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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 129, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/129\hfil An application of the Lyapunov-Schmidt method]
{An application of the Lyapunov-Schmidt method to semilinear
elliptic problems}
\author[Q. A. Ng\^o\hfil EJDE-2005/129\hfilneg]
{Qu\^o\!$^{'}$c Anh Ng\^o}


\address{Qu\^o\!$^{'}$c Anh Ng\^{o} \hfill\break
Department of Mathematics, Mechanics and Informatics,
College of Science, Vietnam National University,
Hanoi, Vietnam}
\email{bookworm\_vn@yahoo.com \; anhngq@yahoo.com.vn}


\date{}
\thanks{Submitted July 12, 2005. Published November 23, 2005.}
\subjclass[2000]{35J50, 35J55}
\keywords{Semilinear; elliptic system; Lyapunov; Schmidt;
fixed-point principle}

\begin{abstract}
 In this paper we consider the existence of nonzero solutions for
 the undecoupling elliptic system
 \begin{gather*}
 -\Delta u=\lambda u+\delta v+f(u,v), \\
 -\Delta v=\theta u+ \gamma v+g(u,v),
 \end{gather*}
 on a bounded domain of $\mathbb{R}^n$, with zero Dirichlet boundary
 conditions. We use the Lyapunov-Schmidt method and the fixed-point
 principle.
\end{abstract}

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

\section{Introduction}

In this present paper we consider the  Dirichlet problem
\begin{equation} \label{btbd}
\begin{gathered}
-\Delta u= \lambda u+\delta v+f(u,v) \mbox{ in } \Omega, \\
-\Delta v=\theta u+\gamma v+g(u,v) \mbox{ in } \Omega, \\
u=v=0 \mbox{ on }\partial \Omega,
\end{gathered}
\end{equation}
where $\Omega \subset \mathbb{R}^{n}$ ($n \geq 3$) is a bounded
domain with smooth boundary and subject to Dirichlet boundary conditions;
 $A=\begin{pmatrix} \lambda & \delta \\ \theta & \gamma \end{pmatrix}$
 is a matrix of real entries; $f,g:\Omega \times \mathbb{R} \to \mathbb{R}$
 are globally Lipschitz functions for $u$, $v$; i.e.
\begin{gather*}
 | {f ( {u,v} ) - f ( {\widetilde u,\widetilde v} )} |
 \leq k_1 ( {| {u - \widetilde u} | + | {v - \widetilde v} |} ),  \\
 | {g ( {u,v} ) - g ( {\widetilde u,\widetilde v} )} |
 \leq k_2 ( {| {u - \widetilde u} | + | {v - \widetilde v} |} ),
\end{gather*}
for all $u, \widetilde u, v, \widetilde v \in \mathbb{R}$.

Our goal is finding non-trivial solutions to the system \eqref{btbd}
under the above hypothesis, and other suitable conditions on the first
two eigenvalues of the Laplacian and on the parameter.

Note that the problem Dirichlet for system \eqref{btbd} have been
studied by many authors. In \cite{T}, under more restrictive
conditions, Hoang has considered the system \eqref{btbd}
in which $\Omega$ is an unbounded domain. In \cite{N3}, the author
has considered the  case of positivity of solutions in a bounded
domain and in \cite{TA}, the positivity of solutions have been
mentioned for an unbounded domain.

Equation \eqref{btbd} represents a steady state case of
reaction-diffusion systems of interest in biology.
Reaction-diffusion systems have been intensively studied during
recent years, see \cite{S2} where many references can be found.
There exists a decoupling technique, which consists of reducing
the system \eqref{btbd} to a single nonlinear equation containing
an integral and a differential term. This technique was introduced
by Rothe \cite{R2}, Lazer \& McKenna \cite{LMc} and Brown \cite{B}
and has been used thereafter by many authors. For the resonant
case many known techniques used to solve the scalar case can be
applied to find solutions and positive solutions. See for example
Ahmad, Lazer \& Paul \cite{ALP}, Ambrosetti \& Mancini \cite{AM},
Anane \cite{A}, Bartolo, Benci \& Fortunato \cite{BBF}, Berestycki
\& De Figueiredo \cite{BF}, Capozzi, Lupo \& Solimini \cite{CLS},
Cesari \& Kannan \cite{CK}, Costa \& Magalh\~{a}es \cite{CM}, De
Figueiredo \& Gossez \cite{FG}, Gossez \cite{G}, Innacci \&
Nkashama \cite{IN1, IN2}, Landesman \& Lazer \cite{LL}, Lupo \&
Solimini \cite{LS}, Omari \& Zanolin \cite{OZ}, Rabinowitz
\cite{R1}, Schechter \cite{S1}, Solimini \cite{S3}, Vargas \&
Zuluaga \cite{VZ1, VZ2}, Zuluaga \cite{Z1, Z2} and the references
therein.

The decoupling technique has some obvious shortcomings, for
example,  it is very difficult to apply to systems with three or
more equations. Even, in the case of two equations is too
restrictive to give conditions to solve the second equation of
\eqref{btbd} for $v$ in terms of $u$.

It is known, see \cite{CM}, using the eigenvalues of the matrix
$A$,  we will be able to give a precise description of kernel of
operator $-\Delta - A$, and easy to see that this kernel is
nonzero if and only if $A - \lambda_j I$ is singular for some
eigenvalue $\lambda_j$ of the operator $-\Delta$.

