\mathcal{J}$ in $\mathbb{R}^{N}$.
\end{lemma}
\begin{proof}
Let us choose a fixed nonzero $u\in \mathbb{R}^{N}$ with $\|u\|=1$. Then
\begin{align*}
\mathcal{J}(tu)
&\leq \frac{1}{2}t^2 \langle Au,u \rangle+\sum_{i=1}^N
\Bigl(g_i(0)- K t^\beta |u_i|^\beta\Bigr)\\
&=c_1 t^2-c_2 t^\beta+c_3 =: h(t),
\end{align*}
where the constants are such that $c_1\geq 0$, $c_2,c_3>0$.
Then $h'(t)<0$ is equivalent to
\begin{equation}\label{e:proof_maximizer}
\frac{2 c_1}{\beta c_2}0$
such that the inequality
\eqref{e:proof_maximizer} is satisfied for all $t\in (0,t_0)$.
Let us define $\gamma(u)= \frac{2 c_1}{\beta c_2}$ for given $u$.
Then the function $\gamma$ is continuous and attains its maximal
value $\gamma_{\rm max}$ on the compact set
$\mathcal{S}=\{u\in X: \|u\|=1\}$. If we choose $t_0$ to be a positive
constant satisfying $t_0<\gamma_{\rm max}$ then the estimate
\eqref{e:proof_maximizer} holds uniformly for all $t\in (0,t_0)$
and for all $u\in \mathcal{S}$. Consequently, the origin $o$ is a
local maximizer of $\mathcal{J}$.
\end{proof}
This result yields directly the existence of at least five solutions.
\begin{theorem}\label{thm:5sol}
Let $A$ be a matrix satisfying {\rm (A1)--(A3)} and let $G$ satisfy
{\rm (G1)--(G6)}. Then there exist at least five solutions of equation
\eqref{e:matrix}.
\end{theorem}
\begin{proof}
Firstly, note that Lemma \ref{lem:maximizer} implies that $o$ is a local
maximizer of $\mathcal{J}$.
Secondly, recall that $e_1=(1,1,\dots,1)^\mathrm{T}$ and $-e_1$ are
global minimizers of $\mathcal{J}$.
Finally, we could apply the arguments from the proof of Lemma \ref{lem:3sol}
to get the existence of the saddle-point type critical point $\tilde{u}$.
Considering Lemma \ref{lem:maximizer}, we see
that $\tilde{u}\neq o$. Taking into account the evenness of $\mathcal{J}$,
$-\tilde{u}$ is a
saddle-point type critical point of $\mathcal{J}$ as well.
\end{proof}
Again, we provide a simple example to demonstrate this result,
especially the assumption (G6).
\begin{example}\label{ex:5sol} \rm
For the sake of brevity and generality, we abstract from a specific discrete
operator (those from Examples \ref{ex:periodic} and \ref{ex:pde} could be
used immediately as well as many others) and
concentrate on the nonlinearity. In order to illustrate the assumption (G6)
we consider potentials
\begin{equation}\label{e:ex:5sol}
g_i(s) = \left|1-| s |^a \right|^b,
\end{equation}
with $a,b>1$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1} % gr1a.pdf
\end{center}
\caption{Functions $g_i(s)=|1-|s|^a|^b$ with $a=2$ and $b=1.1$
(dot-dashed line), $b=2$
(solid line) and $b=6$ (dashed line)}
\label{f:gr1}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2} % gr2a.pdf
\end{center}
\caption{Functions $g_i(s)=|1-|s|^a|^b$ with $b=2$ and $a=1.1$
(dot-dashed line), $a=2$ (solid line) and $a=6$ (dashed line)}
\label{f:gr2}
\end{figure}
Differentiating, we see that these correspond to
$f_i(s) = -ab | s|^{a-1} \operatorname{sign}(s) \big|1-| s |^a\big|^{b-1}
\operatorname{sign}(1-| s |^a)$.
