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\markboth{\hfil A unique continuation property  \hfil EJDE--2001/46}
{EJDE--2001/46\hfil A. Anane, O. Chakrone, Z. El Allali, \&  I. Hadi \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 46, pp. 1--20. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
\vspace{\bigskipamount} \\
%
  A unique continuation property for linear elliptic systems and
  nonresonance problems
%
\thanks{ {\em Mathematics Subject Classifications:} 35J05, 35J45, 35J65.
\hfil\break\indent
{\em Key words:} Unique continuation, eigensurfaces, nonresonance problem.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted January 28, 2000. Published June 20, 2001.} }
\date{}
%
\author{ A. Anane, O. Chakrone, Z. El Allali, \&  I. Hadi }
\maketitle

\begin{abstract}
 The aim of this paper is to study the existence of solutions for a
 quasilinear elliptic system  where the nonlinear term is a Caratheodory
 function on a bounded domain of $\mathbb{R}^N$, by proving the well 
 known unique continuation property for elliptic system in all dimensions:
 1, 2, 3, \dots and the strict monotonocity of eigensurfaces.
 These properties let us to consider the above problem as a nonresonance
 problem.
\end{abstract}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{defn}{Definition}
\newtheorem{rem}{Remark}

\section{Introduction}

We study the existence of solutions for the quasilinear elliptic system
\begin{equation}  \label{A}
\begin{gathered}
-\Delta u_i =\sum_{j=1}^na_{ij}u_j+f_i(x,u_1,\ldots,u_n,\nabla
u_1,\ldots,\nabla u_n) \quad\text{in }\Omega,\\
u_i  =  0 \quad\text{on }\partial\Omega, \; i=1,\ldots,n,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^N$ ($N\geq 1$) is a bounded domain, and the
coefficients $a_{ij}$ ($1\leq i,j \leq n$) are
constants satisfying $a_{ij}=a_{ji}$,  for all $i,j$. The
nonlinearity $f_i:\Omega\times \mathbb{R}^N\times\mathbb{R}^{2N}\to 
\mathbb{R}
(1\leq i\leq n)$ is a Carath\'eodory function. The
case where $n=2$ and $f_i$  $(1\leq i\leq n)$ is independent of
$\nabla u_i$ $(1\leq i\leq n$)  has been studied by several
authors, in particular by Costa and Magalh\~aes in
\cite{C-M}.

This paper is organized as follows. First, we study the unique
continuation property in dimension $N\geq 3$ (section 2),
for systems of differential inequalities of the form
$$
|\Delta u_i(x)|\leq
K\sum_{j=1}^n|u_j(x)|+m(x)|u_i(x)| \quad\text{a.e. }x\in \Omega,
\; 1\leq i\leq n,
$$
where $m\in F^{\alpha,p}$, $0<\alpha<1$ and $p>1$. Here
$F^{\alpha,p}$ denotes the set of  functions of class
Fefferman-Phong. In our proof of Theorem 2, we make  use
of a number results and techniques developed in
\cite{ST,C-SA,L}. Secondly, we study the unique
continuation property in dimension $N=2$ (section 3), for
linear elliptic systems of the  form
$$
-\Delta u_i  = \sum_{j=1}^n a_{ij} u_j + m(x) u_i
\quad\text{in }\Omega\,\; i=1,\ldots,n\,,
$$
where $m$ satisfies the $L_{\log} L$ integrability condition. There
is extensive literature on unique continuation; we refer the reader to
\cite{L,D-G,J-K, K, G-L}.
The purpose of section 4 is to show that strict
monotonicity of eigenvalues for the linear elliptic system
$$
\begin{gathered}
-\Delta u_i = \sum_{j=1}^n a_{ij} u_j +
\mu m(x) u_i \quad\text{in }\Omega,\\
u_i  =  0 \quad\text{on }\partial\Omega ,\;  i=1,\ldots,n
\end{gathered}$$
holds if some unique continuation property is satisfied by the
corresponding eigenfunctions. Here $a_{ij}=a_{ji}$ for all $i\neq
j$, $\mu \in \mathbb{R}$ and
$m\in {\cal{M}} =\{m\in L^{\infty}(\Omega);\mathop{\rm meas}
(x\in \Omega / m(x)>0)\neq 0\}$. This result will be used for
the applications in section 6. In section 5, we study the first
order spectrum for linear elliptic systems and strict monotonicity
of eigensurfaces. This spectrum is defined as the set of couples
$(\beta,\alpha)\in \mathbb{R}^N\times\mathbb{R}$ such that
\begin{equation} \label{B}
\begin{gathered}
-\Delta u_i = \sum_{j=1}^na_{ij}u_j+\alpha
m(x)u_i+\beta\cdot\nabla u_i \quad\text{in }\Omega,\\
u_i  =  0 \quad\text{on }\partial\Omega,\; i=1,\ldots,n
\end{gathered}
\end{equation}
has a nontrivial solution $U=(u_1,\ldots,u_n) \in (H^1_0(\Omega))^n$.
We denote this spectrum  by
$\sigma_1(-\overrightarrow\Delta-A,m)$ where $A=(a_{ij})_{1\leq
i,j\leq n}$ and $m\in {\cal{M}}$. This spectrum is made by an
infinite sequence of eigensurfaces $\Lambda_1, \Lambda_2, \ldots$
(cf. section 5 and \cite{A-C-E} in the case $n=2$). Finally, in
section 6 we apply our results to obtain the existence of
solutions to (\ref{A}) under the condition of nonresonance with
respect to $\sigma_1(-\overrightarrow\Delta-A,1)$.

We use the notation
$$U=\left(\begin{array}{c}
u_1\\ \vdots \\ u_n \end{array}\right),
\quad -\overrightarrow\Delta U=\left(
\begin{array}{c}
-\Delta u_1\\ \vdots \\ -\Delta u_n \end{array}\right),
\quad \nabla U=\left(
\begin{array}{c}
\nabla u_1\\ \vdots \\ \nabla u_n \end{array}\right),
\quad F=\left(
\begin{array}{c}
f_1\\ \vdots \\ f_n \end{array} \right).
$$
We denote by 
$\sigma(-\Delta)=\{\lambda_1,\lambda_2,\ldots,\lambda_j,\ldots\}$
the spectrum of  $-\Delta$ on $H^1_0(\Omega)$.
For $\beta\in \mathbb{R}^N$, we denote
$$(\beta \xi)=\left(
\begin{array}{c}
\beta.\xi_1\\
\vdots \\
\beta.\xi_n
\end{array}
\right) ,\quad  |s|^2=\sum\limits_{i=1}^n|s_i|^2, \quad
|\xi|^2=\sum\limits_{i=1}^n|\xi_i|^2.$$
In the space $(H^1_0(\Omega))^n$ we use the induced inner product
$$\langle U, \Phi\rangle=\sum\limits_{i=1}^n\langle 
u_i,\varphi_i\rangle\quad
\forall U=(u_1,\ldots,u_n), \Phi=(\varphi_1,\ldots,\varphi_n)\in
(H^1_0(\Omega))^n$$ and corresponding norm
$$\|U\|^2_{1,2,\beta}=\int_{\Omega}e^{\beta.x}|\nabla U|^2 
dx=\sum\limits_{i=1}^n\|u_i\|^2_{1,2,\beta},$$
which is equivalent to the original norm.

\section{The unique continuation property for linear elliptic systems
in dimensions $N\geq 3$}

We will say that a family of  functions has the unique
continuation property, if no function, besides possibly the zero
function, vanishes on a set of positive measure.