 Zuluaga \cite{Z3} showed results of existence and nonexistence of
solutions for \eqref{btbd} under the condition $\lambda_1$, the
first eigenvalue of $-\Delta$, is also a eigenvalue of matrix $A$.
In this paper, we will extend these results obtained in \cite{Z3}
under the conditions in which $\lambda_1$ is not a eigenvalue of
$A$.

Our paper is organized as follows. Section 2 provides some
preliminaries and notation including the Lyapunov-Schmidt method.
In Section 3, we consider the problem \eqref{btbd} under some
special case where some parameters and both Lipschitz constants
are equal, problem (\ref{btbd_db}). Our main result for such
problem is the Theorem \ref{dlc_db}. By the similar arguments, in
Section 4, we state our main result of this paper for the problem
\eqref{btbd}.

\section{Preliminaries and Notation}

In $E= L^2(\Omega) \times L^2(\Omega)$ we use the norm
\[
\| U \|_{L^2(\Omega) \times L^2(\Omega)}^2 = \| u
\|_{L^2(\Omega)}^2 + \| v \|_{L^2(\Omega)}^2,
\]
where $U = ( {u,v} )$. To simplify notation, we  use $\|\cdot \|$
to denote the norm in $L^2(\Omega)$ or in $L^2(\Omega) \times
L^2(\Omega)$.

\subsection*{Solutions of \eqref{btbd}}
We say that $U \in H_0^1 ( \Omega ) \times H_0^1 ( \Omega )$ is a
solution of \eqref{btbd} if
\begin{equation}
U = ( { - \Delta } )^{ - 1} ( {AU + G( U )} ),
\label{btbd_k}
\end{equation}
where $G( U ) = ( f( {u,v} ), g( {u,v} ) )$. It is clear that  $(
{ - \Delta } )^{ - 1} :E \to H_0^1 ( \Omega ) \times H_0^1 (
\Omega )$ is a linear, self-adjoint, continuous and bijective
operator. Also, the embedding $H_0^1 ( \Omega ) \times H_0^1 (
\Omega ) \hookrightarrow E$ is compact, thus $( { - \Delta } )^{ -
1} :E \to E$ is compact, self-adjoint and injective as well.
Hence, the operator defined by the right hand side of
(\ref{btbd_k}) is compact.

Throughout this paper we shall denote by $\lambda_1$, $\lambda_2$
the  first two eigenvalues of $-\Delta$ and $\varphi_1$,
$\varphi_2$ are the eigenfunctions associated with the eigenvalue
$\lambda_1$ and $\lambda_2$, respectively.

\subsection*{The Lyapunov-Schmidt method}

We will denote by $X$ the subspace of $H_0^1 ( \Omega )$ spanned
by $\varphi_1$, that is to say
$X = \{ t\varphi _1 : t \in \mathbb{R} \}$. We shall also denote
$Y = X^ \bot = \langle {\varphi _1 } \rangle ^ \bot$. So, we have the
identity
\[
H_0^1 ( \Omega ) = X \oplus Y.
\]
Then all $U = (u, v) \in E$ can be written as
\begin{gather*}
 u = u_0 + z,u_0 \in X,z \in Y, \\
 v = v_0 + w,v_0 \in X,w \in Y,
\end{gather*}
where $u,v \in H_0^1 ( \Omega )$. Let us denote by $P$ and $Q$ the
projection on $X$ and $Y$, respectively. Applying $P$ and $Q$ to both
sides of (\ref{btbd_k}) we obtain a decomposition of it in two systems
as follows
\begin{equation}
\begin{gathered}
u_0 = P( { - \Delta } )^{ - 1} [ {\lambda ( {u_0 + z} ) + \delta ( {v_0 + w} ) + f( {u_0 + z, v_0 + w} )} ],\\
v_0 = P( { - \Delta } )^{ - 1} [ {\theta ( {u_0 + z} ) + \gamma ( {v_0 + w} ) + g( {u_0 + z, v_0 + w} )} ],
\end{gathered}
\label{htx}
\end{equation}
and
\begin{equation}
\begin{gathered}
z = Q( { - \Delta } )^{ - 1} [ {\lambda ( {u_0 + z} ) + \delta ( {v_0 + w} ) + f( {u_0 + z, v_0 + w} )} ], \\
w = Q( { - \Delta } )^{ - 1} [ {\theta ( {u_0 + z} ) + \gamma ( {v_0 + w} ) + g( {u_0 + z, v_0 + w} )} ].
\end{gathered}
\label{hty}
\end{equation}
For each $( {u_0 ,v_0 } ) \in X \times X$ fixed, we solve (\ref{hty})
and have a solution $( {z_0, w_0} ) \in Y \times Y$ which will be plugged
into (\ref{htx}) to get the solution $( {u_0 ,v_0 } )$ of (\ref{htx}).
Thus, the solutions of \eqref{btbd} will be of the form
$( {u_0 + z_0, v_0 + w_0} )$.