Obviously, $G$ satisfies (G1)--(G5), hence Lemma \ref{lem:3sol}
yields the existence of at least three solutions of equation \eqref{e:matrix}
for any difference operator generating a matrix $A$ satisfying (A1)--(A3).
Let us examine the condition (G6) for this case. We choose $\delta=1/2$.
Since the functions $g_i$'s are even, we consider only $s\in [0,1/2)$
which simplifies the corresponding derivatives.
Then $g'_i(s)=f_i(s) = -ab (1-s^{a})^{b-1}s^{a-1}$ and the inequality
$g_i(s)\leq g_i(0)-K|s|^\beta$ holds if $f_i(s) \leq -K \beta s^{\beta-1}$
for all $s\in [0,1/2)$. But one
could make the following estimate for $a\in(1,2)$ and $b>1$:
\begin{align*}
g'_i(s)=f_i(s) &= -ab (1-s^{a})^{b-1}s^{a-1} \\
&\leq - ab (1-2^{-a})^{b-1}s^{a-1}
\leq - K\beta s^{\beta-1},
\end{align*}
if we choose $K\leq b(1-2^{-a})^{b-1}$ and $\beta=a$, holds for all
$s\in [0,1/2)$. Hence the assumption (G6) is satisfied for $a\in(1,2)$ and $b>1$.
Consequently, Theorem \ref{thm:5sol} provides the existence of at least
five solutions of problem \eqref{e:matrix} with
$ g_i(s)=\left|1-| s |^a \right|^b$ with $a\in(1,2)$, $b>1$.
\end{example}
Under additional assumptions on the constant $K$ in (G6), we can
extend this result also to the case with $\beta=2$.
\begin{theorem}\label{thm:5sol:2}
Let $A$ be a matrix satisfying {\rm (A1)--(A3)} and let $G$ satisfy
{\rm (G1)--(G5)} and
\begin{itemize}
\item[(G6')] Thee exist $\delta >0$, $K>\frac{\lambda_{\rm max}}{2}$ such that
for all $s\in \mathbb{R}$ and all $i$: $|s|<\delta$ implies
$g_i(s)\leq g_i(0)-{K}|s|^2$,
\end{itemize}
with $\lambda_{\rm max}$ denoting the largest eigenvalue of $A$.
Then there exist at least five
solutions of the equation \eqref{e:matrix}.
\end{theorem}
\begin{proof}
Following the proof of Lemma \ref{lem:maximizer} we can make the
following estimate for arbitrary vector $u$, $\|u\|=1$.
\begin{align*}
\mathcal{J}(tu)&\leq \frac{1}{2}t^2 \langle Au,u \rangle+\sum_{i=1}^N \Bigl(g_i(0)- K t^2 |u_i|^2\Bigr)\\
&\leq\frac{1}{2} t^2 \lambda_{\rm max}-K t^2 +\sum_{i=1}^N g_i(0) \\
&=\left( \frac{\lambda_{\rm max}}{2}-K\right)t^2 + \sum_{i=1}^N g_i(0) =: h(t).
\end{align*}
Under the assumption $\mathrm{(G6)'}$, the coefficient by $t^2$ is negative
which implies that
$h(t)$, and $\mathcal{J}(tu)$ have local maximizers at the origin $o$.
The arbitrary choice of $u$ implies
the existence of a pair of non-trivial saddle-point type critical points
$\tilde{u}$ and $-\tilde{u}$.
\end{proof}
\begin{remark}\label{rem:5sol} \rm
In the spirit of Remark \ref{rem:3sol} one could rephrase the statements of
Theorems \ref{thm:5sol} and \ref{thm:5sol:2} in the following way.
Under the assumptions $\mathrm{(G6)}$ or
$\mathrm{(G6)'}$ there exist, aside from trivial solutions $o$, $e_1$
and $-e_1$, at least two nontrivial solutions, both being saddle points of $\mathcal{J}$.
