\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 101, 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 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/101\hfil Positivity and negativity of solutions] {Positivity and negativity of solutions to $n\times n$ weighted systems involving the Laplace operator on $\mathbb{R}^N$} \author[B. Alziary, J. Fleckinger, M.-H. Lecureux, N. Wei \hfil EJDE-2012/101\hfilneg] {B\'en\'edicte Alziary, Jacqueline Fleckinger, \\ Marie-H\'el\`ene Lecureux, Na Wei} % in alphabetical order \address{B\'en\'edicte Alziary \newline Institut de Math\'ematique -UMR CNRS 5219- et Ceremath-UT1, Universit\' e de Toulouse, 31042 Toulouse Cedex, France} \email{alziary@univ-tlse1.fr} \address{Jacqueline Fleckinger \newline Institut de Math\'ematique -UMR CNRS 5219- et Ceremath-UT1, Universit\' e de Toulouse, 31042 Toulouse Cedex, France} \email{jfleckinger@gmail.com} \address{Marie-H\'el\`ene Lecureux \newline Institut de Math\'ematique -UMR CNRS 5219- et Ceremath-UT1, Universit\' e de Toulouse, 31042 Toulouse Cedex, France} \email{mhlecureux@gmail.com} \address{Na Wei \newline Dept. Appl. Math., Northwestern Polytechnical Univ. 710072 Xi'an, China \newline School of Stat. \& Math., Zhongnan Univ. Eco. \& Law, 430073, Wuhan, China} \email{nawei8382@gmail.com} \thanks{Submitted February 29, 2012. Published June 15, 2012.} \subjclass[2000]{35B50, 35J05, 35J47} \keywords{Elliptic PDE; maximum principle; fundamental positivity; \hfill\break\indent fundamental negativity; indefinite weight, weighted systems} \begin{abstract} We consider the sign of the solutions of a $n \times n$ system defined on the whole space $\mathbb{R}^N$, $N\geq 3$ and a weight function $\rho$ with a positive part decreasing fast enough, $$ -\Delta U = \lambda \rho(x) MU +F, $$ where $F$ is a vector of functions, $M$ is a $n \times n$ matrix with constant coefficients, not necessarily cooperative, and the weight function $\rho$ is allowed to change sign. We prove that the solutions of the $n\times n$ system exist and then we prove the local fundamental positivity and local fundamental negativity of the solutions when $|\lambda\sigma_1-\lambda_\rho|$ is small enough, where $\sigma_1$ is the largest eigenvalue of the constant matrix $M$ and $\lambda_\rho$ is the ``principal" eigenvalue of $$ -\Delta u = \lambda \rho(x) u , \quad \lim_{|x|\to \infty} u(x) = 0 ; \quad u(x)>0, \quad x\in \mathbb{R}^N. $$ \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Elliptic eigenvalue problems with indefinite weight arise naturally from linearization of many semilinear elliptic equations. A lot of literature in applied mathematics, engineering, physics, and biology treat such problems which occur in the study of transport theory, crystal coloration, laser theory, reaction-diffusion equations, fluid dynamics, etc. (see \cite{BTe} and the references therein). Many researchers studied the indefinite weighted eigenvalue problems under various hypotheses (see \cite{Al,AH,BCF,BTe,FM}). Owing to the importance of such problems, we investigate some systems concerning the Laplace operator. Specially, we will obtain the sign of solutions of some weighted systems in this paper. The positivity of solutions is usually shown with the maximum principles, which are also of great importance for the study of existence and uniqueness of solutions to some linear and nonlinear equations. Moreover, P. Cl\'{e}ment and L. Peletier \cite{ClPe} proved an antimaximum principle for a linear equation $-\Delta u=\lambda u+f$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^N$ with Dirichlet boundary condition $\partial\Omega$. Let $\lambda_1$ be the principal eigenvalue of the Laplace operator $-\Delta$, which is endowed with homogeneous Dirichlet boundary condition. We recall here the maximum principle as well as the antimaximum principle for a bounded domain. \begin{proposition}[Hopf Maximum principle] \label{prop1} Assume $f\in L^p(\Omega)$, $p>N$ and $f\geq0$, $f\not \equiv 0$. Let $u$ be a solution of the equation $-\Delta u=\lambda u+f$, $u=0$ on the boundary $\partial\Omega$. Then for $\lambda < \lambda_1$, $$u(x)>0 \quad\text{on } \Omega; \quad \frac{\partial u }{\partial n}(x)\big|_{\partial \Omega}<0, $$ where $\frac{\partial u }{\partial n}$ is the outward normal derivative on $\partial\Omega$. \end{proposition} \begin{proposition}[Antimaximum principle] \label{prop2} Assume $f\in L^p(\Omega)$, $p>N$ and $f\geq0$, $f\not \equiv 0$. $u$ solves $-\Delta u=\lambda u+f$ in $\Omega$, $u|_{\partial\Omega}=0$. Moreover, $\partial \Omega$ is smooth enough. Then there exists $\delta>0$, depending on $f$, such that for $\lambda_1 < \lambda < \lambda_1+\delta$, $$ u(x)<0 \quad\text{on } \Omega;\quad \frac{\partial u }{\partial n}(x)\big|_{\partial \Omega}>0. $$ \end{proposition} These results are classical. However, many problems in mechanic, physic or biology lead to some more general problems as the following equation defined on the whole space $\mathbb{R}^N$, $N>2$: $$ L u=\lambda\rho(x)u+f, \quad x\in \mathbb{R}^N. $$ Here $L$ is an elliptic operator as e.g. the Laplacian or the p-Laplacian. The weight $\rho$ ensures the discreteness of the spectrum (e.g. $\rho^+=max(\rho,0)$ tends to zero fast enough). There are several results with regard to such kinds of problems, such as the maximum principle \cite{FS}, the anti-maximum principle (\cite{FGoT04} or \cite{StT}), the (local) fundamental positivity or negativity of solutions \cite{FW}. Fleckinger, Gossez and de Th\'{e}lin also proved in \cite{FGoT04} a local antimaximum principle for the p-Laplacian on the whole space $\mathbb{R}^N$, which follows the approach introduced in \cite{FGoTaT}. Maximum and antimaximum principles have been extended into notions of ``fundamental positivity" and ``fundamental negativity" by Alziary and Tak\'{a}\v{c} \cite{AT1} or \cite{ATa}, who introduced these definitions for the solutions of Schr\"{o}dinger equation. We say that a function $u$ satisfies the ``fundamental positivity" if there exists a constant $c>0$ such that $u\geq c\varphi_\rho>0$ a.e. in $\mathbb{R}^N$ and a function $u$ satisfies the ``fundamental negativity" if there exists a constant $c>0$ such that $u\leq -c\varphi_\rho<0$ a.e. in $\mathbb{R}^N$, where $\varphi_\rho$ denotes a positive principal eigenfunction associated to the principal eigenvalue $\lambda_\rho$. We will explain it in detail in the next section. If these results are obtained on balls, we have the definitions of ``local fundamental positivity" and "local fundamental negativity" respectively. There are also some results for systems in \cite{AFL,FiMi,FHeT1995,FHeT2004,FMST,FS, Hess}, concerning the existence of principal eigenvalue and maximum principle. Alziary, Tak\'{a}\v{c} \cite{AT1} investigate equations involving Schr\"{o}dinger operators \begin{equation}\label{schrodinger} -\Delta u+q(x)u=\lambda u+f(x) \end{equation} in $L^2(\mathbb{R}^N)$. They proved a pointwise lower bound for the solution of \eqref{schrodinger} for a function $0\leq f \not \equiv 0$. Alziary, Fleckinger, and Tak\'{a}\v{c} showed in \cite{AFT} and \cite{AFT2} the fundamental negativity of solutions to \eqref{schrodinger} when $\lambda$ is slightly above the principal eigenvalue. Alziary, Fleckinger,L\'{e}cureux \cite{AFL} obtained the fundamental positivity and fundamental negativity of a $2\times 2$ systems of Schr\"{o}dinger equations on $\mathbb{R}^N$. Then L\'{e}cureux \cite{Lec} extended the results to the non cooperative $n\times n$ system $$\mathcal{L}U=\lambda U+MU+F$$ on $\mathbb{R}^N$, where $\mathcal{L} $ is a diagonal matrix of Schr\"{o}dinger operators of the form $\mathcal{L}:=-\Delta +q$, $q$ is a potential growing fast enough and $M$ is a constant matrix. In this paper, we combine our methods with those of L\'{e}cureux \cite{Lec} to study a $n\times n$ weighted systems defined on the whole space ${\mathbb{R}^N},~N\geq 3$: \begin{equation*} -\Delta U = \lambda \rho (x) MU + F; \end{equation*} here $\Delta$ denotes the standard Laplace operator, $\lambda$ is a real spectral parameter; $U=(u_1,u_2,\dots ,u_n)^\top$ and $F=(f_1,f_2,\dots ,f_n)^\top$. The functions $f_i$ are measurable and bounded. The function $\rho$ is allowed to change sign; i.e., $\rho$ is an indefinite weight function. Moreover, $\rho^+=\max(\rho,0)$ tends to $0$ fast enough as $|x|\to\infty$. The matrix $M$ is a $n\times n$ matrix with real constant coefficients $m_{ij}$ and real eigenvalues $\sigma_k\neq 0$; it is not necessarily cooperative, that is that the terms outside the diagonal may have any sign. Recall that a matrix is said to be cooperative if the coefficients outside the diagonal are positive. We derive our results from results for the case of one equation and of the $2\times 2$ system which have been studied in \cite{FW}. This paper is organized as follows: in Section 2, we recall or prove results for the case of one equation. Then we give our main results for a positive weight in Section 3 and we prove them in Section 4; finally we extend our results to indefinite weight in Section 5. \section{Equations with positive weight} We introduce some notations and hypotheses in this section; then we recall and prove some results on local fundamental positivity and local fundamental negativity which we will use