\section{A special case of Problem \eqref{btbd}}

Before stating our main result, in this Section, we consider
 problem \eqref{btbd} in which $\gamma = \lambda = \lambda_1$,
  $k_1 = k_2 = k$ and $\delta = \theta > 0$, i.e. we shall deal with
the existence of non-trivial solutions of the problem
\begin{equation} \label{btbd_db}
\begin{gathered}
-\Delta u= \lambda_1 u+\delta v+f(u,v) \quad \mbox{in } \Omega, \\
-\Delta v=\delta u+\lambda_1 v+g(u,v) \quad \mbox{in } \Omega, \\
u=v=0 \quad \mbox{on }\partial \Omega,
\end{gathered}
\end{equation}
where $\lambda_1$ is the first eigenvalue of $-\Delta$.
Applying the Lyapunov-Schmidt method we obtain a decomposition of it
 in two systems as follows
\begin{equation}
\begin{gathered}
u_0 = P( { - \Delta } )^{ - 1} [ {\lambda _1 ( {u_0 + z} )
+ \delta ( {v_0 + w} ) + f( {u_0 + z, v_0 + w} )} ],\\
v_0 = P( { - \Delta } )^{ - 1} [ {\delta ( {u_0 + z} )
+ \lambda _1 ( {v_0 + w} ) + g( {u_0 + z, v_0 + w} )} ],
\end{gathered}
\label{htx_db}
\end{equation}
and
\begin{equation}
\begin{gathered}
z = Q( { - \Delta } )^{ - 1} [ {\lambda _1 ( {u_0 + z} )
+ \delta ( {v_0 + w} ) + f( {u_0 + z, v_0 + w} )} ], \\
w = Q( { - \Delta } )^{ - 1} [ {\delta ( {u_0 + z} )
+ \lambda _1 ( {v_0 + w} ) + g( {u_0 + z, v_0 + w} )} ].
\end{gathered}
\label{hty_db}
\end{equation}
Fixing $( {u_0 ,v_0 } ) \in X \times X$, we shall consider (\ref{hty_db}).
Letting
\[
F_Q ( {z,w} ) = ( {F_Q^{( 1 )} ( {z,w} ),F_Q^{( 2 )} ( {z,w} )} ),
\]
where
\begin{gather*}
 F_Q^{( 1 )} ( {z,w} ): = Q( { - \Delta } )^{ - 1}
 [ {\lambda _1 ( {u_0 + z} ) + \delta ( {v_0 + w} ) + f( {u,v} )}], \\
 F_Q^{( 2 )} ( {z,w} ): = Q( { - \Delta } )^{ - 1}
 [ {\delta ( {u_0 + z} ) + \lambda _1 ( {v_0 + w} ) + g( {u,v} )} ].
\end{gather*}

\begin{lemma}\label{bd1}
If
\begin{equation}\label{dk1}
( {\lambda _1 + k} )^2 + ( {\delta + k} )^2 < \frac{{\lambda _2^2 }}{2}
\end{equation}
then $F_Q$ is a contraction in $Y \times Y$.
\end{lemma}

\begin{proof}
Let $z,\widetilde z,w,\widetilde w \in Y$, by the definition of
$F_Q^{( 1 )} ( {z,w} )$, we find
\begin{align*}
F_Q^{( 1 )} ( {z,w} ) - F_Q^{( 1 )} ( {\widetilde z,\widetilde w} )
=& Q( { - \Delta } )^{ - 1} \big( \lambda _1 ( {z - \widetilde z} )
+ \delta ( {w - \widetilde w} ) \\
&+ f( {u_0 + z,v_0 + w} ) - f( {u_0 + \widetilde z,v_0 + \widetilde w} ) \big).
\end{align*}
Therefore, from the characterization of $\lambda_1$, we get
\begin{align*}
\|F_Q^{( 1 )} ( {z,w} ) - F_Q^{( 1 )} ( {\widetilde z,\widetilde w} ) \|
\leq& \frac{1}{\lambda _2} \big( \lambda _1 \| {z - \widetilde z} \|
+ \delta \| {w - \widetilde w} \| \\
&+ \| {f( {u_0 + z,v_0 + w} ) - f( {u_0 + \widetilde z,v_0 +
\widetilde w} )} \| \big).
\end{align*}
By using the Minkowski's inequality and our Lipschitzian assumptions,
we obtain
\[
\| {f( {u_0 + z,v_0 + w} ) - f( {u_0 + \widetilde z,v_0 +
\widetilde w} )} \| \leq k( {\| {z - \widetilde z} \| + \| {w -
\widetilde w} \|} ).
\]
So
\[
\| {F_Q^{( 1 )} ( {z,w} ) - F_Q^{( 1 )} ( {\widetilde z,\widetilde
w} )} \| \leq \frac{1}{{\lambda _2 }}( {( {\lambda _1 + k} )\| {z
- \widetilde z} \| + ( {\delta + k} )\| {w - \widetilde w} \|} ).
\]
Therefore,
\[
\| {F_Q^{( 1 )} ( {z,w} ) - F_Q^{( 1 )} ( {\widetilde z,\widetilde
w} )} \|^2 \leq \frac{2}{{\lambda _2^2 }}( {( {\lambda _1 + k} )^2
\| {z - \widetilde z} \|^2 + ( {\delta + k} )^2 \| {w - \widetilde
w} \|^2 } ).
\]
Similarly, we claim that
\[
\| {F_Q^{( 2 )} ( {z,w} ) - F_Q^{( 2 )} ( {\widetilde z,\widetilde
w} )} \|^2 \leq \frac{2}{{\lambda _2^2 }}\Big( {( {\delta + k} )^2 \|
{z - \widetilde z} \|^2 + ( {\lambda_1 + k} )^2 \| {w - \widetilde
w} \|^2 } \Big).
\]
Now we obtain
\[
\| {F_Q ( {z,w} ) - F_Q ( {\widetilde z,\widetilde w} )} \|^2 \leq
\frac{2} {{\lambda _2^2 }}\Big( {( {\lambda _1 + k} )^2 + ( {\delta +
k} )^2 } \Big) \Big( {\| {z - \widetilde z} \|^2 + \| {w - \widetilde w}
\|^2 } \Big).
\]
Hence, the assertion follows.
\end{proof}