\end{remark}
\begin{example}\label{ex:5solbeta2} \rm
In this example, we study the behaviour of functions $g_i$'s at $0$
and its consequences for solution multiplicity for $\beta=2$.
Let us assume that nonlinear terms $g_i$ are defined as
$g_i(s)=\frac{1}{\varepsilon^2}|1-|s|^a|^b$ with $\varepsilon>0$.
For a given $\delta>0$ we could repeat the procedure of
Example \ref{ex:5sol} to show that $\mathrm{(G6)'}$ is satisfied for
$$
K \leq \frac{b}{\varepsilon^2}(1-\delta^a)^{b-1} \quad \beta = a=2.
$$
Considering the arbitrary choice of $\delta$, we could see that the
assumption $\mathrm{(G6)'}$ is satisfied for any
$$
\varepsilon<\sqrt{\frac{2b}{\lambda_{\rm max}}},\quad\text{or equivalently}\quad
b>\frac{\lambda_{\rm max}\varepsilon^2}{2}
$$
which, if satisfied, guarantees the existence of at least five solutions
of problem \eqref{e:matrix}.
\end{example}
We conclude this section with the study of a specific boundary-value problem.
\begin{example}\label{ex:5sol:periodic} \rm
Let us consider the one-dimensional discrete problem with periodic
boundary conditions (see \eqref{e:ex:periodic})
\begin{equation}\label{e:ex:5sol:periodic}
\begin{gathered}
-\Delta^2 x_{i-1} + g_i'(x_i) = 0, \quad i=1,\ldots,N \\
x_0=x_N,\ \Delta x_0=\Delta x_N,
\end{gathered}
\end{equation}
with $g_i(s)=\frac{1}{\varepsilon^2}|1-|s|^a|^b$.
The eigenvalues of the corresponding matrix $A$
(cf. \eqref{e:ex:periodic:matrix}) satisfy (see e.g. \cite[Chapter 11]{agarwal})
\[
\lambda_{\rm max}=\begin{cases}
4 & \text{if $N$ is even,} \\
4\sin^2\left( \frac{N-1}{N}\frac{\pi}{2}\right)
& \text{if $N$ is odd.}
\end{cases}
\]
Hence, the results listed in this section imply that
problem \eqref{ex:5sol:periodic} has at
least five solutions if either $a\in(1,2)$ and $b>1$
(Example \ref{ex:5sol}) or $a=2$ and $b>\frac{\lambda_{\rm max}\varepsilon^2}{2}$
(Example \ref{ex:5solbeta2}).
\end{example}
\section{Application to the discrete $p$-Laplacian}\label{s:plap}
In this section we extend the results to the problems with $p$-Laplacian;
i.e., we consider a situation in which the left-hand side discrete
operator is not linear.
Let $p>1$, and define Banach space
$$
X= \bigl\{ u=(u_0,u_1,\ldots,u_{N+1})^T:u_0=u_1,\ u_N=u_{N+1} \bigr\}
\subset\mathbb{R}^{N+2}
$$
equipped with norm
$$
\|u\|_p=\Big(\sum_{i=0}^{N+1}|u_i|^p\Big)^{1/p}.
$$
We define a nonlinear functional $\mathcal{J}_p: u \in X \mapsto \mathcal{J}_p(u)\in \mathbb{R}$ by
\begin{equation}\label{e:plapl_energy}
\mathcal{J}_p(u):=\sum_{i=1}^N
\Bigl( \frac{|\Delta u_{i-1}|^p}{p}+g_i(u_i) \Bigr)
+ \frac{|\Delta u_N|^p}{p}.