In this section, we proceed to establish the unique continuation
property when $m \in F^{\alpha,p}$, $0<\alpha<1$ and $p>1$ in
dimension $N\geq 3$.
The proof of the main result is based on the Carleman's inequality
with weight.

\begin{thm}[Carleman's inequality with weight]
Let $m\in F^{\alpha,p}$, $0<\alpha\leq \frac{2}{N-1}$ and $p>1$.
Then there exists  a constant $c=c(N,p)$ such that
\begin{equation}
\label{C} \left(\int_{\mathbb{R}^N}|e^{\tau x_N}f|^sm\right)^{1/s}
\leq c\|m\|^{2/s}_{F^{\alpha,p}} \left(\int_{\mathbb{R}^N}|e^{\tau
x_N}\Delta f|^rm^{1-r}\right)^{1/r},
\end{equation}
for all $\tau\in \mathbb{R}\backslash\{0\}$,  and all $f\in S(\mathbb{R}^N)$ 
where
$\frac{1}{r}-\frac {1}{s}=\frac{2}{N+1}$ and
$\frac{1}{r}+\frac{1}{s}=1$.
\end{thm}
For the proof of this theorem see \cite{L}.

\begin{thm}
Let $X$ be an open subset in $\mathbb{R}^N$ and $U=(u_1,\ldots,u_n) \in
(H^{2,r}_{{\rm loc}}(X))^n$ ($r=\frac{2(N+1)}{N+3}$) be a solution of
the following differential inequalities:
\begin{equation}
\label{D} |\Delta u_i(x)|\leq
K\sum_{j=1}^n|u_j(x)|+m(x)|u_i(x)| \quad\text{a.e. }x\in X\; 1\leq i\leq n,
\end{equation}
where $K$ is a constant and $m$ is a locally positive function in
$F^{\alpha,p}$, with $\alpha=\frac{2}{N-1}$ and $p>1$, i.e.
$$\lim_{r\to 0}\|\chi_{\{x:|x-y|<r\}}m\|_{F^{\alpha,p}}\leq
c(N,p) \quad \forall y\in X.$$
Then, if $U$ vanishes on an open $X
\subset \Omega$, $U$ is identically null in $\Omega$.
\end{thm}

\begin{lem}
Let $U=(u_1,\ldots,u_n)\in (H^{2,r}_{{\rm loc}}(X))^n$
($r=\frac{2(N+1)}{N+3}$) be a solution of (\ref{D}) in a
neighborhood of a sphere $S$. If $U$ vanishes in one side of $S$,
then $U$ is identically null in the neighborhood of $S$.
\end{lem}

\paragraph{Proof.}
We may assume without loss generality that $S$ is centered at
$-1=(0,\ldots,-1)$ and has radius 1.
By the reflection principle (see \cite{ST}), we can also suppose
that $U=0$ in the exterior
neighborhood of $S$.

Now, let $\varepsilon >0$  small enough such that $U(x)=0$ when
$|(x+1)|>1$ and $|x|<\varepsilon$. Set $f_i(x)=\eta(|x|)u_i(x)$
for each $i=1,\ldots,n$ where $\eta\in
C^{\infty}_0([-\varepsilon,\varepsilon])$, $\eta(|x|)=1$ if
$|x|<\varepsilon/2$. For fixed $\rho$ such that
$0<\rho<\frac{\varepsilon}{2}$, let $B_{\rho}$ the ball
of radius $\rho$ centered at zero.
By the Carleman inequality, theorem 1 yields
\begin{equation}
\label{E} \Big(\int_{B_{\rho}}|e^{\tau
x_N}f_i|^sm\Big)^{1/s}\leq c
\|\chi_{B_{\rho}}m\|^{2/s}_{F^{\alpha,p}}\Big(\int_{\mathbb{R}^N}|e^{\tau
x_N}\Delta f_i|^rm^{1-r}\Big) ^{1/r}, \quad\forall \tau
>0
\end{equation}
for $i=1,\ldots,n$. Inequality (\ref{E}) implies
\begin{eqnarray} \label{F}
\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^s m\Big)^{1/s}
& \leq & c\|\chi_{B_{\rho}}m\|^{2/s}_{F^{\alpha,p}}
\Big\{\Big(\int_{\mathbb{R}^N\backslash B_{\rho}}|e^{\tau x_N}\Delta 
f_i|^rm^{1-r}\Big)
^{1/r} \nonumber \\
&&+\Big(\int_{B_{\rho}}|e^{\tau x_n}\Delta f_i|^rm^{1-r}\Big)
^{1/r}\Big\}.
\end{eqnarray}
>From (\ref{D}), we have
\begin{eqnarray}
\Big(\int_{B_{\rho}}|e^{\tau x_N} \Delta f_i|^rm^{1-r}\Big)^{1/r}
&\leq&  c\sum_{j=1}^n\Big(\int_{B_{\rho}}|e^{\tau x_N}f_j|^r
m^{1-r}\Big) ^{1/r} \nonumber \\
&&+\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^rm\Big)^{1/r} \label{G}
\end{eqnarray}
for each $i=1,\ldots,n$. Using the H{\"o}lder's inequality, we obtain
\begin{eqnarray}
\label{H} \Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^r m\Big)^{1/r}
& =  & \Big(\int_{B_{\rho}}|e^{\tau 
x_N}f_i|^rm^{r/s}m^{1-r/s}\Big)^{1/r}\nonumber\\
& \leq & \Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^s m\Big)^{1/s}
\Big(\int_{B_{\rho}}m\Big)^{\frac{1}{r}-\frac{1}{s}}.
\end{eqnarray}
As $m \in F^{\alpha,p}_{{\rm loc}}(X)$, it follows that
\begin{equation}
\label{I} \int_{B_{\rho}}m \leq c\rho
^{N-\alpha}\|\chi_{B_{\rho}}m\|_{F^{\alpha,p}}.
\end{equation}
Indeed, if $m \in F^{\alpha,p}_{{\rm loc}}(X)$ then
\begin{eqnarray}
\int_{B_{\rho}}m
& \leq & 
\Big(\int_{B_{\rho}}m^p\Big)^{1/p}|B_{\rho}|^{1-\frac{1}{p}}\nonumber\\
& \leq & |B_{\rho}|^{1-\alpha /N}\Big(|B_{\rho}|^{\alpha /N}
\Big(\frac{1}{|B_{\rho}|}\int_{B_{\rho}}m^p\Big)^{1/p}\Big)\nonumber\\
& \leq & c\rho ^{N-\alpha }\|\chi_{B_{\rho}}m\|_{F^{\alpha,p}}.\nonumber
\end{eqnarray}
It follows from (\ref{H}) and (\ref{I}) that
\begin{eqnarray}
\label{J} \Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^rm\Big)^{1/r}
& \leq & c\rho^{(N-\alpha)(\frac{1}{r}-\frac{1}{s})}\|\chi_{
B_{\rho}}m\|_{F^{\alpha,p}}^{\frac{1}{r}-\frac{1}{s}}
\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^sm\Big)^{1/s},\nonumber\\
& \leq & 
c\rho^{\frac{2(N-\alpha)}{N+1}}\|\chi_{B_{\rho}}m\|_{F^{\alpha,p}}^{\frac{1}{r}-\frac{1}{s}}
\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^sm\Big)^{1/s},
\end{eqnarray}
for each $i=1,\ldots,n$.\\
We may assume without loss generality that $m\geq 1$, then
$$\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^rm^{1-r}\Big)^{1/r}
\leq \Big(\int_{B_{\rho}}|e^{\tau
x_N}f_i|^rm\Big)^{1/r} \quad \forall 1\leq i\leq n.$$
>From (\ref{J}), we deduce
\begin{equation}
\label{K} \Big(\int_{B_{\rho}}|e^{\tau
x_N}f_i|^rm^{1-r}\Big)^{1/r} \leq
c\rho^{\frac{2(N-\alpha)}{N+1}}\|\chi_{B_{\rho}}m\|_{F^{\alpha,p}}^{\frac{1}{r}-\frac{1}{s}}
\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^sm\Big)^{1/s}.
\end{equation}
Therefore from (\ref{J}) and (\ref{K}), we have
\begin{eqnarray}
\label{L} \Big(\int_{B_{\rho}}|e^{\tau
x_N}f_i|^sm\Big)^{1/s}
& \leq & c\|\chi_{B_{\rho}}m\|^{2/s}_{F^{\alpha,p}}
  \Big(\int_{\mathbb{R}^N\backslash B_{\rho}}|e^{\tau x_N}\Delta f_i|^r
       m^{1-r}\Big)^{1/r}\nonumber\\
&      & + 
c\rho^{\frac{2(N-\alpha)}{(N+1)}}\|\chi_{B_{\rho}}m\|_{F^{\alpha,p}}
\sum_{j=1}^n\Big(\int_{B_{\rho}}|e^{\tau x_N}f_j|^sm\Big)^{1/s}\nonumber\\
&      & + 
c\rho^{\frac{2(N-\alpha)}{(N+1)}}\|\chi_{B_{\rho}}m\|_{F^{\alpha,p}}
\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^sm\Big)^{1/s},
\end{eqnarray}
for each $i=1,\ldots,n$.
Replacing $\alpha$ by $\frac{2}{N-1}$ in (\ref{L}), we obtain
\begin{eqnarray}
\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^sm\Big)^{1/s}
& \leq & c\|\chi_{B_{\rho}}m\|^{2/s}_{F^{\alpha,p}}
\Big(\int_{\mathbb{R}^N\backslash B_{\rho}}|e^{\tau x_N}\Delta f_i|^r
   m^{1-r}\Big)^{1/r}\nonumber\\
&      & +c\rho^{\frac{2(N-2)}{(N-1)}}\|\chi_{B_{\rho}}m\|_{F^{\alpha,p}}
    \sum_{j=1}^n\Big(\int_{B_{\rho}}|e^{\tau 
x_N}f_j|^sm\Big)^{1/s}\nonumber\\
&      & +c\rho^{\frac{2(N-2)}{(N-1)}}\|\chi_{B_{\rho}}m\|_{F^{\alpha,p}}
\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^sm\Big)^{1/s},\nonumber
\end{eqnarray}
for each $=1,\ldots,n$.
Let us choose $\rho$ small enough, such that
$$\|\chi_{B_{\rho}}m\|_{F^{\alpha,p}} \leq \frac{1}{2nc}.$$
So for $i=1,\ldots,n$,
\begin{eqnarray*}
\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^sm\Big)^{1/s}
&  \leq &  c \Big(\int_{\mathbb{R}^N\backslash B_{\rho}}|e^{\tau x_N}\Delta 
f_i|^r
m^{1-r}\Big)^{1/r} \\
&&+\frac{1}{2n}\sum_{j=1}^n\Big(\int_{B_{\rho}}|e^{\tau 
x_N}f_j|^sm\Big)^{1/s}\\
&&+\frac{1}{2n}\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^sm\Big)^{1/s},
\end{eqnarray*}
Since $f_i(x)=0$ for all $1\leq i \leq n$ when $|(x+1)|>1$ or
$|x|>\varepsilon$, we deduce that
$$\frac{n-1}{2n}\sum_{i=1}^n\Big(\int_{B_{\rho}}|e^{\tau x_N}f_i|^s
m\Big)^{1/s} \leq c e^{-\rho
\tau}\sum_{i=1}^n\Big( \int_{\mathbb{R}^N\backslash
B_{\rho}}|\Delta f_i|^rm^{1-r}\Big)^{1/r}.$$ So
\begin{equation}
\label{M} \frac{n-1}{2n}\sum_{i=1}^n
\Big(\int_{B_{\rho}}|e^{\tau(\rho+x_N)}f_i|^sm\Big)^{1/s}
\leq c\sum_{i=1}^n\Big(\int_{\mathbb{R}^N}|\Delta f_i|^r m^{1-r}\Big)^{1/r}.
\end{equation}
Taking $\tau \to +\infty$ in (\ref{M}), we conclude that
$U=0$ in $B_{\rho}$. \quad$\Box$