By using fixed-point principle, we conclude that (\ref{hty_db})
has a unique solution $( {z_0( {u_0 ,v_0 } ) , w_0( {u_0 ,v_0 } )} )$
for each $( {u_0 ,v_0 } ) \in X \times X$ fixed.
The assertion of the above Lemma let us to define
\begin{align*}
 F:X \times X&\to Y \times Y,\\
 ( {u_0 ,v_0 } ) &\mapsto F( {u_0 ,v_0 } ) := ( {z_0 ,w_0 } ),
\end{align*}
be the function such that $( {z_0 ,w_0 } )$ is the only fixed point of $F_Q$.

\begin{lemma}\label{bd2}
If
\begin{equation}\label{dk2}
( {\lambda _1 + k} )^2 + ( {\delta + k} )^2 < \frac{{\lambda _2^2 }}{4}
\end{equation}
then
\begin{equation}\label{dk3}
\begin{split}
&\| F( {u_0 ,v_0 } )  - F( {\widetilde u_0 ,\widetilde v_0 } )  \|^2 \\
& \leq \frac{{8k^2 }}{{\lambda _2^2 - 4( {( {\lambda _1 + k} )^2 +
( {\delta + k} )^2 } )}}( \| {u_0 - v_0 } \|^2 + \| {\widetilde
u_0 - \widetilde v_0 } \|^2 ).
\end{split}
\end{equation}
for every $( {u_0 ,v_0 } )$ and $( {\widetilde u_0 ,\widetilde v_0 } )$ in $X \times X$.
\end{lemma}

\begin{proof}
Suppose that $F( {u_0 ,v_0 } ) = ( {z_0 ,w_0 } )$ and
$F( {\widetilde u_0 ,\widetilde v_0 } ) = ( {\widetilde z_0 ,\widetilde w_0 } )$.
 By the definition of $F$, we have
\begin{gather*}
 z_0 = Q( { - \Delta } )^{ - 1} [ {\lambda _1 ( {u_0 + z_0} )
 + \delta ( {v_0 + w_0} ) + f( {u_0 + z_0, v_0 + w_0} )} ], \\
 w_0 = Q( { - \Delta } )^{ - 1} [ {\delta ( {u_0 + z_0} )
 + \lambda _1 ( {v_0 + w_0} ) + g( {u_0 + z_0, v_0 + w_0} )} ],
\end{gather*}
and
\begin{gather*}
 \widetilde z_0 = Q( { - \Delta } )^{ - 1}
 [ {\lambda _1 ( {\widetilde u_0 + \widetilde z_0} )
 + \delta ( {\widetilde v_0 + \widetilde w_0} )
 + f( {\widetilde u_0 + \widetilde z_0, \widetilde v_0
 + \widetilde w_0} )} ], \\
 \widetilde w_0 = Q( { - \Delta } )^{ - 1}
 [ {\delta ( {\widetilde u_0 + \widetilde z_0} )
 + \lambda _1 ( {\widetilde v_0 + \widetilde w_0} )
 + g( {\widetilde u_0 + \widetilde z_0, \widetilde v_0 + \widetilde w_0} )} ].
\end{gather*}
Because
\begin{gather*}
Q( { - \Delta } )^{ - 1} ( {u_0 } ) = Q( { - \Delta } )^{ - 1} ( {v_0 } )
 = 0, \\
Q( { - \Delta } )^{ - 1} ( {\widetilde u_0 } )
= Q( { - \Delta } )^{ - 1} ( {\widetilde v_0 } ) = 0,
\end{gather*}
we have
\begin{align*}
\| {z_0 - \widetilde z_0 } \| \leq & \frac{1}{{\lambda _2 }}
\Big( \lambda _1 \| {z_0 - \widetilde z_0} \|
+ \delta \| {w_0 - \widetilde w_0} \| \\
&+ \| {f( {u_0 + z_0 ,v_0 + w_0 } ) - f( {\widetilde u_0
+ \widetilde z_0 ,\widetilde v_0 + \widetilde w_0 } )} \| \Big)\\
\leq & \frac{1}{{\lambda _2 }}\Big( ( {\lambda _1 + k} )\| {z_0 -
\widetilde z_0} \|+ ( {\delta + k} )\| {w_0 - \widetilde w_0} \| \\
&+ k ( {\| {u_0 - \widetilde u_0 } \| + \| {v_0 - \widetilde v_0 }
\|} ) \Big).
\end{align*}
Thus
\begin{align*}
\| {z_0 - \widetilde z_0 } \|^2
\leq&  \frac{4} {{\lambda _2^2 }}\Big( ( {\lambda_1 + k} )^2 \| {z_0
- \widetilde z_0 } \|^2  + ( {\delta + k} )^2 \| {w_0 - \widetilde w_0 } \|^2 \\
& + k^2 \big( \| {u_0 - \widetilde u_0 } \|^2 + \| {v_0 -
\widetilde v_0 } \|^2 \big) \Big).
\end{align*}
Similarly, we find
\begin{align*}
\| {w_0 - \widetilde w_0 } \|^2
\leq & \frac{4}
{{\lambda _2^2 }}\Big( ( {\delta + k} )^2 \| {z_0 - \widetilde z_0 } \|^2
 + ( {\lambda _1 + k} )^2 \| {w_0 - \widetilde w_0 } \|^2 \\
& + k^2 \big( \| {u_0 - \widetilde u_0 } \|^2 + \| {v_0 -
\widetilde v_0 } \|^2 \big) \Big).
\end{align*}
Hence
\begin{align*}
\| {z_0 - \widetilde z_0 } \|^2 + \| {w_0 - \widetilde w_0 } \|^2
\leq & \frac{4}
{\lambda _2^2 }\Big( ( {( {\delta + k} )^2 + ( {\lambda _1 + k} )^2 } )( {\| {z_0 - \widetilde z_0 } \|^2 + \| {w_0 - \widetilde w_0 } \|^2 } ) \\
&+ 2k^2 ( {\| {u_0 - \widetilde u_0 } \|^2 + \| {v_0 - \widetilde
v_0 } \|^2 } ) \Big).
\end{align*}
Therefore
\begin{align*}
&\| {z_0 - \widetilde z_0 } \|^2 + \| {w_0 - \widetilde w_0 } \|^2 \\
&\leq \frac{{8k^2 }} {{\lambda _2^2 - 4( {( {\lambda _1 + k} )^2 +
( {\delta + k} )^2 } )}}( {\| {u_0 - v_0 } \|^2 + \| {\widetilde
u_0 - \widetilde v_0 } \|^2 } ).
\end{align*}
\end{proof}