\end{equation}
The critical point $u$ of \eqref{e:plapl_energy}
corresponds to the solution of the problem
\begin{equation}\label{e:plapl}
\begin{gathered}
-\Delta(\varphi_p(\Delta u_{i-1}))+f_i(u_i)=0 \quad \text{for } i=1,\dots,N,\\
\Delta u_0=\Delta u_{N}=0,
\end{gathered}
\end{equation}
where $\varphi_p: s\mapsto |s|^{p-2}s$ for $s\neq 0$ and $\varphi_p(0):=0$.
\begin{remark} \label{rem:triv_plapl} \rm
One can observe that $e_1=(1,1,\dots,1)^T$, $-e_1$ and $o$ are solutions of
\eqref{e:plapl}.
\end{remark}
As in the linear case we proceed by proving the existence of at least three
solutions under the assumption (G1)--(G5). In order to obtain the existence
of at least five solutions, we modify (G6) and assume that
\begin{itemize}
\item[(G6P)] There exist $\delta>0$, $K>0$, $\beta\in(1,p)$ such that
for all $s \in \mathbb{R}$ and all $i$: $|s|<\delta$ implies
$g_i(s)\leq g_i(0)-{K}|s|^\beta$
\end{itemize}
holds instead. Assumption {\rm (G6P)} plays equivalent role as (G6)
in the linear case. It ensures that $\mathcal{J}_p$ attains its local maximum at $o$.
\begin{lemma}\label{lem:maximizer_plap}
Let $G$ satisfy {\rm (G6P)}. Then $o$ is a local maximizer of $\mathcal{J}_p$ in $X$.
\end{lemma}
\begin{proof}
We follow the steps of Lemma \ref{lem:maximizer}. First, we choose
a fixed nonzero $u\in X$. Then
\begin{align*}
\mathcal{J}_p(tu)
&\leq \sum_{i=1}^N\Bigl(t^p \frac{|\Delta u_i|^p}{p}
+g_i(0)- K t^\beta |u_i|^\beta\Bigr) + t^p\frac{|\Delta u_N|^p}{p}\\
&=c_1 t^p-c_2 t^\beta+c_3=:h(t),
\end{align*}
with $c_1\geq 0$, $c_2,c_3>0$. Rewriting $h'(t)<0$ we obtain
\begin{equation}\label{e:proof_maximizer_plap}
\gamma(u):= \frac{c_1 p}{c_2 \beta}\beta$ implies that there exists some $t_0>0$ such that
inequality \eqref{e:proof_maximizer_plap} is satisfied for all $t\in (0,t_0)$.
Since the function $\gamma$ is continuous on the compact set
$\mathcal{S}=\{u\in X: \|u\|_p=1\}$, it attains its maximal value
$\gamma_{\rm max}$. If we choose $t_0$ as $00$
($\tilde{z}$ is a nonconstant function). Thus $\mathcal{J}_p$ satisfies the saddle-point
geometry. To show that the Palais-Smale condition holds true, we
literally follow the proof of Lemma \ref{lem:3sol}. Finally, the direct
application of the abstract Theorem \ref{thm:saddlePt} gives a critical
point $u$ of saddle-point type and the existence of at least three solutions.
Moreover, if $G$ satisfies {\rm (G6P)}, Lemma \ref{lem:maximizer_plap}
implies that the critical point $o$ is a local maximizer thus there has
to exist a critical point $\tilde{u}\neq o$. The evenness of $\mathcal{J}_p$ implies
that also $-\tilde{u}$ is a critical point of the saddle-point type.
\end{proof}
\begin{example} \rm
Focusing on the role of the parameter $p$ in the $p$-Laplace operator,
we choose the standard double-well function $g_i(s)=(1-s^2)^2$. Consequently,
the problem \eqref{e:plapl} has the form
\begin{equation}\label{e:ex:plapl}
\begin{gathered}
-\Delta(|\Delta u_{i-1})|^{p-2}\operatorname{sign}
(\Delta u_{i-1}))-4u_i+4u_i^3=0 \quad \text{for } i=1,\dots,N,\\
\Delta u_0=\Delta u_{N}=0.