later. \subsection*{Notation and Hypotheses for a positive weight} For a positive and bounded function $r$, we denote $L^2_r(\mathbb{R}^N)$ or shortly $L^2_r$ the space $$ L^2_r(\mathbb{R}^N)=\big\{u:\int_{\mathbb{R}^N}ru^2dx<\infty\big\} \quad \text{with norm } \|u\|_{L^2_r(\mathbb{R}^N)}= \Big(\int_{\mathbb{R}^N}ru^2dx \Big)^{1/2}. $$ Let us consider the equation \begin{equation}\label{eq:1} -\Delta u = \lambda \rho(x) u + f , \quad \lim_{|x|\to \infty} u(x) = 0, \quad x\in \mathbb{R}^N, \quad N\geq 3, \end{equation} where $f\in L^{\infty}(\mathbb{R}^N) \cap L^2_{1/\rho}$ and $\rho(x)$ is a positive smooth function decreasing faster than $1/|x|^2$ as $|x|\to \infty$. More precisely we assume that $\rho$ satisfies the hypothesis: \begin{itemize} \item[(H1)] $\rho(x) >0$, is smooth, bounded and there exists some constants $K>0$ and $\alpha >1$ such that $ \rho(x) \leq Kp_{\alpha}(x) $ where \begin{equation} p_{\alpha}(x):= (1+|x|^2)^{-\alpha}. \label{Pa} \end{equation} \end{itemize} \begin{remark} \label{rmk1} \rm It follows from {\rm (H1)} that $\rho \in L^{N/2}(\mathbb{R}^N)$ but is not necessarily in $L^1$. \end{remark} \noindent\textbf{The space $\mathcal{D}^{1,2}$ }: We denote by $\mathcal{D}^{1,2}$ the closure of $\mathcal{C}^{\infty}_0$ with respect to the norm $$ \|u\|_{{\mathcal{D}^{1,2}} }=\Big( \int_{\mathbb{R}^N} {| \nabla u |}^2 dx \Big)^{1/2}. $$ It can be shown (see \cite[Proposition 2.4]{KS91}) that $$ \mathcal{D}^{1,2}= \{ u \in L^{\frac{2N}{N-2}}(\mathbb{R}^N): \nabla u \in (L^2(\mathbb{R}^{N}))^N \}, $$ and that there exists $K>0$ such that for all $u \in \mathcal{D}^{1,2}$, $$ \|u\|_{ L^{\frac{2N}{N-2}}} \leq K \|u\|_{\mathcal{D}^{1,2}}. $$ We also use Hardy's inequality and deduce that there exists two positive constants $\gamma$ and $\gamma'$ such that \begin{equation}\label{eq:hardy} \int_{\mathbb{R}^N}|\nabla u|^2 dx \geq \gamma \int_{\mathbb{R}^N} p_{\alpha} |u|^2 dx \geq \gamma' \int_{\mathbb{R}^N} \rho |u|^2 dx,\; {\rm \forall } \; u\in \mathcal{D}^{1,2}. \end{equation} Hence any $u \in \mathcal{D}^{1,2}$ is in $L^2_{\rho}$. Moreover the embedding $\mathcal{D}^{1,2}$ into $L^2_{\rho}$ is compact \cite[Lemma 2.3]{FGoT04}. \begin{remark} \label{rmk2}\rm Note that by \eqref{eq:hardy}, $\mathcal{D}^{1,2}$ does not depend on $\rho$. Moreover note that the constant function is not in $\mathcal{D}^{1,2}$. \end{remark} For any $\rho$ satisfying (H1), we seek weak solutions to \eqref{eq:1} in $\mathcal{D}^{1,2}$. \noindent\textbf{The principal eigenpair:} Denote by $(\lambda_\rho,\varphi_{\rho})$ the ``principal" eigenpair which exists \cite{BCF,BTe} and satisfies \begin{equation} -\Delta \varphi_{\rho} = \lambda_{\rho} \rho(x) \varphi_{\rho} , \quad x\in \mathbb{R}^N; \quad \varphi_{\rho}(x)>0, \quad \forall x \in \mathbb{R}^N; \quad \int_{\mathbb{R}^N} \rho \varphi_{\rho}^2 = 1, \label{EV} \end{equation} with $\varphi_{\rho}$ in $L^2_{\rho}$. Here $\lambda_\rho$ is the smallest eigenvalue of $(EV)$; it is simple and it is called the ``principal eigenvalue". The eigenfunction $\varphi_\rho$ corresponding to $\lambda_\rho$ is called the ``groundstate" or ``principal eigenfunction". We define the subspaces $X$ of $L^2_{\rho}(\mathbb{R}^N)$ and $Y$ of $L^2_{1/\rho}(\mathbb{R}^N)$ as: \begin{gather}\label{eq:X} X:=\{u\in L^2_{\rho}(\mathbb{R}^N):u/\varphi_\rho\in L^\infty(\mathbb{R}^N)\},\\ \label{eq:Y} Y:=\{u\in L^2_{1/\rho}(\mathbb{R}^N):u/(\rho\varphi_\rho)\in L^\infty(\mathbb{R}^N)\}, \end{gather} respectively, which are Banach spaces with the norms $\|u\|_{X}:=\operatorname{ess\,sup}_{\mathbb{R}^N}(|u|/\varphi_\rho)$ and $\|u\|_{Y}:=\operatorname{ess\,sup}_{\mathbb{R}^N}(|u/(\rho\varphi_\rho)|)$, respectively. The function $u \in \mathcal{D}^{1,2}$ is a weak solution of Equation \eqref{eq:1} if $$ \int_{\mathbb{R}^N} (\nabla u \cdot \nabla \eta) = \lambda \int_{\mathbb{R}^N} \rho(x) u \eta + \int_{\mathbb{R}^N} f \eta, \;\; \forall \eta \in \mathcal{D}^{1,2}. $$ This solution is also classical by regularity properties. \begin{theorem}[{\cite[Sec. 4]{BCF}, \cite[Lemma 4]{FS}}] \label{thm1} Assume {\rm (H1)}. There exists a positive principal eigenvalue $\lambda_{\rho}$ which is given by \begin{equation}\label{eq:eigenvalue} 0 < \lambda_{\rho} = \inf_{\{u \in \mathcal{D}^{1,2}, \int_{\mathbb{R}^N} \rho u^2 =1\}} \int |\nabla u|^2. \end{equation} The equality in \eqref{eq:eigenvalue} holds if and only if $u$ is proportional to $\varphi_{\rho}$. \end{theorem} \begin{remark} \label{rmk3}\rm The existence of a principal eigenvalue and Equation \eqref{eq:eigenvalue} are still valid, under some conditions, for a non negative weight and for an indefinite weight (see