Now, we consider the system (\ref{htx_db}). First, the fact that $F$ is
a contraction mapping yields
\begin{equation}
\begin{gathered}
u_0 = P( { - \Delta } )^{ - 1} [ {\lambda _1 ( {u_0 + z_0} )
+ \delta ( {v_0 + w_0} ) + f( {u_0 + z_0, v_0 + w_0} )} ],\\
v_0 = P( { - \Delta } )^{ - 1} [ {\delta ( {u_0 + z_0} )
+ \lambda _1 ( {v_0 + w_0} ) + g( {u_0 + z_0, v_0 + w_0} )} ].
\end{gathered}
\end{equation}
Because
\begin{gather*}
P( { - \Delta } )^{ - 1} ( {z_0 } ) = P( { - \Delta } )^{ - 1}
( {w_0 } ) = 0,
\\
P( { - \Delta } )^{ - 1} ( {\lambda _1 u_0 } ) = u_0 ,
P( { - \Delta } )^{ - 1} ( {\lambda _1 v_0 } ) = v_0,
\end{gather*}
we deduce that
\begin{equation}
\begin{gathered}
0 = P( { - \Delta } )^{ - 1} [ \delta v_0 + f( {u_0 + z_0, v_0 + w_0} ) ],\\
0 = P( { - \Delta } )^{ - 1} [ \delta u_0 + g( {u_0 + z_0, v_0 + w_0} ) ].
\end{gathered}
\end{equation}
On the other hand, from the definition of subspace $X$,
\[
P( { - \Delta } )^{ - 1} ( {\delta u_0 } )
= \frac{\delta }{{\lambda _1 }}u_0, \quad
P( { - \Delta } )^{ - 1} ( {\delta v_0 } ) = \frac{\delta }{{\lambda _1 }}v_0.
\]
This yields
\begin{equation}\label{bd2_1}
\begin{gathered}
0 =  \frac{\delta }{{\lambda _1 }}v_0 + P( { - \Delta } )^{ - 1} [ f( {u_0 + z_0, v_0 + w_0} ) ],\\
0 =  \frac{\delta }{{\lambda _1 }}u_0 + P( { - \Delta } )^{ - 1} [ g( {u_0 + z_0, v_0 + w_0} ) ].
\end{gathered}
\end{equation}
Now, (\ref{bd2_1}) is equivalent to
\begin{equation}
\begin{gathered}
u_0 = -  \frac{\lambda _1}{\delta} P( { - \Delta } )^{ - 1} [ g( {u_0 + z_0, v_0 + w_0} ) ],\\
v_0 = -  \frac{\lambda _1}{\delta} P( { - \Delta } )^{ - 1} [ f( {u_0 + z_0, v_0 + w_0} ) ].
\end{gathered}
\end{equation}
Letting
\[
F_P ( {u_0, v_0} ) = ( {F_P^{( 1 )} ( {u_0, v_0} ),F_P^{( 2 )} ( {u_0, v_0} )} ),
\]
where
\begin{gather*}
 F_P^{( 1 )} ( {u_0, v_0} ): = - \frac{\lambda _1}{\delta}
 P( { - \Delta } )^{ - 1} [ g( {u_0 + z_0, v_0 + w_0} ) ], \\
 F_P^{( 2 )} ( {u_0, v_0} ): = - \frac{\lambda _1}{\delta}
 P( { - \Delta } )^{ - 1} [ f( {u_0 + z_0, v_0 + w_0} ) ].
\end{gather*}

\begin{lemma}\label{bd3}
If
$( {\lambda _1 + k} )^2 + ( {\delta + k} )^2 < \lambda _2^2 /4$
and
\begin{equation}\label{dk4}
\frac{{8k^2 }}{{\delta ^2 }}\Big( {1 + \frac{{8k^2 }}{{\lambda _2^2
- 4( {( {\delta + k} )^2 + ( {\lambda _1 + k} )^2 } )}}} \Big) < 1
\end{equation}
then $F_P$ is a contraction in $X \times X$.
\end{lemma}