\end{gathered}
\end{equation}
Taking into account the first part of Theorem \ref{thm:plap} we see that
the problem \eqref{e:ex:plapl} has at least three solutions for arbitrary $p>1$.
Moreover, following the procedure from Example \ref{ex:5sol} one could
check that $g_i(s)\leq g_i(0)-{K}|s|^\beta$ with
$\beta=2$. Hence, the assumption {\rm (G6P)} is satisfied for $p>2$ and
the problem \eqref{e:ex:plapl} has at least five solutions.
\end{example}
As in the linear case, we are able to extend the existence of five solutions
to the case with $p=\beta$.
\begin{theorem}\label{thm:plap:2}
Let $p>1$, function $G$ satisfy {\rm (G1)--(G5)} and
\begin{itemize}
\item[(G6P')] there exist $\delta >0$, $K>\frac{2^{p+1}}{p}$ such that for all
$s\in \mathbb{R}$ and all $i$: $|s|<\delta$ implies $g_i(s)\leq g_i(0)-{K}|s|^p$.
\end{itemize}
Then there exist at least five solutions of equation \eqref{e:plapl}.
\end{theorem}
\begin{proof}
We follow the proof of Lemma \ref{lem:maximizer}.
Let us fix some $u$, $\|u\|_p\neq 0$. Then using
Minkovski inequality, employing built-in Neumann conditions in the space
$X$ and the assumption
(G6P') we get following upper bound of $\mathcal{J}_p$ on $X$:
\begin{align*}
\mathcal{J}_p(tu)
&=\sum_{i=1}^{N+1} \frac{|\Delta tu_{i-1}|^p}{p}+\sum_{i=1}^{N}g_i(t u_i) \\
&=\frac{t^p}{p}\sum_{i=1}^{N+1}{|u_{i}-u_{i-1}|^p}+\sum_{i=1}^{N}g_i(t u_i)\\
&\leq
\frac{t^p}{p}\bigg(\Bigl(\sum_{i=1}^{N+1}{|u_{i}|^p}\Bigr)^{1/p}
+\Bigl(\sum_{i=0}^{N}{|u_{i}|^p}\Bigr)^{1/p}\bigg)^p
+\sum_{i=1}^{N}g_i(t u_i)\\
&\leq \frac{2^p t^p}{p} \sum_{i=0}^{N+1}{|u_{i}|^p}+\sum_{i=1}^{N}g_i(t u_i)\\
&=\frac{2^p t^p}{p}\Big(|u_{1}|^p+|u_{N}|^p + \sum_{i=1}^{N}{|u_{i}|^p}\Big)
+\sum_{i=1}^{N}g_i(t u_i)\\
&\leq \frac{2^{p+1} t^p}{p} \sum_{i=1}^{N}{|u_{i}|^p}+\sum_{i=1}^{N}g_i(t u_i)\\
&\leq \frac{2^{p+1} t^p}{p} \sum_{i=1}^{N}{|u_{i}|^p}
+\sum_{i=1}^{N}\bigl(g_i(0)-K t^p|u_i|^p\bigr)\\
&= t^p\Big(\frac{2^{p+1} }{p}-K \Big)\sum_{i=1}^{N}{|u_{i}|^p}
+\sum_{i=1}^{N}g_i(0).
\end{align*}
The assumption (G6P') implies the negativity of $\frac{2^{p+1} }{p}-K$
which guarantees that function $t \mapsto \mathcal{J}(tu)$ attains a local maximimum
at $t=0$. Since we can choose $u$ arbitrarily, there exists at least one pair
of non-trivial saddle-point type critical points $\tilde{u}$ and $-\tilde{u}$.
Moreover, constant solutions $o$, $e_1$ and $-e_1$ also solve
\eqref{e:plapl}. This completes the proof.