e.g. \cite{BCF,BTe}). We recall the proof later (Lemma \ref{lem5.1}). \end{remark} \subsection{Results for one equation with positive weight} First, we recall a maximum principle and a (local) antimaximum principle for the case of one equation. \begin{proposition}[Maximum Principle, \cite{FS}] \label{Prop:2} We suppose that $\rho$ satisfies Hypothesis {\rm (H1)}, $f\in L^{\infty}$, $f\geq 0$, $f \not \equiv 0$. A necessary and sufficient condition for having a (strong) maximum principle, (that is $f$ bounded, $f\geq 0$, $f \not \equiv 0$ implies any solution $u$ to $\eqref{eq:1}$ is positive), is $0<\lambda < \lambda_{\rho}$. Moreover, if $0<\lambda < \lambda_{\rho} $, then there exists a solution to \eqref{eq:1} and this solution is unique and positive. \end{proposition} \begin{corollary}\label{coroA} Assume that $f\in Y$. For $0<\lambda<\lambda_\rho$, $u$ exists and \begin{equation*} 0 \leq u(x) \leq \frac{\|f\|_{Y}}{\lambda_\rho-\lambda}\varphi_\rho(x), \quad \forall x \in \mathbb{R}^N . \end{equation*}. \end{corollary} \begin{proposition}[{Local antimaximum principle, \cite[Theorem 5.2]{StT} or \cite[Theorem 3.3]{FGoT04}}], \label{Prop:3} Assume {\rm (H1)} and $f\in L^{\infty}$, $f\geq 0$, $f\not \equiv 0$. Let $R>0$ be given; there exists $\delta>0$ depending on $R$, $f$ and $\rho$ such that for $\lambda_{\rho} <\lambda< \lambda_{\rho}+\delta$, then $u <0$ on the ball $B_R=\{x\in \mathbb{R}^N/ |x|0. \end{equation} Then, for any given $R>0$, there exist positive numbers $k', k'',K', K'', \delta, \delta'$ (depending on $R$, $\rho$ and $f$) such that on $B_R=\{x\in \mathbb{R}^N:|x|0$, on $B_R$, $ \frac{|u|}{ \varphi_{\rho} } \to+\infty$ as $\lambda \to \lambda_{\rho}$. \end{corollary} This corollary plays an important role when studying systems of equations. \section{Main results for a system with positive weight} We consider now a $n\times n$ system: \begin{equation}\label{n system} -\Delta U = \lambda \rho(x) MU +F, \quad x\in\mathbb{R}^N, \quad U\to 0 \quad\text{as } |x|\to +\infty, \end{equation} where $\rho$ satisfies Hypothesis (H1). We assume the hypothesis \begin{itemize} \item[(H3)] $M$ is a $n\times n$ non-degenerate matrix which has constant coefficients and has only real eigenvalues. Moreover, the largest one which is denoted by $\sigma_1$ is positive and algebraically and geometrically simple. \end{itemize} Of course some of the other eigenvalues can be equal. Therefore we write them in decreasing order $\sigma_1>\sigma_2 \geq \dots \geq \sigma_n$. The eigenvalues of $M=(m_{ij})_{1\leq i,j\leq n}$, $\sigma_1$, $\sigma_2$, \dots , $\sigma_n$ are the roots of the associated characteristic polynomial \begin{equation}\label{polynomial} p_M(\sigma)=\det(\sigma I_n-M)=\prod(\sigma-\sigma_j), \end{equation} where $I_n$ is the $n\times n$ identity matrix. \begin{remark} \label{rmk4}\rm By \eqref{polynomial}, $\sigma> \sigma_1\Rightarrow p_M(\sigma)>0$. \end{remark} In fact, Matrix $M$ can be expressed as $M=PJP^{-1}$, where $P=(p_{ij})$ is the change of basis matrix of $M$ and $J$ is the Jordan canonical form (upper triangular matrix) associated with $M$. The diagonal entries of $J$ are the ordered eigenvalues of $M$ and $p_M(\sigma)=p_J(\sigma)$. In the following, we denote by $Q$ the eigenspace associated with $\sigma_1$ ($\dim Q=1$) and by $T$ the hyperplain spanned by the other column vectors of matrix $P$. By Hypothesis (H3), $\mathbb{R}^N=Q \bigoplus T$. Now we define another hypothesis. \begin{itemize} \item[(H4)] Assume that $F=(f_i) = (\rho h_i) \in Y$, $1\leq i \leq n$, and $h_i \in L_{\rm loc}^{\infty} \cap L^2_{\rho}$ for any $i=1,\dots ,n$. \end{itemize} \begin{theorem} \label{thm3} Assume that {\rm (H1), (H3), (H4)} are satisfied. We also assume that there exists an eigenvector $\Theta\in Q$ associated with $\sigma_1$ such that $F(x)= F_Q(x) + F_T(x) $ with $$ F_Q(x)=\widetilde{f_1}(x)\Theta\not\equiv 0 \quad\text{and}\quad \int \widetilde{f_1}\varphi_{\rho} > 0 ; $$ also $F_Q(x)\in Q$ and $F_T(x)\in T$. Then, for any $R>0$, there exist two positive real numbers $\delta$ and $\delta'$, depending on $F, M, \rho, R$ such that: (I) If $\lambda_\rho-\delta<\lambda\sigma_1<\lambda_\rho$, System \eqref{n system} has a unique solution $U=(u_i)$. Moreover, on $B_R$, for each integer $i\in [1,n]$, $u_i$ has the sign of $p_{i1}$, the $i^{th}$ item of the first column vector of the matrix $P=(p_{ij})$. (II) If $\lambda_\rho<\lambda\sigma_1<\lambda_\rho+\delta'$, System \eqref{n system} has a unique solution $U=(u_i)$. Moreover, on $B_R$, for each integer $i\in [1,n]$, $u_i$ has the sign of $-p_{i1}$, where $p_{i1}$ is as above. \end{theorem} \begin{remark} \label{rmk5} \rm From \eqref{polynomial} , we can also derive (always for $\lambda >0$) that $\lambda\sigma_1<\lambda_\rho$ implies $\det(B)>0$, where \begin{equation}\label{cooperative} B:=(\lambda_\rho I_n-\lambda M) \end{equation} \end{remark} \noindent\textbf{Cooperative system:} We make comments on the case of a cooperative system; that is, a system where the coefficients outside the diagonal of $M$ are positive: $m_{ij}>0, i\neq j$. \begin{remark} \label{rmk6} \rm For the case of a cooperative system and $\lambda=1$, Theorem \ref{thm3} extends earlier results in \cite[Thm. 7]{FS}. \end{remark} \begin{remark} \label{rmk7} \rm Always for a cooperative system, the first column $p_{i1}$, which consists in the components of the first eigenvector, by Perron-Frobinius theorem (\cite[p. 27]{BePl}, \cite{Gan}), is of one sign hence all components $u_i$ have the same sign for $1\leq i\leq n$. \end{remark} From \cite{FS}, we derive the following result. \begin{corollary} \label{coro3.1} Assume {\rm (H1), (H3)} are satisfied and $M$ is a cooperative matrix. $F=(f_i)\in L^\infty\cap L^2_{1/\rho}$. The cooperative system satisfies the maximum principle if and only if $B:=(\lambda_\rho I_n-\lambda M)$ is a nonsingular M-matrix. \end{corollary} A nonsingular square matrix $\mathcal{B}=(b_{ij})$ is a M-matrix if $b_{ij}\leq 0$ for $i\neq j$, $b_{ii}>0$ and if all principal minors extracted from $\mathcal{B}$ are positive (see \cite{BePl,FHeT1995}). \begin{remark} \label{rmk8}\rm For a cooperative system, the condition $0<\lambda \sigma_1 <\lambda_\rho$ is equivalent to Condition \eqref{cooperative}; that is, \begin{itemize} \item[(a)] $b_{ij}=-\lambda m_{ij}\leq 0$ for $i\neq j$ by $m_{ij}\geq 0$ ($M$ is cooperative). \item[(b)] $\lambda_\rho-\lambda\sigma_1, \dots , \lambda_\rho-\lambda\sigma_n$ are eigenvalues of $B$. They are real and positive $$ \Leftrightarrow 0<\lambda\sigma_1<\lambda_\rho \Leftrightarrow \lambda\sigma_n\leq \dots \leq\lambda\sigma_2<\lambda\sigma_1<\lambda_\rho. $$ \end{itemize} \end{remark} \section{Proofs for a positive weight} \begin{proof}[Proof of Corollary \ref{coroA}] It follows simply from the maximum principle. Consider the two equations \begin{gather*} -\Delta (c \varphi_{\rho}) = \lambda_{\rho} \rho (c \varphi_{\rho}), \\ -\Delta u = \lambda \rho u + f, \end{gather*} where $c$ is a constant. By substraction $$ -\Delta ( c \varphi_{\rho} - u )= \lambda \rho ( c \varphi_{\rho} - u ) + (\lambda_{\rho} - \lambda ) \rho (c \varphi_{\rho}) - f ; $$ Choosing $c= \|f\|_Y/(\lambda_{\rho} - \lambda)$ we have $ (\lambda_{\rho} - \lambda ) \rho (c \varphi_{\rho}) - f \geq 0$, $\not \equiv 0$ and we derive the result by the classical maximum principle. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] This is the same proof as in \cite{FW}, but our hypothesis here are more general. We study Equation \eqref{eq:1}: $$ -\Delta u = \lambda \rho(x) u + f , \quad \lim_{|x|\to \infty} u(x)= 0, \quad x\in \mathbb{R}^N, \quad N\geq 3, $$ when (H1) and (H2) are satisfied as well as \eqref{eq:f}: $ \int f \varphi_{\rho}>0$. We decompose $u$. Set $u=u^* \varphi_{\rho} + u^{\perp}$ where $\int u^{\perp} \rho \varphi_{\rho} = 0$. Also $u^* = \int \rho u \varphi_{\rho}$. Analogously set $f=\rho h$ and decompose $h$: $h=h^* \varphi_{\rho}+ h^{\perp}$. Equation \eqref{eq:1} becomes $$ (\lambda_{\rho} - \lambda) \rho u^* \varphi_{\rho} -\Delta u^{\perp} = \lambda \rho u^{\perp} + h^* \rho \varphi_{\rho} + \rho h^{\perp}. $$ Multiply by $\varphi_{\rho}$ and integrate. We obtain $ (\lambda_{\rho} - \lambda) u^* = h^*$. Hence $$ u= \frac{h^*}{\lambda_{\rho} - \lambda} \varphi_{\rho} + u^{\perp}. $$ Also $u^{\perp}$ is solution to \begin{equation}\label{perp} -\Delta u^{\perp} = \lambda \rho u^{\perp} + \rho h^{\perp}. \end{equation} Since $\int |\nabla u^{\perp}|^2 \geq \lambda_2 \int \rho (u^{\perp})^2$, $\lambda_2$ being the second eigenvalue of our problem, we compute: $$ (\lambda_2 - \lambda) \int_{\mathbb{R}^N} \rho(x)(u^{\perp})^2 \leq \int_{\mathbb{R}^N} \rho(x)h^{\perp}u^{\perp} \leq \Big[\int_{\mathbb{R}^N} \rho(x)(h^{\perp})^2 \int_{\mathbb{R}^N} \rho(x)(u^{\perp})^2\Big]^{1/2}. $$ Hence $$ \int_{\mathbb{R}^N} \rho(x)(u^{\perp})^2 \leq \big(\frac{1}{\lambda_2 - \lambda} \big)^2 \int_{\mathbb{R}^N} \rho(x)(h^{\perp})^2. $$ When $\lambda < \lambda_{\rho} <\lambda_2$, $ \lambda_2 - \lambda > \lambda_2 - \lambda_{\rho}$. Hence there exists $K>0$, depending on $h$, such that $$ \int_{\mathbb{R}^N} \rho(x)(u^{\perp})^2 \leq K (\lambda_2 - \lambda_{\rho})^{-2}; $$ this upper bound is independent of $\lambda$. When $\lambda_{\rho} <\lambda <\lambda_2$, we choose $\lambda_2-\lambda > \frac{1}{4}(\lambda_2 - \lambda_{\rho})$. Again there exists $K'>0$ such that $$ \int_{\mathbb{R}^N} \rho(x)(u^{\perp})^2 \leq K' (\lambda_2 - \lambda_{\rho})^{-2}; $$ and again the upper bound is independent of $\lambda$. Now choose $R>0$. In both cases, $u^{\perp}$ being bounded in $L^2_{\rho}(\mathbb{R}^N)$, is also bounded on $L^2_{\rho}(B_R)$ and then in $L^2(B_R)$. Therefore, since $\lambda < \lambda_2$ the right-hand side term in \eqref{perp}: $\lambda \rho u^{\perp} + \rho h^{\perp}$ is also bounded in $L^2(B_R)$. \par \noindent By a classical bootstrap method we derive (e.g. as in \cite{FHeT2011}) that $u^{\perp}$ is bounded in $L^q(B_R) $ and finally in $B_R$ with respect to the sup-norm. On $B_R $, the groundstate $\varphi_{\rho}$ is bounded below. For $\lambda_\rho-\delta<\lambda<\lambda_\rho<\lambda_2$, it is possible to choose $\lambda_{\rho} - \lambda>0$ small enough so that $$ |u^{\perp}| \leq K'\frac{h^*}{\lambda_2 -\lambda_{\rho} } \varphi_{\rho} = K'' \frac{h^*}{\lambda_\rho -\lambda} \varphi_{\rho} <\frac{h^*}{\lambda_\rho -\lambda} \varphi_{\rho} . $$ Hence we obtain $$ u = \frac{h^*}{\lambda_{\rho} - \lambda} \varphi_{\rho} + u^{\perp}\leq \frac{h^*}{\lambda_{\rho} - \lambda} \varphi_{\rho}+K'' \frac{h^*}{\lambda_\rho -\lambda} \varphi_{\rho} = (1+K'')\frac{h^*}{\lambda_\rho -\lambda} \varphi_{\rho}, $$ and $$ u= \frac{h^*}{\lambda_{\rho} - \lambda} \varphi_{\rho} + u^{\perp} \,\geq\, \frac{h^*}{\lambda_{\rho} - \lambda} \varphi_{\rho}-K'' \frac{h^*}{\lambda_\rho -\lambda} \varphi_{\rho} = (1-K'')\frac{h^*}{\lambda_\rho -\lambda} \varphi_{\rho}, $$ then \eqref{eq:u} is obtained. For $\lambda_\rho<\lambda<\lambda_2$, it is possible to choose $\lambda - \lambda_{\rho}>0$ small enough so that $|u^{\perp}| \leq K' \frac{h^*}{\lambda_2 -\lambda_{\rho} } \varphi_{\rho}\leq K'' \frac{h^*}{\lambda -\lambda_\rho} \varphi_{\rho}$. Then \eqref{eq:7} can be obtained similarly to the discussion above. \end{proof} \begin{proof}[Proof of Theorem \ref{thm3}] We study $-\Delta U = \lambda \rho(x) MU +F$ when $M$ is a $n \times n$ matrix and $\rho$ a non negative weight. The case of a $2\times 2$ system is studied in \cite{FW}. We set $$ U=P\widetilde{U}\Leftrightarrow \widetilde{U}=P^{-1}U, \quad F=P\widetilde{F}\Leftrightarrow \widetilde{F}=P^{-1}F, $$ here $\widetilde{U}$ and $\widetilde{F}$ are column vectors with components $\widetilde{u_i}$ and $\widetilde{f_i}$, respectively, $1\leq i\leq n$. $P$ has constant coefficients. System \eqref{n system} can be written as \begin{equation}\label{eq:8} -\Delta \widetilde{U} = \lambda \rho(x) J\widetilde{U} +\widetilde{F}, \end{equation} where $J$ is a Jordan canonical form which has $p$ Jordan blocks $J_i$ ($1\leq j\leq p \leq n$). Every Jordan block is a square $k_i\times k_i$ matrix of the form: $$ J_i=\begin{pmatrix} \sigma_i&1 &0 & \\ &\ddots&\ddots& \\ &0&\ddots&1\\ & & &\sigma_i \end{pmatrix}. $$ By Hypothesis (H3), the first block is $1\times 1$: $J_1=(\sigma_1)$. We will adapt the method in \cite{Lec}. By Hypothesis (H3), we obtain the first equation \begin{equation}\label{eq:9} -\Delta\widetilde{u}_1=\lambda \rho(x)\sigma_1\widetilde{u}_1+\tilde{f}_1, \end{equation} where $\int \tilde{f}_1\varphi_{\rho} > 0$, and $\tilde{f}_1\in Y$. \noindent(\textbf{I}) If $\lambda \sigma_1<\lambda_{\rho}$. If $\lambda_\rho-\delta<\lambda \sigma_1<\lambda_{\rho}$, by Theorem \ref{thm2}, and \eqref{eq:9}, we get the local fundamental positivity for $\widetilde u_1$: \begin{equation}\label{eq:12} 0< \frac{k'}{(\lambda_{\rho} - \lambda\sigma_1 )} \varphi_{\rho} \leq \widetilde{u}_1 \leq \frac{k''}{(\lambda_{\rho} - \lambda\sigma_1 )} \varphi_{\rho}, \end{equation} on the ball $B_R=\{x\in \mathbb{R}^N: |x|\frac{\lambda_\rho}{2}$, hence let $\lambda_{\rm min}=\frac{\lambda_\rho}{2\sigma_1}$. Then we obtain $$ |\widetilde {u}_{s_{i+1}}| \leq \frac{\|\tilde{f}_{s_{i+1}}\|_{Y}}{\lambda(\sigma_1-\sigma_i)} \varphi_\rho\leq\frac{\|\tilde{f}_{s_{i+1}}\|_{Y}}{\lambda_{\rm min} (\sigma_1-\sigma_i)}\varphi_\rho. $$ So we know that $\widetilde {u}_{s_{i+1}}\in X$ and $\|\widetilde {u}_{s_{i+1}}\|_{X}\leq c_{s_{i+1}}$, where $c_{s_{i+1}}$ depends only on $F,M$. In line $j=s_{i+1}-1$, we have \begin{equation*} -\Delta\widetilde{u}_j=\lambda \rho(x) \sigma_i\widetilde{u}_j +\lambda \rho(x) \widetilde{u}_{s_{i+1}} + \tilde{f}_j~~(i\geq 2) \end{equation*} from \eqref{eq:10}. Using Corollary \ref{coroA}, we obtain the existence of $\widetilde{u}_j$ and \begin{align*} |\widetilde{u}_j| &\leq \frac{\|\lambda \rho(x)\widetilde{u}_{s_{i+1}} +\tilde{f}_{j}\|_{Y}}{\lambda_\rho-\lambda\sigma_i}\varphi_\rho \leq \frac{\lambda \|\rho(x)\widetilde{u}_{s_{i+1}}\|_Y +\|\tilde{f}_{j}\|_{Y}}{\lambda(\sigma_1-\sigma_i)}\varphi_\rho \\ &\leq \frac{\lambda c_{s_{i+1}}+m_j}{\lambda(\sigma_1-\sigma_i)}\varphi_\rho \leq (\frac{ c_{s_{i+1}}}{\sigma_1-\sigma_i}+\frac{m_j}{\lambda_{\rm min} (\sigma_1-\sigma_i)})\varphi_\rho \end{align*} where $\|\rho(x)\widetilde{u}_{s_{i+1}}\|_Y\leq c_{s_{i+1}}$, $\|\tilde{f}_{j}\|_{Y}\leq m_j$. So for $j=s_{i+1}-1$, we have $\widetilde{u}_j\in X$ and $\|\widetilde{u}_j\|_X\leq c_j$, where $c_{j}$ depends only on $F,~M, ~\rho$. Similar discussion from line $s_{i+1}$ to line $s_i+1$, we obtain that, for each integer $j$ with $s_i+1\leq j\leq s_{i+1}$, $\|\widetilde{u}_j\|_X\leq c_j$ in each Jordan's block $J_i$, where the real number $c_{j}$ depends only on $F,M,\rho$. In conclusion, we have that for $2\leq j\leq n$, \begin{equation}\label{eq:13} |\widetilde{u}_j|\leq c_j\varphi_\rho, \end{equation} where $c_{j}$ depends only on $F,M,\rho$. For $j=1$, \begin{equation}\label{eq:14} \widetilde{u}_j\geq c \varphi_\rho, \end{equation} where $c$ depends on $F,M,\lambda$; moreover $\widetilde{u}_1/\varphi_\rho\to +\infty$ as $\lambda\sigma_1\to\lambda_\rho$. Now we go back to the functions $u_i$. For any given $R>0$, $U=P\tilde{U}=(u_i)$ implies that for each $u_i$, $1\leq i\leq n$, we have \begin{equation}\label{eq:15} u_i=p_{i1}\widetilde{u}_1+\sum_{j=2}^np_{ij}\widetilde{u}_j. \end{equation} So $\widetilde{u}_1/\varphi_\rho\to +\infty$ as $\lambda\sigma_1\to\lambda_\rho$; by \eqref{eq:13}, $\sum_{j=2}^np_{ij}\widetilde{u}_j$ is bounded by a constant times $\varphi_\rho$. Therefore, on any ball $B_R$, for any $i\in \{1,2,\dots ,n\}$, there exists $\delta=\inf_i\delta_i> 0$ such that for $\lambda_\rho-\delta<\lambda\sigma_1<\lambda_\rho$, $u_j$ has the sign of $p_{ij}$ . \noindent\textbf{(II)} If $\lambda\sigma_1>\lambda_{\rho} $ and $\lambda\sigma_1-\lambda_{\rho} $ is small enough, then there exists $\delta'>0$ such that $\lambda\sigma_k<\lambda_{\rho} <\lambda\sigma_1< \lambda_{\rho}+\delta'$. In this case, we also have Equations \eqref{eq:9}, \eqref{eq:10} and \eqref{eq:11}. By Theorem \ref{thm2} and \eqref{eq:9}, we can get the local fundamental negativity: On any ball $B_R$, there exists $\delta_1>0$ such that if $\lambda_{\rho} <\lambda\sigma_1<\lambda_{\rho}+\delta_1< \lambda_{\rho}+\delta'$, then $$ -\frac{K''}{\lambda\sigma_1 - \lambda_{\rho}} \varphi_{\rho} \leq \widetilde{u}_1 \leq -\frac{K'}{\lambda\sigma_1 - \lambda_{\rho}} \varphi_{\rho} < 0, $$ and $\widetilde{u}_1/\varphi_\rho\to -\infty$ as $\lambda\sigma_1\to\lambda_\rho$. In the equations $-\Delta\widetilde {u_i}=\lambda \rho(x) \sigma_k \widetilde {u_i} +\widetilde {f_i}$, we have $\lambda\sigma_k<\lambda_\rho$, so we can use the local fundamental positivity results and by \eqref{eq:13}, $\sum_{j=2}^np_{ij}\widetilde{u}_j$ is bounded by a constant times $\varphi_\rho$. In $u_i=p_{i1}\widetilde{u}_1+\sum_{j=2}^np_{ij}\widetilde{u}_j$, we have $\sum_{j=2}^np_{ij}\widetilde{u}_j$ bounded by a constant times $\varphi_\rho$ and $\widetilde{u}_1/\varphi_\rho\to -\infty$ as $\lambda\sigma_1\to\lambda_\rho$. So, on any $B_R$, there exists $\delta'>0$ such that for $\lambda_{\rho} <\lambda\sigma_1< \lambda_{\rho}+\delta'$, we obtain that $u_i$ has the sign of $-p_{i1}$. \end{proof} \section{The case of an indefinite weight} We suppose now that $\rho$ changes sign and we set $\rho^+(x)=max\{\rho(x), 0\}$ and $\rho=\rho^+-\rho^-$. We assume the hypothesis \begin{itemize} \item[(H1')] \begin{itemize} \item $\rho$ is smooth, bounded and positive somewhere; that is, $\Omega_+:= \{x \in \mathbb{R} ^N / \rho(x) >0\}$ is with positive Lebesgue measure. \item $\rho^+$ satisfies (H1) with some $\alpha >1$. \end{itemize} \end{itemize} We seek for solutions in the space $\mathcal{D}_{\rho^-}^{1, 2} := \{ u \in \mathcal{D}^{1, 2}\text{ s.t. } \int_{\mathbb{R}^N} \rho^- u^2 < \infty \}$. \noindent\textbf{The principal eigenpair} We denote by $(\lambda_{\rho}>0, \varphi_{\rho}>0)$ the associated principal eigenpair. From the hypothesis on $\rho$, we know that a non-positive eigenvalue may exist. Since $\lambda_{\rho} = (-\lambda) (-\rho)$, we only consider the case of positive $\lambda >0$; also \eqref{eq:eigenvalue} adapted to this case is stated as: \begin{lemma} \label{lem5.1} Assume {\rm (H1')}. Then \begin{equation}\label{eq:eigenvalue'} 0 < \lambda_{\rho} = \inf_{\{u \in \mathcal{D}_{\rho^-}^{1,2}: \int_{\mathbb{R}^N} \rho u^2 =1\}} \int |\nabla u|^2. \end{equation} The equality holds if and only if $u$ is proportional to $\varphi_{\rho}$. \end{lemma} Now define the hypothesis \begin{itemize} \item[(H2')] There exist $\alpha>1$, $\epsilon >0$ with $f=(\rho^+ + \epsilon p_{\alpha}) h$ such that $h\in