\begin{proof}
Letting $( {\widetilde u_0, \widetilde v_0} )$ in $X \times X$.
Corresponding to $( \widetilde u_0, \widetilde v_0 )$, from
Lemma \ref{bd1}, we have $( {\widetilde z_0, \widetilde w_0} )$ in
$Y \times Y$. From the definition of $F_P^{( 1 )} ( {u_0, v_0} )$ we find
\begin{align*}
&F_P^{( 1 )} ( {u_0 ,v_0 } ) - F_P^{( 1 )} ( {\widetilde u_0 ,
\widetilde v_0 } )\\
&= - \frac{{\lambda _1 }}{\delta }P( { - \Delta } )^{ - 1}
[ {g( {u_0 + z_0 ,v_0 + w_0 } ) - g( {\widetilde u_0 + \widetilde z_0 ,
\widetilde v_0 + \widetilde w_0 } )} ].
\end{align*}
Using our Lipschitzian assumptions we obtain
\begin{align*}
&\| {F_P^{( 1 )} ( {u_0 ,v_0 } ) - F_P^{( 1 )} ( {\widetilde u_0 ,
\widetilde v_0 } )} \| \\
& \leq \frac{{\lambda _1 }}{\delta }
\frac{1}{{\lambda _1 }}\| {g( {u_0 + z_0 ,v_0 + w_0 } )
- g( {\widetilde u_0 + \widetilde z_0 ,\widetilde v_0
+ \widetilde w_0 } )} \|\\
& \leq \frac{k}{\delta }( {\| {u_0 - \widetilde u_0 } \| + \| {v_0
- \widetilde v_0 } \| + \| {z_0 - \widetilde z_0 } \| + \| {w_0 -
\widetilde w_0 } \|} ).
\end{align*}
By (\ref{dk3}), we have
\begin{align*}
& \| {F_P^{( 1 )}( {u_0 ,v_0 } ) - F_P^{( 1 )} ( {\widetilde u_0 ,\widetilde v_0 } )} \|^2 \\
&\leq \frac{{4k^2 }}{{\delta ^2 }}
\Big( {\| {u_0 - \widetilde u_0 } \|^2 + \| {v_0 - \widetilde v_0 } \|^2
 + \| {z_0 - \widetilde z_0 } \|^2 + \| {w_0 - \widetilde w_0 } \|^2 } \Big)
 \\
&\leq \frac{{4k^2 }}{{\delta ^2 }}
\Big( {\| {u_0 - \widetilde u_0 } \|^2 + \| {v_0 - \widetilde v_0 } \|^2 } \Big) \\
&\quad+ \frac{{4k^2 }}{{\delta ^2 }}\frac{{8k^2 }}{{\lambda _2^2 - 4( {( {\lambda _1 + k} )^2 + ( {\delta + k} )^2 } )}}( {\| {u_0 - \widetilde u_0 } \|^2 + \| {v_0 - \widetilde v_0 } \|^2 } )\\
&\leq \frac{{4k^2 }}{{\delta ^2 }}\Big( 1 + \frac{{8k^2
}}{{\lambda _2^2 - 4( {( {\lambda _1 + k} )^2 + ( {\delta + k} )^2
} )}} \Big) ( {\| {u_0 - \widetilde u_0 } \|^2 + \| {v_0 - \widetilde
v_0 } \|^2 } ).
\end{align*}
Similarly, we have
\begin{align*}
& \| {F_P^{( 2 )} ( {u_0 ,v_0 } ) - F_P^{( 2 )} ( {\widetilde u_0 ,
\widetilde v_0 } )} \|^2 \\
&\leq  \frac{{4k^2 }}{{\delta ^2 }}\Big( 1 + \frac{{8k^2
}}{{\lambda _2^2 - 4( {( {\lambda _1 + k} )^2 + ( {\delta + k} )^2
} )}} \Big) ( {\| {u_0 - \widetilde u_0 } \|^2 + \| {v_0 - \widetilde
v_0 } \|^2 } ).
\end{align*}
Thus
\begin{align*}
& \| {F_P ( {u_0 ,v_0 } ) - F_P ( {\widetilde u_0 ,\widetilde v_0 } )} \|^2 \\
&\leq \frac{{8k^2 }}{{\delta ^2 }}\Big( 1 + \frac{{8k^2
}}{{\lambda _2^2 - 4( {( {\lambda _1 + k} )^2 + ( {\delta + k} )^2
} )}} \Big) ( {\| {u_0 - \widetilde u_0 } \|^2 + \| {v_0 - \widetilde
v_0 } \|^2 } ).
\end{align*}
So, the proof is complete.
\end{proof}

The main result in this section is the following theorem, whose
proof follows the arguments above.

\begin{theorem}\label{dlc_db}
If
$( {\lambda _1 + k} )^2 + ( {\delta + k} )^2 < \lambda _2^2/4$,
and
\[
\frac{{8k^2 }}{{\delta ^2 }}\Big( {1 + \frac{{8k^2 }}{{\lambda _2^2
- 4( {( {\delta + k} )^2 + ( {\lambda _1 + k} )^2 } )}}} \Big) < 1,
\]
then (\ref{btbd_db}) has a solution. Furthermore, this solution is unique.
\end{theorem}


\section{Main results}

In this Section, we establish existence result for the cases in
which $A - \lambda_1 I$ is regular.
Letting
\[
l := ( {| \lambda | + k_1 } )^2 + ( {| \delta | + k_1 } )^2 + ( {| \theta | + k_2 } )^2 + ( {| \gamma | + k_2} )^2.
\]
Our main result is as follows.

\begin{theorem}\label{dlc}
Suppose that $\lambda_1$ is not a eigenvalue of matrix $A$,
 $l < \lambda _2^2/2$, and
\[
\frac{{4( {k_1^2 + k_2^2 } )\big( {( {\lambda _1 - \lambda } )^2
+ ( {\lambda _1 - \gamma } )^2 + \theta ^2 + \delta ^2 } \big)}}
{{( {( {\lambda _1 - \lambda } )
( {\lambda _1 - \gamma } ) - \theta \delta } )^2 }}
\Big( {1 + \frac{{4( {k_1^2 + k_2^2 } )}}
{{\lambda _2^2 - 2l}}} \Big) < 1.
\]
Then, \eqref{btbd} has a unique solution $( {u,v} )$ in
$L^2 ( \Omega ) \times L^2 ( \Omega )$.
\end{theorem}