\end{proof}
\begin{remark} \rm
Note that the assumption (G6P') in the linear case $p=2$ requires $K>4$, which is
stricter than the assumption $K>\frac{\lambda_{\rm max}}{2}$ from
$\mathrm{(G6)'}$. This implies that
it could be possible to get a better lower bound on $K$ by employing more
subtle estimates in the proof of Theorem \ref{thm:plap:2}.
\end{remark}
\begin{example} \rm
We consider
\[
g_i(s)=\frac{1}{\varepsilon^p}|1-|s|^p|^b=\frac{1}{\varepsilon^2} \tilde{g}_i(s)
\]
with $\varepsilon>0$ as in Example \ref{ex:5solbeta2} to illustrate the
influence of parameter $p$ on the solution multiplicity. Following the procedure
employed in Example \ref{ex:5solbeta2} we get
$K\leq \frac{b}{\varepsilon^p}(1-\delta^{p})^{b-1}$.
The assumption (G6P') is then reduced to
$$
\varepsilon<\frac{1}{2}\sqrt[p]{\frac{bp}{2}}.
$$
\end{example}
\begin{remark} \rm
In the continuous case, the multiplicity of solutions was studied in
Otta \cite{otta1d}. For $p>\beta$ one can prove that there exists a sequence
of solutions of the continuous version of \eqref{e:plapl}.
Whereas for $p\leq\beta$ the set of solutions is finite and the minimal number of
solutions is three -- constant solutions $\pm 1$, $0$. On the one hand,
the discrete results presented here studies only the case with $p\geq\beta$.
On the other hand, the assumptions are weaker in the sense that no growth
conditions on $g_i$'s are required with the exception of $p=\beta$.
\end{remark}
\section{Boundedness of solutions}
In this section, we focus on the boundedness of solutions to \eqref{e:plapl}. Returning back to the
application of similar problems in the image processing, there is a reasonable question of whether
the solutions stay between -1 and 1 (recall that $u$ described the gray scale between -1 and 1 there).
Let us study solutions of initial-value problem
\begin{equation} \label{e:ivp}
\begin{gathered}
-\Delta(\varphi_p(\Delta u_{i-1}))+f_i(u_i)=0 \quad \text{for }
i=1,\dots,N,\\
u_0=\tilde{u}_0,\quad u_1=\tilde{u}_1,
\end{gathered}
\end{equation}
under additional assumption on the nonlinear functions $g_i$:
\begin{itemize}
\item[(G7)] for each $i$, $s\mapsto g_i(s)$ is an increasing function
for $s>1$.
\end{itemize}
Note that all $g_i$'s mentioned above in this paper satisfy (G7).
In the following, we use
$\varphi_{p'}(\varphi_{p}(s))=s$, where $p$ and $p'$ are conjugate
exponents (i.e. $\frac{1}{p}+\frac{1}{p'}=1$).
\begin{theorem}\label{thm:boundededness}
Let us assume that $g_i$ satisfy {\rm (G1), (G2), (G3), (G7)}.
If $u$ is a solution of \eqref{e:plapl} then $|u_i|\leq 1$ for all
$i=1,2,\dots N$.
\end{theorem}
\begin{proof}
To prove the boundedness of solutions to \eqref{e:plapl} we use solutions
of related initial-value problem \eqref{e:ivp}.
Let us expand difference terms to get recursively defined solution $u$ to
\eqref{e:ivp} in the form
\begin{equation}\label{e:rec}
\begin{gathered}
u_{i+1}=u_{i}+\varphi_{p'}(\varphi_{p}(u_{i}-u_{i-1})+f_i(u_i)) \quad
\text{for } i=1,\dots,N,\\
u_0=\tilde{u}_0,\quad u_1=\tilde{u}_1.
\end{gathered}
\end{equation}
Assumption (G7) guarantees the positivity of $g_i'(s)=f_i(s)$ for all $s>1$.