\paragraph{Proof of Theorem 2}
We assume that $U\not\equiv 0$ on X. Let $\Omega$ be a maximal
open set on which $U$ vanishes and $\Omega\neq X$, then there
exists a sphere $S$ which its interior is contained in $\Omega$,
such that there exists $x\in \partial\Omega \cap S$. As $U$
vanishes in one side of $S$, it follows that $x\in \Omega$, which
is absurd.\quad$\Box$

\section{The unique continuation property for linear elliptic systems in
dimension $N = 2$}

In this section we prove the unique continuation property where
$m\in L_{\log}L$ in lower dimension by using the zero of infinite
order theory.
\begin{defn}\rm
Let $\Omega$ be an open subset in $\mathbb{R}^N$. A function
$U=(u_1,\ldots,u_n) \in (L^2_{{\rm loc}}(\Omega))^n$ has a zero of
infinite order at $x_0\in \Omega$, if for each $l\in \mathbb{N}$
$$\lim_{R\to 0}R^{-l}\int_{|x-x_0|<R}|U(x)|^2dx=0.$$
Let us denote by $\psi$ the N-function
$$\psi(t)=(1+t)\log(1+t)-t, \quad t\geq 0$$
and by $L^{\psi}$ the corresponding Orlicz space (see \cite{K-R}).
\end{defn}

\begin{thm}
Let $\Omega$ be a bounded open subset in $\mathbb{R}^2$ and $m\in
L^{\psi}_{{\rm loc}}(\Omega)$. Let $U=(u_1,\ldots,u_n)\in
(H^1_{{\rm loc}}(\Omega))^n$ be a solution of the linear
elliptic system
\begin{equation}
\label{N} -\Delta u_i  =  \sum_{j=1}^n a_{ij} u_j +
m(x) u_i \quad\text{in }\Omega ; \quad i=1,\ldots,n
\end{equation}
where the coefficients $a_{ij} (1\leq i,j \leq n)$ are assumed to
be constants satisfying $a_{ij}=a_{ji} \; \forall i,j$. If $U$
vanishes on a set $E\subset \Omega$ of positive measure, then
almost every point of $E$ is a zero of infinite order for $U$.
\end{thm}
The proof of this theorem is done in several lemmas.

\begin{lem}
Let $\omega$ be a bounded open subset in $\mathbb{R}^2$ and $m\in
L^{\psi}(\omega)$. Then for any $\varepsilon$ there exists
$c_{\varepsilon}=c_{\varepsilon}(\omega,m)$ such that
\begin{equation}
\label{O} \int_{\omega}m u^2 \leq \varepsilon \int_{\omega}|\nabla
u|^2+c_{\varepsilon} \int_{\omega}u^2
\end{equation}
for all $u\in H^1_0(\omega)$.
\end{lem}
For a proof of this lemma, see \cite{B-K}.

\begin{lem}
Let $U$ be a solution of system (\ref{N}), $B_r$ and $B_{2r}$ be
two concentric balls contained in $\Omega$. Then
\begin{equation}
\label{P} \int_{B_r}|\nabla U|^2 \leq
\frac{c}{r^2}\int_{B_{2r}}|U|^2,
\end{equation}
where the constant $c$ does not depend on $r$.
\end{lem}