For the proof of the above theorem we need some lemmas.

\begin{lemma}\label{bd4}
For each $( {u_0 ,v_0 } ) \in X \times X$ fixed, if $l < \lambda _2^2$
then  (\ref{hty}) has a unique solution $( {z_0 , w_0 } ) \in Y \times Y$.
\end{lemma}

As in Lemma \ref{bd1}, it is easy to verify the statement of the above lemma.
This result let us to define
\begin{align*}
 T:X \times X&\to Y \times Y,\\
 ( {u_0 ,v_0 } ) &\mapsto T( {u_0 ,v_0 } ) := ( {z_0 ,w_0 } ),
\end{align*}
where $( {z_0 ,w_0 } )$ is the unique solution of (\ref{hty}).

\begin{lemma}\label{bd5}
If $l < \lambda _2^2/2$ then
\begin{equation}\label{dk5}
\| {T( {u_0 ,v_0 } ) - T( {\widetilde u_0 ,\widetilde v_0 } )}
\|^2 \leq \frac{{4( {k_1^2 + k_2^2 } )}}{{\lambda _2^2 - 2l}}
\big( {\|
{u_0 - v_0 } \|^2 + \| {\widetilde u_0 - \widetilde v_0 } \|^2 }
\big).
\end{equation}
\end{lemma}

It is easy to check the statement of the above lemma.

\begin{lemma}\label{bd6}
If $l < \lambda _2^2/2$ and
\begin{equation}\label{dk6}
\frac{{4( {k_1^2 + k_2^2 } )( {( {\lambda _1 - \lambda } )^2
+ ( {\lambda _1 - \gamma } )^2 + \theta ^2 + \delta ^2 } )}}
{{( {( {\lambda _1 - \lambda } )( {\lambda _1 - \gamma } )
- \theta \delta } )^2 }}\Big( {1 + \frac{{4( {k_1^2 + k_2^2 } )}}
{{\lambda _2^2 - 2l}}} \Big) < 1
\end{equation}
then (\ref{htx}) has a unique solution in $X \times X$.
\end{lemma}