We choose $\tilde{u}_0=\tilde{u}_1>1$. Using assumptions
(G1), (G2), (G3) and (G7) one can obtain following
estimates:
\begin{gather*}
u_{2}=u_{1}+\varphi_{p'}(\varphi_{p}(u_{1}-u_{0})+f_1(u_1))=u_0+\varphi_{p'}(f_1(u_1))=u_0+\delta,\\
u_{3}=u_{2}+\varphi_{p'}(\varphi_{p}(u_{2}-u_{1})+f_2(u_2))\geq
u_2+\varphi_{p'}(\varphi_{p}(u_2-u_1))=u_2+\delta,\\
\dots\\
\begin{aligned}
u_{N+1}&=u_{N}+\varphi_{p'}(\varphi_{p}(u_{N}-u_{N-1})+f_N(u_N))\\
&\geq u_N+\varphi_{p'}(\varphi_{p}(u_{N}-u_{N-1}))=u_N+\delta.
\end{aligned}
\end{gather*}
Since $u_{N+1}\geq u_{N}+\delta$ the vector $u$ can not satisfy
homogenous Neumann boundary conditions
at $x_N$ and $x_{N+1}$. The same solution behavior can be observed for
$\tilde{u}_0=\tilde{u}_1<-1$. This implies that a solution of the
boundary-value problem \eqref{e:plapl}
has to satisfy initial condition $u_0=u_1\in[-1,1]$.
It remains to show that there exists no index $k$,
$k\in \{ 1,2,\dots,N\}$ such that $|u_k|>1$. Let us consider $u$
to be a solution of \eqref{e:plapl} and such index $k$ do
exist. Without any loss of generality, we can assume that
$u_{k-1}\leq 1$, $u_k>1$ hold. Then a solution satisfies \eqref{e:rec} and
$$
u_{k+1}=u_{k}+\varphi_{p'}(\varphi_{p}(u_{k}-u_{k-1})+f_k(u_k))\geq u_k+\delta
$$
where $\delta=u_k-u_{k-1}>0$. By induction, following steps from previous
paragraph, we get $u_{N+1}\geq u_{N}+\delta$ which is a contradiction
to $u$ as a solution of \eqref{e:plapl}. By
similar arguments, one can show that there exists no index $k$ such that
$u_k<-1$. This completes the proof.
\end{proof}
\begin{remark} \rm
In the continuous case, Dr\'abek et al. \cite{DMT} studied the following problem
\begin{equation} \label{e:evol-DMT}
\begin{gathered}
\varepsilon^p (|u'(x)|^{p-2}u'(x))' - g'(u(x))=0,\quad x \in (0,1),\\
u'(0)=u'(1)=0,
\end{gathered}
\end{equation}
with $g(s)=|1-s^2|^b$, $b>1$. Interestingly, they proved that for $p>b$
there exist dead-core solutions touching the values $\pm 1$. Moreover,
for $p=4$ and $b=2$, they proved that these solutions forms continua of
saddle points of the related functional in $W^{1,p}(0,1)$.
\end{remark}
\subsection*{Final remarks}
Multiplicity results for discrete equations with double-well potentials offer
many interesting questions. This paper contains some basic answers but there
are many issues which could be followed further.
On the one hand, there is some space in improving the presented results.
General multiplicities of the eigenvalue $\lambda=0$ (see (A2)) or more
intricate double well potentials (see (G2) and (G3)) could be considered.
On the other hand, different approaches could extend some of the results as well.
More complicated operators (like multidimensional $p$-Laplacian) could
be analyzed. Furthermore, the boundedness of solutions has been proven by
iteration technique and is thus restricted to the one-dimensional
problems. As our numerical experiments suggest some other techniques
could generalize this to partial difference equations as well.
\subsection*{Acknowledgements}
This work was supported by the European Regional Development Fund
(ERDF), project “NTIS – New Technologies for the Information
Society”, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.
Petr Stehl\'{\i}k thanks for the support by the Czech Science Foundation,
Grant No. 201121757.
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