\paragraph{Proof.}
Let $\varphi$, with $\mathop{\rm supp} \varphi \subset B_{2r}, \varphi(x) = 
1
\quad\text{for } x\in B_r \quad \text{and}\quad |\nabla \varphi|
\leq \frac{2}{r}$.
Using $\varphi^2 U$ as test function in (\ref{N}), we get
$$\int_{\Omega}-\overrightarrow \Delta 
U.(\varphi^2U)=\int_{\Omega}AU.(\varphi^2U)
+\int_{\Omega}mU.(\varphi^2U).$$ So
\begin{equation}
\label{Q} \int_{\Omega}|\nabla
U|^2\varphi^2=\int_{\Omega}(AU.U)\varphi^2-2\int_{\Omega}
\langle\varphi \nabla U,\nabla \varphi\
U\rangle+\int_{\Omega}m\varphi^2U^2.
\end{equation}
On the other hand, we have
$$AU(x).U(x)\leq \rho(A) U(x).U(x) \quad\text{a.e. }x\in \Omega,$$
where $\rho(A)$ is the largest eigenvalue of the matrix $A$. Using
Schwartz and Young's inequalities, we have
\begin{equation}
\label{R} 2|\langle \varphi\nabla U, \nabla\varphi U\rangle|\leq
\varepsilon|\varphi \nabla U|^2 + \frac{|\nabla\varphi\
U|^2}{\varepsilon}\quad\text{for }\varepsilon>0.
\end{equation}
Thus, by lemma 2, we have for any $\varepsilon >0$, there exists
$c_{\varepsilon}=c_{\varepsilon}(\Omega,m)$ such that
\begin{equation}
\label{S} \int_{\Omega}m|\varphi U|^2\leq \varepsilon
\int_{\Omega}|\nabla(\varphi U)|^2+
c_{\varepsilon}\int_{\Omega}|\varphi U|^2.
\end{equation}
It follows from (\ref{Q}), (\ref{R}) and (\ref{S}) that
\begin{eqnarray}
\int_{B_{2r}}\varphi^2|\nabla U|^2 & \leq &
\rho(A)\int_{B_{2r}}|\varphi U|^2+\varepsilon\int_{B_{2r}}
|\nabla 
U|^2\varphi^2+\frac{1}{\varepsilon}\int_{B_{2r}}|\nabla\varphi|^2U^2\nonumber\\
&      & +\varepsilon\int_{B_{2r}}|\nabla(\varphi 
U)|^2+c_{\varepsilon}\int_{B_{2r}}|\varphi U|^2,\nonumber
\end{eqnarray}
and therefore
$$(1-(\varepsilon^2+2\varepsilon))\int_{B_{2r}}\varphi^2|\nabla U|^2\leq 
(\varepsilon+1+\frac{1}
{\varepsilon})\int_{B_{2r}}|(\nabla\varphi\
U)|^2+(\rho(A)+c_{\varepsilon})\int_{B_{2r}} |\varphi U|^2.$$
Using the fact that $|\nabla \varphi|\leq \frac{2}{r}$,
$|\varphi|\leq\frac{c}{r}$ and $\varphi =1$ in $B_r$, we have
immediately (\ref{P}). \quad$\Box$

\begin{rem} \rm
If $U$ has a zero of infinite order at $x_0\in \Omega$, then
$\nabla U$ has also a zero of infinite order at $x_0$.
\end{rem}

\begin{lem}[\cite{L-U}]
Let $u\in W^{1,1}(B_r)$, where $B_r$ is the ball of radius $r$ in
$\mathbb{R}^N$ and let $E=\{x\in B_r: u(x)=0\}$. Then there exists a
constant $\beta$ depending only on $N$ such that
$$\int_{D}|u| \leq \beta \frac{r^N}{|E|}|D|^{1/N}\int_{B_r}|\nabla u|$$
for all $B_r$, $u$ as above and all measurable sets $D\subset
B_r$.
\end{lem}

\paragraph{Proof of Theorem 3.}
Let $U=(u_1,\ldots,u_n)\in(H^1_{{\rm loc}}(\Omega))^n$ be a solution of
(\ref{N}) which vanishes on a set $E$ of positive measure. We know
that almost every point of $E$ is a point of density of $E$. Let $x_0$ be 
such a point, i.e.
\begin{equation} \label{T}
\frac{|E^c\cap B_r|}{|B_r|}\to 0\quad\text{and}\quad
\frac{|E\cap B_r|}{|B_r|}\to 1\quad\text{as }r\to 0,
\end{equation}
where $B_r$ is the ball of radius $r$ centered at $x_0$. So, for a
given $\varepsilon>0$ there exists $r_0=r_0(\varepsilon)>0$ such
that for $r\leq r_0$
$$\frac{|E^c\cap B_r|}{|B_r|}<\varepsilon\quad\text{and}\quad
\frac{|E\cap B_r|}{|B_r|}>1-\varepsilon,$$
where $E^c$ denotes the complement of $E$ in $\Omega$. Taking
$r_0$ smaller if necessary, we may assume that
$\overline{B}_{2r_0}\subset \Omega$. By lemma 4 we have
\begin{eqnarray}
\int_{B_r}|u_i|^2=\int_{B_r\cap E^c}|u_i|^2& \leq  &\beta
\frac{r^2}{|E\cap B_r|}|E^c\cap B_r|^{1/2}
     \int_{B_r}|\nabla (u_i)^2|\nonumber\\
   &  \leq  & 2\beta \frac{r^2}{|E\cap B_r|}|E^c\cap B_r|^{1/2}
\int_{B_r}|u_i||\nabla u_i|\nonumber
\end{eqnarray}
for each $i=1,\ldots,n$.
The H\"{o}lder and Youngs's inequalities lead to
\begin{eqnarray}
\label{U} \int_{B_r}|u_i|^2 & \leq & 2\beta \frac{r^2}{|E\cap
B_r|}|E^c\cap B_r|^{1/2}
       \Big(\int_{B_r}|u_i|^2\Big)^{1/2}\Big(\int_{B_r}|\nabla u_i|^2
           \Big)^{1/2}\nonumber\\
    & \leq & \beta \frac{r^2}{|E\cap B_r|}|E^c\cap B_r|^{1/2}
    \Big(\frac{1}{r}\int_{B_r}u_i^2+r\int_{B_r}|\nabla u_i|^2\Big),
\end{eqnarray}
for each $i=1,\ldots,n$.
We take the sum for $i=1$ to $n$, we obtain
$$\int_{B_r}|U|^2 \leq \beta \frac{r^2}{|E\cap B_r|}|E^c\cap B_r|^{1/2}
\Big(\frac{1}{r}\int_{B_r}|U|^2+r\int_{B_r}|\nabla U|^2\Big).
$$
It follows from (\ref{P}) and (\ref{T}) that
\begin{eqnarray}
\label{V} \int_{B_r}|U|^2 & \leq & \beta
\frac{r^2|B_r|}{|B_r|^{1/2}|E\cap B_r|}\frac{|E^c\cap
B_r|^{1/2}}{|B_r|^{1/2}}
\Big(\frac{1}{r}\int_{B_{2r}}|U|^2+\frac{c}{r}\int_{B_{2r}}|U|^2\Big)\nonumber\\
& \leq & \beta \frac{c 
r}{|B_r|^{1/2}}\frac{\varepsilon^{1/2}}{1-\varepsilon}\int_{B_{2r}}|U|^2 
\nonumber\\
& \leq & c\frac{\varepsilon^{1/2}}{1-\varepsilon}\int_{B_{2r}}|U|^2,
\quad\text{for } r\leq r_0.
\end{eqnarray}
Set $f(r)=\int_{B_r}|U|^2$. Let us fix $n\in \mathbb{N}$, we
have $\varepsilon >0$ such that
$\frac{c\varepsilon^{1/2}}{1-\varepsilon}=2^{-n}$.
Observe that now
$r_0$ depend on $n$.
>From (\ref{V}), we deduce that
\begin{equation}
\label{W} f(r)\leq 2^{-n}f(2r),\quad\text{for } r\leq r_0.
\end{equation}
Iterating (\ref{W}), we get
\begin{equation}
\label{X} f(\rho)\leq 2^{-kn} f(2^k\rho)\quad\text{if }2^{k-1}\rho\leq r_0.
\end{equation}
Thus, given $0<r<r_0(n)$ and choosing $k\in \mathbb{N}$ such that
$$2^{-k}r_0\leq r \leq 2^{-(k-1)}r_0.$$
>From  (\ref{X}), we conclude that
$$f(r)\leq 2^{-kn}f(2^kr)\leq 2^{-kn}f(2r_0),$$
and since $2^{-k}\leq \frac{r}{r_0}$, we get
$$f(r)\leq \left(\frac{r}{r_0}\right)^n f(2r_0).$$
This shows that $f(r)=0(r^n) \quad\text{as } r\to 0$.
Consequently $x_0$ is a zero of infinite order for $U$. \quad$\Box$