\begin{proof}
By Lemma \ref{bd4}, we obtain
\begin{equation}
\begin{gathered}
u_0 = P( { - \Delta } )^{ - 1} [ {\lambda ( {u_0 + z_0} )
+ \delta ( {v_0 + w_0} ) + f( {u_0 + z_0, v_0 + w_0} )} ],\\
v_0 = P( { - \Delta } )^{ - 1} [ {\theta ( {u_0 + z_0} )
+ \gamma ( {v_0 + w_0} ) + g( {u_0 + z_0, v_0 + w_0} )} ].
\end{gathered}
\end{equation}
It follows from the properties of $P$ that
\begin{equation}
\begin{gathered}
u_0 = P( { - \Delta } )^{ - 1} [ {\lambda u_0 + \delta v_0
+ f( {u_0 + z_0, v_0 + w_0} )} ],\\
v_0 = P( { - \Delta } )^{ - 1} [ {\theta u_0 + \gamma v_0
+ g( {u_0 + z_0, v_0 + w_0} )} ],
\end{gathered}
\end{equation}
which implies
\begin{equation}\label{bd4_1}
\begin{gathered}
u_0 = \frac{\lambda }{\lambda _1} u_0 + \frac{\delta}{\lambda _1} v_0
+ P( { - \Delta } )^{ - 1} [f( {u_0 + z_0, v_0 + w_0} ) ],\\
v_0 = \frac{\theta }{\lambda _1} u_0 + \frac{\gamma}{\lambda _1} v_0
+ P( { - \Delta } )^{ - 1} [g( {u_0 + z_0, v_0 + w_0} ) ].
\end{gathered}
\end{equation}
By solving the system (\ref{bd4_1}), we have
\begin{gather*}
 u_0 = \frac{{\lambda _1 ( {\lambda _1 - \gamma } )P( { - \Delta } )^{ - 1} [ {f} ] + \lambda _1 \delta P( { - \Delta } )^{ - 1} [{g} ]}}
{{( {\lambda _1 - \lambda } )( {\lambda _1 - \gamma } ) - \theta \delta }}
= :F_P^{( 1 )} ( {u_0 ,v_0 } ), \\
 v_0 = \frac{{\lambda _1 ( {\lambda _1 - \lambda } )P( { - \Delta } )^{ - 1} [ {g} ] + \lambda _1 \theta P( { - \Delta } )^{ - 1} [ {f} ]}}
{{( {\lambda _1 - \lambda } )( {\lambda _1 - \gamma } ) - \theta \delta }}
= :F_P^{( 2 )} ( {u_0 ,v_0 } ).
\end{gather*}
Hence, we obtain
\begin{align*}
& F_P^{( 1 )} ( {u_0 ,v_0 } ) - F_P^{( 1 )} ( {\widetilde u_0 ,
\widetilde v_0 } ) \\
&= \frac{\lambda _1 ( {\lambda _1 - \gamma } )}{{( {\lambda _1 - \lambda } )( {\lambda _1 - \gamma } ) - \theta \delta }} P( { - \Delta } )^{ - 1} [ {f( {u_0 + z_0 ,v_0 + w_0 } ) - f( {\widetilde u_0 + \widetilde z_0 ,\widetilde v_0 + \widetilde w_0 } )} ] \\
& + \frac{\lambda _1 \delta}{{( {\lambda _1 - \lambda } )( {\lambda _1 - \gamma } ) - \theta \delta }} P( { - \Delta } )^{ - 1} [ {g( {u_0 + z_0 ,v_0 + w_0 } ) - g( {\widetilde u_0 + \widetilde z_0 ,\widetilde v_0 + \widetilde w_0 } )} ].
\end{align*}
Thus
\begin{align*}
&\| {F_P^{( 1 )} ( {u_0 ,v_0 } ) - F_P^{( 1 )} ( {\widetilde u_0 ,\widetilde v_0 } )} \| \\
&\leq \frac{| {\lambda _1 - \gamma } |}{{| {( {\lambda _1 - \lambda } )( {\lambda _1 - \gamma } ) - \theta \delta } |}} \| {f( {u_0 + z_0 ,v_0 + w_0 } ) - f( {\widetilde u_0 + \widetilde z_0 ,\widetilde v_0 + \widetilde w_0 } )} \| \\
&\quad +\frac{ | \delta |}{{| {( {\lambda _1 - \lambda } )( {\lambda _1
- \gamma } ) - \theta \delta } |}}\| {g( {u_0 + z_0 ,v_0 + w_0 } )
- g( {\widetilde u_0 + \widetilde z_0 ,\widetilde v_0 + \widetilde
w_0 } )} \|.
\end{align*}
Similarly, we have
\begin{align*}
&\| {F_P^{( 2 )} ( {u_0 ,v_0 } ) - F_P^{( 2 )} ( {\widetilde u_0 ,\widetilde v_0 } )} \| \\
&\leq  \frac{| {\lambda _1 - \lambda } |}{{| {( {\lambda _1 - \lambda } )( {\lambda _1 - \gamma } ) - \theta \delta } |}} \| {f( {u_0 + z_0 ,v_0 + w_0 } ) - f( {\widetilde u_0 + \widetilde z_0 ,\widetilde v_0 + \widetilde w_0 } )} \| \\
&\quad +\frac{ | \theta |}{{| {( {\lambda _1 - \lambda } )( {\lambda _1
- \gamma } ) - \theta \delta } |}}\| {g( {u_0 + z_0 ,v_0 + w_0 } )
- g( {\widetilde u_0 + \widetilde z_0 ,\widetilde v_0 + \widetilde
w_0 } )} \|.
\end{align*}
Thus
\begin{align*}
&\| {F_P ( {u_0 ,v_0 } ) - F_P ( {\widetilde u_0 ,\widetilde v_0 } )}
 \|^2 \\
&\leq  \frac{{( {\lambda _1 - \gamma } )^2
 + ( {\lambda _1 - \lambda } )^2 + \delta ^2
 + \theta ^2 }}{{( {( {\lambda _1 - \lambda } )( {\lambda _1 - \gamma } )
 - \theta \delta } )^2 }}
\Big( \| {f( {u_0 + z_0 ,v_0 + w_0 } ) - f( {\widetilde u_0
+ \widetilde z_0 ,\widetilde v_0 + \widetilde w_0 } )} \|^2 \\
&\quad+ \| {g( {u_0 + z_0 ,v_0 + w_0 } ) - g( {\widetilde u_0 +
\widetilde z_0 ,\widetilde v_0 + \widetilde w_0 } )} \|^2 \Big),
\end{align*}
where
\[
F_P ( {u_0, v_0} ) = ( {F_P^{( 1 )} ( {u_0, v_0} ),F_P^{( 2 )}
( {u_0, v_0} )} ).
\]
Using our Lipschitzian assumptions,
\begin{align*}
& \| {F_P ( {u_0 ,v_0 } ) - F_P ( {\widetilde u_0 ,\widetilde v_0 } )} \|^2 \\
&\leq  4( {k_1^2 + k_2^2 } ) \frac{{( {\lambda _1 - \gamma } )^2
+ ( {\lambda _1 - \lambda } )^2 + \delta ^2
+ \theta ^2 }}{{( {( {\lambda _1 - \lambda } )( {\lambda _1 - \gamma } )
- \theta \delta } )^2 }}
 \Big( {1 + \frac{{4( {k_1^2 + k_2^2 } )}}{{\lambda _2^2 - 2l}}} \Big) \\
&\quad\times ( {\| {u_0 - \widetilde u_0 } \|^2 + \| {v_0 - \widetilde v_0 }
\|^2 } )
\end{align*}
which completes the proof.
\end{proof}

The proof of Theorem \ref{dlc} is similar to the proof of
Theorem \ref{dlc_db}; therefore, we omit it.

\subsection*{Acknowledgments}
Part of this research was done in Spring of 2005, while
I was a student at the Department of Mathematics,
Mechanics and Informatics, College of Science, Vietnam National
University. I thank the department members for their hospitality.
The author would like to thank also Professor
Ho\`{a}ng Quoc Toan, for his interest, encouragement,
fruitful discussions and helpful comments.
Special thanks goes to Professor Tr\^{a}n Huy Ho for his interest
 in my work and valuable conversations.
This work cannot be done without the support and understanding from my family. I would like to thank my daddy Ngo D. Quoc, my mommy Phan T. T. Huyen and my younger sister Ngo T. H. Trang for their endless love.
Also, I want to thank the anonymous referee for his/her
valuable  comments and suggestions.

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\end{document}