\begin{thm}
Let $\Omega$ be an open subset in $\mathbb{R}^2$. Assume that
$U=(u_1,\ldots,u_n)\in (H^2_{{\rm loc}}(\Omega))^n$ has a zero of
infinite order at $x_0\in \Omega$ and satisfies
\begin{equation}
\label{Y} |\Delta u_i(x)|\leq
K\sum_{j=1}^n|u_j(x)|+m(x)|u_i(x)| \quad\text{a.e. }x\in \Omega,\; 1\leq 
i\leq n,
\end{equation}
where $m$ is a positive function belong to a class of
$L_{\log}L_{{\rm loc}}(\Omega)$. Then $U$ is identically null in
$\Omega$.
\end{thm}

\paragraph{Proof.}
The technique used here is due to S. Chanillo and E. Sawyer (see
\cite{C-SA}), we may assume that $m\geq 1$. Since $m+1$ also
satisfy the hypotheses of theorem 4 when $m\in
L_{\log}L_{{\rm loc}}(\mathbb{R}^{N})$, we have
\begin{equation}
\label{Z} \int_{\mathbb{R}^2}|I_1f|^2m \leq c\| 
m\|\int_{\mathbb{R}^2}f^2\quad
\forall f\in C^{\infty}_0(\mathbb{R}^2),
\end{equation}
(cf. \cite{F,STE}), where $I_{\alpha} f$ denotes the Riesz
potential of order $0<\alpha <n$, defined by
$$I_{\alpha}(x)=\int_{\mathbb{R}^N}|x-y|^{-n+\alpha}f(y)dy,$$
where one posed to simplify $\|m\|=\|m\|_{L_{\log}L}$. The
inequality (\ref{Z}) is equivalent to the dual inequality
\begin{equation}
\label{AB} \int_{\mathbb{R}^2}|I_2f|^2m \leq c\|m\|^2
\int_{\mathbb{R}^2}|f|^2m^{-1} \quad\forall f\in 
C^{\infty}_{0}(\mathbb{R}^2).
\end{equation}
(cf. \cite{FO}) where $I_2f=\phi_2\ast f$ denotes the Newton
Potential with $\phi_2$ is the elementary solution of $-\Delta$.
On the other hand, from the result of E.Sawyer (cf. \cite{S}), if
$$\phi_2(x)=\frac{1}{2\pi}\log|x|,$$
then
\begin{equation} \label{BB}
|\phi_2(x-y)-\sum_{j=0}^{l-1}\frac{1}{j!}\left(\frac{\partial}{\partial
s}\right)^j\phi_2(sx-y)|_{s=0}|\leq c\left(
\frac{|x|}{|y|}\right)^l \phi_2(x-y) \quad\forall l\in \mathbb{N}.
\end{equation}
The constant $c$ does not depend on $l, x$ and $y$.
Let $U=(u_1,\ldots,u_n)$ be a solution of (\ref{Y}) and has a zero
of infinite order at $x_0\in \Omega$. We may suppose without loss
generality that $0\in \Omega$ and $x_0=0$. Let also $\eta$ and
$\psi$ be two functions such that $\eta \in C^{\infty}_0(B_{2r}),
\eta=1$ on $B_r$ and $\psi=0$ on $B_1$,
$\psi=1\quad\text{outside }B_2$ and $0\leq\psi\leq 1$.
Set $\psi_k(x)=\psi(kx), k\geq 0.$
We also assume that $k\geq 4/r$ and $r<1/2$. Then
by (\ref{AB}, \ref{BB}) and \cite[Theoremm 4.3]{G-T}, for $l\geq 1$ we have
\begin{eqnarray}
\label{CB}
\lefteqn{\int_{B_r}\frac{|\psi_k(x)u_i(x)|}{|x|^{2l}}m(x)dx}\\
  &=& \int_{B_r}|x|^{-2l}|\int\phi_2(x-y)\Delta(\eta 
\psi_ku_i)dy|^2m(x)dx\nonumber\\
  &=& \int_{B_r}|x|^{-2l}|\int(\phi_2(x-y)-\sum_{j=0}^{l-1}\frac
       {1}{j!}\big(\frac{\partial}{\partial 
s}\big)^j\phi_2(sx-y)|_{s=0})\Delta(\eta\psi_ku_i)
       dy|^2 mdx\nonumber\\
  &\leq& c\int_{B_r}\Big(\frac{\int\phi_2(x-y)|\Delta
  (\psi_k\eta u_i)|}{|y|^l}\Big)^2m(x)dx\nonumber\\
  &\leq& c\int_{B_r}|I_2(\frac{\Delta (\psi_k\eta 
u_i)}{|y|^l})|^2m(x)dx\nonumber\\
  &\leq& c\|\chi_{B_r}m\|^2\int_{B_{2r}}\frac{|\Delta(\psi_k\eta 
u_i)|^2}{|x|^{2l}}m^{-1}(x)dx\nonumber\\
  &\leq& 
c\|\chi_{B_r}m\|^2(\int_{B_{2r}}\frac{|\Delta\psi_k|^2u_i^2}{|x|^{2l}}m^{-1}(x)dx+\int_{B_{2r}}
     \frac{|\nabla \psi_k|^2|\nabla u_i|^2}{|x|^{2l}}m^{-1}(x)dx\nonumber\\
  & &+\int_{B_r}\frac{|\psi_k|^2|\Delta(\eta 
u_i)|^2}{|x|^{2l}}m^{-1}dx)+\int_{|x|>r}\frac{|\psi_k|^2|\Delta(\eta 
u_i)|^2}{|x|^{2l}}m^{-1}dx)\nonumber\\
  &=&c\|\chi_{B_r}m\|^2(I^i_k+II_k^i+III_k^i+IV^i_k),
\end{eqnarray}
for each $i=1,\ldots,n$.
Choosing $c\|\chi_{B_r}m\|^2<\frac{1}{2n}$ (this is possible since
the measure $L_{\log}L$ is absolutely continuous) it follows that
\begin{eqnarray*}
III^i_k & \leq & \frac{1}{2n}\int_{|x|<r}\frac{|\psi_k|^2|\Delta
u_i|^2}{|x|^{2l}} m^{-1}(x)dx \\
& \leq & \frac{1}{2n}\Big(\sum_{j=1}^n\int_{|x|<r}\frac{|\psi_k|^2|u_j|^2}
{|x|^{2l}}m^{-1}(x)dx + 
\int_{|x|<r}\frac{|\psi_k|^2|u_i|^2}{|x|^{2l}}m(x)dx\Big).
\end{eqnarray*}
We have
$$\int_{|x|<r}\frac{|\psi_k|^2|u_j|^2}{|x|^{2l}}m^{-1}(x)dx \leq 
\int_{|x|<r}\frac{|\psi_k|^2|u_j|^2}
{|x|^{2l}}m(x)dx\quad\text{whenever }m\geq 1.$$ So
\begin{equation}
\label{DB}
III_k^i\leq\frac{1}{2n}\Big(\sum_{j=1}^n\int_{|x|<r}\frac{|\psi_k||u_j|^2}
{|x|^{2l}}m(x)dx+\int_{|x|<r}\frac{|\psi_k|^2|u_i|^2}{|x|^{2l}}m(x)dx\Big).
\end{equation}
As $U=(u_1,\ldots,u_n)$ is a solution of (\ref{Y}), from
(\ref{CB}) and (\ref{DB}), we conclude that
\begin{equation}
\label{EB}
(1-\frac{1}{2n})\int_{B_r}\frac{|\psi_k|^2|u_i|^2}{|x|^{2l}}m(x)dx-\frac{1}{2n}
\sum_{j=1}^n\int_{B_r}\frac{|\psi_k|^2|u_j|^2}{|x|^{2l}}m(x)dx
\leq I^i_k+II^i_k.
\end{equation}
On the other hand, we have
$$I^i_k\leq \int_{\frac{1}{k}\leq 
|x|\leq\frac{2}{k}}\frac{|\psi_k|^2|u_i|^2}{|x|^{2l}}
m^{-1}(x)dx\leq ck^{2l+4}\int_{|x|\leq \frac{2}{k}}|u_i|^2dx,$$
for each $i=1,\ldots,n$.
Hence $\lim_{k\to +\infty}
I^i_k=0\quad\forall 1\leq i\leq n$, since $U$
has a zero of infinite order at $0$ by hypothesis.
On the other side
$$II^i_k\leq ck^{2l+2} \int_{|x|\leq\frac{2}{k}}|\nabla u_i|^2dx.$$
By Remark 1, it follows that $\lim_{k\to
+\infty}
II_k^i=0\quad \forall 1\leq i\leq n$.
The sum from $i=1$ to $n$ in the inequality (\ref{EB}), yields
$$\frac{n-1}{2n}\sum_{i=1}^n\int_{B_r}\frac{|\psi_k u_i|^2}{|x|^{2l}}
m(x)dx\leq \sum_{i=1}^n(I_k^i+II_k^i+IV^i_k).$$
So that
\begin{equation}
\label{FB} \int_{|x|<r}|\psi_k U|^2m\leq
r^{2l}\int_{B_r}\frac{|\psi_k U|^2}{|x|^{2l}} \leq
\frac{2n}{n-1}r^{2l}\sum_{i=1}^n(I^i_k+II_k^i+IV^i_k).
\end{equation}
Taking the limit as $k \ \mbox{and}\ l \to +\infty$ in (\ref{FB}), we 
conclude
that $U=0$ on $B_r$.\quad$\Box$

\begin{rem} \rm
In the following sections we take $m$ in $\cal{M}$ which
is obviously a subspace of $F^{\alpha,p}$ and $L_{\log}L$. Also for those 
bounded
potential we can use the Carleman inequality of N. Arnsajn \cite{Ar}.

\end{rem}

\section{Strict monotonicity of eigenvalues for linear elliptic systems}

In this section we  study the strict monotonicity of eigenvalues
for the  linear elliptic system
\begin{equation}\label{GB}
\begin{gathered}
-\Delta u_i = \sum_{j=1}^na_{ij}u_j+\mu m(x)u_i \quad\text{in }\Omega,\\
u_i  =  0 \quad\text{on }\partial\Omega,\; i=1,\ldots,n\,.
\end{gathered}
\end{equation}
We will assume that
\begin{equation}
\label{HB} \lambda_1>\rho(A),
\end{equation}
where $\rho(A)$ is the largest eigenvalue of the matrix $A$ and
$\lambda_1$ the smallest eigenvalue of $-\Delta$ .

As it is well Known \cite{A1,H,C-S,A-C-E}, that the eigenvalues
in (\ref{GB}) form a sequence of positive eigenvalues, which can be written 
as
$$\mu_1(m)<\mu_2(m)\leq \ldots\,. $$

Here we use the symbol $\precneqq$ to indicate inequality a.e.
with strict inequality on a set of positive measure.

\begin{prop}
Let $m_1$ and $m_2$ be two weights of $M$ with $m_1\precneqq m_2$
and let $j\in \mathbb{N}$. If the eigenfunctions associated to
$\mu_j(m_1)$ enjoy the unique continuation property, then
$\mu_j(m_1)>\mu_j(m_2)$.
\end{prop}

\paragraph{Proof.}
We proceed by the similar arguments which has been developed by
D.G. de Figueiredo and J.P. Gossez \cite{D-G}. $\mu_j(m_1)$ is
given by the variational characterization
\begin{equation}
\label{IB}
\frac{1}{\mu_j(m_1)}=\sup_{F_j}\inf\{\int_{\Omega}m_1|U|^2dx;
U\in F_j \quad\text{and }  {\cal{L}}(U,U)=1\},
\end{equation}
where ${\cal {L}}(U,U)=\int_{\Omega}|\nabla
U|^2-\int_{\Omega}AU.Udx$ and $F_j$ varies over all
$j$-dimensional subspace of $(H_0^1(\Omega))^n$ (cf. \cite{A1,H,C-S}).
Since the extrema in (\ref{IB}) are achieved \cite{DF},
there exists $F_j\subset (H^1_0(\Omega))^n$ of dimension $j$ such
that
\begin{equation}
\label{JB} \frac{1}{\mu_j(m_1)}=\inf\{\int_{\Omega}m_1|U|^2dx;
U\in F_j\quad\text{and}\quad {\cal{L}}(U,U)=1\}.
\end{equation}
Pick $U\in F_j$ with ${\cal {L}}(U,U)=1$. Either $U$ achieves tits
infimum in (\ref{JB}) or not. In the first case, $U$ is an
eigenfunction associated to $\mu_j(m_1)$ (cf. \cite{DF}), and so,
by the unique continuation property
$$\frac{1}{\mu_j(m_1)}=\int_{\Omega}m_1|U|^2< \int_{\Omega}m_2|U|^2.$$
In the second case
$$\frac{1}{\mu_j(m_1)}<\int_{\Omega}m_1|U|^2\leq \int_{\Omega}m_2|U|^2.$$
Thus, in any case
$$\frac{1}{\mu_j(m_1)}<\int_{\Omega}m_2|U|^2.$$
It follows, by a simple compactness argument that
$$\frac{1}{\mu_j(m_1)} < \inf\{\int_{\Omega}m_2|U|^2; U\in F_j
\text{ and } {\cal {L}}(U,U)=1\}.$$
This yields the desired inequality
$$\frac{1}{\mu_j(m_1)}<\frac{1}{\mu_j(m_2)}.$$

\section{Spectrum for linear elliptic systems}
\subsection*{First order spectrum}
\begin{thm} \begin{enumerate}
\item[{\bf a)}] $\Lambda_n(.,A,m):\mathbb{R}^N \to \mathbb{R} $ is the
positive function characterized in a variational form by
$$\frac{1}{\Lambda_n(\beta,A,m)}=\sup _{F_n\in{\cal F}_n((H^1_0(\Omega))^n)}
\min \big\{
\int_{\Omega}e^{\beta.x}m(x)|U|^2dx,\, U\in F_n\cap S_{\beta}(A)\big\}
$$
$\text{for all } \beta \in \mathbb{R}^N$, with
$$S_{\beta}(A)=\left\{ U\in (H^1_0(\Omega))^n:
\|U\|^2_{1,2,\beta}-\int_{\Omega} e^{\beta.x}AU.U dx=1 \right\},
$$
and  ${\cal F}_n((H^1_0(\Omega))^n)$ is  the  set of $n$-dimensional
subspaces  of $(H^1_0(\Omega))^n$.

\item[{\bf b)}] For all $U\in (H^1_0(\Omega))^n$,
$$\Lambda_1(\beta,A,m)\int_{\Omega}e^{\beta.x}m(x)|U|^2dx\leq\|U\|^2_{1,2,\beta}-\int_{\Omega}e^{\beta 
.x}
AU.U\; dx \,.$$
\item[{\bf c)}] For all $\beta \in \mathbb{R}^N$,
$\lim_{n\to +\infty}\Lambda_n(\beta,A,m)=+\infty$.
\end{enumerate}
\end{thm}
For the proof of this theorem see \cite{A-C-E}.

\subsection*{Strict monotonicity of eigensurfaces for linear elliptic 
systems}

By theorem 5 it seems that the following result may be
proved by arguments similar to those in proposition 1
(see section 4).

\begin{prop}
Let $m_1, m_2 \in M$, if $m_1\precneqq m_2$ then
$\Lambda_j(\beta,A,m_1)>\Lambda_j(\beta,A,m_2)$ for all $j\in
\mathbb{N}^*$.
\end{prop}

\section{Nonresonance between consecutives eigensurfaces}

In this section, we study the existence of solutions for
the  quasilinear elliptic system
\begin{equation} \label{KB}
\begin{gathered}
-\overrightarrow\Delta U  =  AU + F(x,U,\nabla U) \quad\text{in } \Omega,\\
U  =  0 \quad\text{on }\partial\Omega.
\end{gathered}
\end{equation}
Let us consider the situation where the nonlinearity $F$ is asymptotically
between two consecutive eigensurfaces in the following sense:
we assume that there exists $\alpha_1<\alpha_2\in \mathbb{R}$, $\beta\in
\mathbb{R}^N$ and for all $\delta >0$ there exist $a_{\delta}\in
L^2(\Omega)$ such that
\begin{eqnarray} \label{LB}
\alpha_1|s|^2+(\beta\xi).s-\delta(|\xi|^2+a_{\delta}(x))|s|
&\leq& s.F(x,s,\xi)\\
&\leq& \alpha_2|s|^2+(\beta\xi).s+\delta(|\xi|^2+a_{\delta}(x))|s| \nonumber
\end{eqnarray}
a.e.  $\in \Omega$ and for all $(\xi,s)\in 
\mathbb{R}^{2N}\times\mathbb{R}^2$.

A function $U$ in $(H^1_0(\Omega))^n$ is said to be a solution of (\ref{KB}) 
if
$U$ satisfies (\ref{KB}) in the sense of distributions.
With this definition, we state the  main result of this section.

\begin{thm}
Let (\ref{LB}) be satisfied  with
$\Lambda_k(\beta,A,1)< \alpha_1<\alpha_2<\Lambda_{k+1}(\beta,A,1)$
for some $k\geq 1$, then (\ref{KB}) admits a solution.
\end{thm}

\begin{rem} \rm
It is clear that by (\ref{LB}) there exist
$b_1>0$ such that for all $\delta>0$ there exists $a_{\delta}\in 
L^2(\Omega)$
such that
\begin{equation}
\label{MB} |F(x,s,\xi)-(\beta\xi)|\leq b_1|s|+\delta
b_1(|\xi|+a_{\delta}(x))
\end{equation}
a.e. $x\in \Omega$ and for all $(\xi,s)\in 
\mathbb{R}^{2N}\times\mathbb{R}^2.$
\end{rem}

\paragraph{Proof of Theorem 6}
Let $(T_t)_{t\in[0,1]}$ be a family of operators from
$(H_0^1(\Omega))^n$ to $(H^{-1}(\Omega))^n$:
$$
T_t(U)=-\overrightarrow\Delta^{\beta}(U)-e^{\beta.x}(t(F(x,U,\nabla
U)+(1-t)\alpha U-t(\beta\nabla U))
$$
where  $\alpha_1<\alpha<\alpha_2$.
Since $F$ verifies (\ref{LB}), the operator $T_t$
is of the type $(S_+)$.
Now, we  show the a priori estimate:
$$
\exists r>0 \text{ such that } \forall t\in [0,1], \forall
U\in(\partial B(0,r))^n, \text{ we have } T_t(U)\neq 0.
$$
We proceed by contradiction, if the a priori estimate
is not true, then
$$
\forall n\in\mathbb{N},\; \exists t_n\in [0,1],\; \exists U_n\in(\partial
B(0,n))^n\; (\|U_n\|_{1,2}=n), \text{such that }
T_{t_n}(U_n)=0,
$$
so that
\begin{equation}
\label{NB} -\overrightarrow\Delta^{\beta}(U_n)=e^{\beta.x}(t_n
F(x,U_n,\nabla U_n)+ (1-t_n)\alpha U_n-t_n(\beta\nabla U_n)).
\end{equation}
Set $V_n=\frac{U_n}{\|U_n\|_{1,2}}$, the sequence
$(V_n)$ is bounded in  $(H_0^1(\Omega))^n$. Therefore, there
exists a subsequence of $(V_n)$ (also noted $(V_n)$) such that:
$V_n\rightharpoonup V \text{in } (H_0^1(\Omega))^n$,
$V_n\to V \quad\text{in } (L^q(\Omega))^n$ for all $q\in
[1,2^*[$, with $2^*=\frac{2N}{N-2}$.
Then we proceed in several steps.

\paragraph{Step 1:} The sequence of functions defined a.e.
$x\in \Omega$ by
$$
G_n(x)=\frac{F(x,U_n,\nabla U_n)}{\|U_n\|_{1,2}}-(\beta\nabla V_n)
$$
is bounded in  $(L^2(\Omega))^n$.

To prove this statement we divide (\ref{MB}) by $\|U_n\|_{1,2}$. Then
$$
|G_n(x)|\leq b_1|V_n|+\delta b_1(|\nabla
V_n|+\frac{a_{\delta}(x)}{n}).
$$
and
\begin{eqnarray}
\|G_n\|&\leq& b_1\|V_n\|_2+\delta 
b_1(\|V_n\|_{1,2}+\frac{\|a_{\delta}\|_2}{n})\nonumber\\
&\leq& \frac{b_1}{(\lambda_1)^{1/2}}+\delta
b_1(1+\frac{\|a_{\delta}\|_2}{n}),\nonumber
\end{eqnarray}
which proves step 1.

Since $(L^2(\Omega))^n$ is a reflexive space, there exists a
subsequence of $(G_n)$, also denoted by $(G_n)$, and $\tilde{F}\in
(L^2(\Omega))^n$ such that
\begin{equation}
\label{PB} G_n\rightharpoonup \tilde{F}\quad\text{in }
(L^2(\Omega))^n.
\end{equation}

\paragraph{Step 2.}
$\tilde{F}(x)=0$ a.e. in
${\cal {A}} :=\{x\in\Omega :  V(x)=0  \text{ a.e.}\}$

To prove this statement, we define $\phi (x)=\mathop{\rm 
sgn}(\tilde{F}(x))\chi_{\cal {A}}$.
By (\ref{MB}), we have
$$
|G_n(x)\phi (x)|\leq b_1(|V_n|+\delta (|\nabla
V_n|+\frac{a_{\delta}(x)}{n}))\chi_{\cal {A}}(x)
$$
and
$$
\|G_n\phi\|_2\leq a(\|V_n\chi_A\|_2+\delta
(1+\frac{\|a_{\delta}\chi_A\|_2}{n})).
$$
Since $V_n\to V$ in $(L^2(\Omega))^n$, we have $V_n\chi_{\cal {A}}\to 0 $
in $(L^2(\Omega))^n$. Passing to the limit, we obtain
$$
\limsup\, \|G_n\phi\|_2\leq \delta b_1.
$$
As $\delta$ is arbitrary, it follows that
$$G_n\phi\to 0 \quad \text{in } (L^2(\Omega))^n.$$
On the other hand, (\ref{PB}) implies
$$
\int_{\Omega} G_n.\phi\to \int_{\Omega}
\tilde{F}.\phi=\int_{\Omega} |\tilde{F}(x)|\chi_{\cal {A}}(x).
$$
So $\int_{\cal {A}} |\tilde{F}(x)|=0$, which completes the proof of step 2.

Now, we define the function
$$
D(x)=\left\{
\begin{array}{ll}
\frac{\tilde{F}(x).V(x)}{|V(x)|^2} & \text{a.e. }x\in\Omega\setminus 
{\cal {A}},\\
\alpha & \text{a.e. }x\in {\cal {A}}.
\end{array}
\right.
$$

\paragraph{Step 3.}  $\alpha_1\leq D(x)\leq \alpha_2 $ a.e. $x\in\Omega$.

First, we prove that $\alpha_1\leq\frac{\tilde{F}(x).V(x)}{|V(x)|^2}$ a.e.
$x\in \Omega\setminus {\cal{A}}$. then analogously we prove that
$\frac{\tilde{F}(x).V(x)}{|V(x)|^2}\leq \alpha_2$
a.e. $x\in \Omega\setminus {\cal{A}}$).

Set  $B=\{x\in\Omega\setminus {\cal {A}} : 
\alpha_1|V(x)|^2>\tilde{F}(x).V(x)
\text{ a.e.}\}$.
It is sufficient to show that $\mathop{\rm meas}B=0$.
Indeed, the assumption (\ref{LB}) yields
\begin{equation}
\label{OB} \alpha_1|U_n|^2-\delta(|\nabla
U_n|+a_{\delta}(x))|U_n|\leq U_n.F(x,U_n,\nabla U_n) -(\beta\nabla
U_n).U_n,
\end{equation}
dividing by $\|U_n\|_{1.2}^2$, we obtain
$$
\alpha_1|V_n|^2-\delta(|\nabla
V_n|+\frac{a_{\delta}(x)}{n})|V_n|\leq V_n.G_n(x).
$$
Multiplying (\ref{OB}) by $\chi_B$ and integrating over $\Omega$,
we have
\begin{eqnarray*}
\lefteqn{ \alpha_1\int_{\Omega} |V_n|^2\chi_B }\\
&\leq
&\delta\int_{\Omega} (|\nabla
V_n|+\frac{a_{\delta}(x)}{n})
|V_n|\chi_B+\int_{\Omega} V_n. G_n(x)\chi_B\nonumber\\
&\leq & \int_{\Omega} V_n.G_n\chi_B+
\delta\Big((\int_{\Omega} |\nabla
V_n|^2)^{1/2}(\int_{\Omega} |V_n|^2)^{1/2}+
\frac{\|a_{\delta}\|_2}{n}\|V_n\|_2\Big)\nonumber\\
&\leq & \int_{\Omega}
V_n.G_n(x)\chi_B+\delta\Big(\frac{1}{\lambda_1^{1/2}}+
\frac{\|a_{\delta}\|_2}{n\lambda_1^{1/2}}\Big).\nonumber
\end{eqnarray*}
Passing to the limit (Knowing that $G_n\rightharpoonup \tilde{F}$ and
$V_n\to V$ in $(L^2(\Omega))^n$), we get
$$
\alpha_1\int_{\Omega} |V(x)|^2\chi_B\leq
\int_{\Omega}
V(x).\tilde{F}(x)\chi_B+\frac{\delta}{\lambda_1^{1/2}},
$$
for all $\delta>0$. Then
$$\int_{\Omega} \left(V(x). \tilde{F}(x)-\alpha_1|V(x)|^2\right)\chi_Bdx\geq 
0.
$$
Finally, by the definition of $B$, we deduce that $\mathop{\rm meas} B=0$,
and the proof of step 3 concludes.

It is clear that we can suppose that $t_n\to t$. Set $m(x)=t 
D(x)+(1-t)\alpha$.

\paragraph{Step 4.}
{\bf 1)}\quad The function  $V$ is a solution of
$$ \displaylines{
-\overrightarrow\Delta^{\beta} U =e^{\beta.x}AU+e^{\beta.x}m(x)U
\quad\text{in } \Omega  ,\cr
U=0 \quad\text{on } \partial\Omega.
}$$
{\bf 2)}\quad   $\alpha_1\leq m(x)\leq \alpha_2$   a.e. $x\in\Omega$.

To prove 1), we dividing (\ref{NB}) by $n=\|U_n\|_{1,2}$. Then
\begin{equation}
\label{QB} -\overrightarrow\Delta^{\beta}V_n=
e^{\beta.x}AV_n+e^{\beta.x}(t_nG_n(x)+(1-t_n)\alpha V_n).
\end{equation}
Since $V_n\rightharpoonup  V$ in $(H_0^1(\Omega))^n$,
$$
\int_{\Omega} e^{\beta.x}\nabla
V_n.\nabla\Phi\to \int_{\Omega}
e^{\beta.x}\nabla V.\nabla\Phi\quad \text{for all }\Phi\in
(H_0^1(\Omega))^n.
$$
On the other hand, multiplying (\ref{QB}) by $\Phi\in
(H_0^1(\Omega))^n$, as  $n\to +\infty$ we obtain
\begin{eqnarray*}
\int_{\Omega} e^{\beta.x}\nabla V.\nabla\Phi
&=&\int_{\Omega}AV.\Phi + \int_{\Omega}
e^{\beta.x}(t\tilde{F}(x)+(1-t)\alpha V(x)).\Phi, \nonumber\\
&=&
\int_{\Omega}e^{\beta.x}AV.\Phi+\int_{\Omega}e^{\beta.x}
\Big(t\frac{\tilde{F}(x).V(x)}{|V(x)|^2}+(1-t)\alpha\Big)V(x).\Phi\nonumber\\
&=&
\int_{\Omega}e^{\beta.x}AV.\Phi+\int_{\Omega}e^{\beta.x}
\left(tD(x)+(1-t)\alpha\right)V(x).\Phi\nonumber
\end{eqnarray*}
From the second step and the definition of $D(x)$ it follows that
$$-\overrightarrow\Delta^{\beta}V= e^{\beta.x}AV+e^{\beta.x}m(x)V
\quad\text{in  }  (H^{-1}(\Omega))^n.$$
Then  assertion 1) follows.

To prove 2), we combine the result of step 3 and the fact that
$\alpha_1<\alpha<\alpha_2$.

\paragraph{Step 5.}  $V\not\equiv 0$.

To prove this statement, we multiplying  (\ref{QB}) by $V_n$. Then
$$
\int_{\Omega} e^{\beta.x}(t_nG_n(x).V_n+(1-t_n)\alpha
|V_n|^2) +\int_{\Omega}
e^{\beta.x}AV_n.V_n=\int_{\Omega}e^{\beta.x}|\nabla
V_n|^2\geq M,
$$
where $M=\min\limits_{\overline{\Omega}}\, e^{\beta.x}>0 $.
Passing to the limit, we get
$$\int_{\Omega}e^{\beta.x}(t\tilde{F}(x)+(1-t)\alpha |V(x)|^2)+
\int_{\Omega}e^{\beta.x}AV.V\geq M>0.$$ This
which completes the proof of step 5.

Finally, from the step 4 and step 5, we conclude that
$(\beta,1)$ is a first order eigenvalue of the problem, with
$$\Lambda_k(\beta,A,1)<\alpha_1\leq m(x)\leq 
\alpha_2<\Lambda_{k+1}(\beta,A,1).$$
By the strict monotonicity with respect to  weight (see
proposition 2.), we have
$$\Lambda_k(\beta,A,m)<1<\Lambda_{k+1}(\beta,A,m),$$
which is absurd, and present proof is complete.\quad$\Box$

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\noindent\textsc{A. Anane} (e-mail: anane@sciences.univ-oujda.ac.ma)\\
\textsc{O. Chakrone} (e-mail: chakrone@sciences.univ-oujda.ac.ma)\\
\textsc{Z. El Allali} (e-mail: zakaria@sciences.univ-oujda.ac.ma) \\
\textsc{Islam Eddine Hadi} (e-mail: hadi@sciences.univ-oujda.ac.ma)\\[5pt]
D\'epartement de Math\'ematiques et Informatique Facult\'e des Sciences,\\
Universit\'e Mohamed 1, Oujda, Maroc

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