L^{\infty}_{\rm loc} \cap L^2_{p_{\alpha}}$. \end{itemize} \noindent\textbf{Results for a weight changing sign:} We state our results first for one equation, then for an $n \times n$ system. It is easy to prove that Proposition \ref{Prop:3} and Corollary \ref{coroA} are still valid for this indefinite weight. \begin{theorem} \label{thm4} Assume {\rm (H1'), (H2')} and \eqref{eq:f}: $\int_{\mathbb{R}^N} f \varphi_{\rho} > 0 $. Then, for any $R>0$, there exists $\delta>0$ (depending on $R$, $f$ and $\rho$) such that \begin{itemize} \item for $\lambda < \lambda_{\rho}$, and $\lambda_{\rho} - \lambda <\delta$, there exist positive constants $k'$ and $k''$, depending on $h$ and $R$, such that on $B_R=\{x\in \mathbb{R}^N : |x|< R \}$, \begin{equation} 0< \frac{k'}{\lambda_{\rho} - \lambda } \varphi_{\rho} \leq u \leq \frac{k''} {\lambda_{\rho} - \lambda } \varphi_{\rho}. \label{Iu} \end{equation} ("Local Fundamental Positivity") \item for $\lambda_{\rho} <\lambda < \lambda_{\rho}+\delta$, there are positive constants $K'$, $K''$, depending on $f$, $\varphi_{\rho}$ and $R$ such that on the ball $B_R=\{x\in \mathbb{R}^N: |x| 0 $ where $F_Q(x)\in Q$, $F_T(x)\in T$. Then, for any $R>0$, there exist two positive real numbers $\delta$ and $\delta'$, depending on $F, M, R, \rho$ such that (I) If $\lambda_\rho-\delta<\lambda\sigma_1<\lambda_\rho$, System \eqref{n system} has a unique solution $U=(u_i)$. Moreover, on any $B_R$, for each integer $i\in [1,n]$, we have $u_i\in X$ and $u_i$ has the sign of $p_{i1}$, the $i^{th}$ item of the first column vector of the matrix $P=(p_{ij})$. (II) If $\lambda_\rho<\lambda\sigma_1<\lambda_\rho+\delta'$, System \eqref{n system} has a unique solution $U=(u_i)$. Moreover, on any $B_R$, for each integer $i\in [1,n]$, we have $u_i\in X$ and $u_i$ has the sign of $-p_{i1}$, where $p_{i1}$ is given as above. \end{theorem} \begin{proof}[Proof of Lemma \ref{lem5.1}] First, we recall for completeness the proof of the existence of a positive principal eigenvalue, denoted as in Section 1, $\lambda_{\rho}$ associated with a principal eigenfunction $\varphi_{\rho}$. We follow e.g. \cite{FHeT2004}. The equation $$ -\Delta u = \lambda \rho(x) u \quad\text{on } \mathbb{R}^N $$ can be rewritten as \begin{gather*} -\Delta u + \lambda \rho^-(x) u = \lambda \rho^+(x) u \quad\text{on } \mathbb{R}^N \; \Leftrightarrow \\ -\Delta u + \lambda \rho^-_{\alpha}(x) u = \lambda \rho^+_{\alpha}(x) u \quad\text{on } \mathbb{R}^N, \end{gather*} where $\rho^{\pm}_{\alpha} := \rho^{\pm} + \epsilon p_{\alpha}$. This shift is introduced since $\rho^+$ is not necessarily positive but only $\geq 0$. We study now, for $\mu>0$, \begin{equation}\label{one equation} -\Delta u + \mu \rho^-_{\alpha}(x) u = \lambda \rho^+_{\alpha}(x) u\quad\text{on } \mathbb{R}^N. \end{equation} For the rest of this article, we set $L_{\mu} u:= -\Delta u + \mu \rho^-_{\alpha}(x) u$. Since $\rho^+_{\alpha}$ decreases fast enough, we have an analogous to Theorem \ref{thm1}: there exists a principal eigenpair $(\lambda^*(\mu), \psi_{\mu})$ such that \begin{gather*} L_{\mu} \psi_{\mu} = \lambda^*(\mu) \rho^+_{\alpha}(x) \psi_{\mu} \quad\text{on } \mathbb{R}^N ,\\ \lambda^*(\mu) = \inf_{\{u\in \mathcal{D}_{\rho^-_{\alpha}}^{1,2} : \int_{\mathbb{R}^N}\rho^+_{\alpha}u^2=1\}} \int (|\nabla u|^2+ \mu \rho^-_{\alpha}(x) u^2) dx. \end{gather*} Note that, by \eqref{eq:hardy}, $ \mathcal{D}_{\rho^-_{\alpha}}^{1,2}= \mathcal{D}_{\rho^-}$. $\lambda^*(\mu)$ varies continuously and monotonically with respect to $ \mu$; it increases. Moreover $\lambda^*(0)$ is the principal eigenvalue satisfying $$ -\Delta u = \lambda \rho^+_{\alpha}(x) u \quad\text{on } \mathbb{R}^N $$ where the weight is positive. Hence we know that $\lambda^*(0)$ exists and is positive. Also, by properties of monotonicity of eigenvalues w.r.t. the domain (for Dirichlet boundary conditions) , since $\Omega^+ \subset \mathbb{R}^N $, $\lambda^*(\mu)$ is always less than $\tau^*_{\mu}$ the positive principal eigenvalue of $L_{\mu}u =\tau \rho^+_{\alpha} u$ defined on $\Omega_+$ with Dirichlet boundary conditions. This curve cuts the bisectrix at $\tau^0$, the positive principal eigenvalue of $-\Delta u = \tau^0 \rho^+ u$ on $\Omega^+$. Hence , the (continuous) curve $(\mu, \lambda^*(\mu))$ intersects the bisectrix at $\lambda^0$ such that, for some $u$: $$ -\Delta u + \lambda^0 \rho^-_{\alpha}(x) u = L_{\lambda^0} u = \lambda^0 \rho^+_{\alpha}(x) u \; \Leftrightarrow \; -\Delta u =\lambda^0 \rho(x) u \quad\text{on } \mathbb{R}^N. $$ $\lambda^0$ is the (unique) positive principal eigenvalue $\lambda_{\rho}$ of $$ -\Delta u = \lambda \rho(x) u,~ x\in \mathbb{R}^N, $$ and $\lambda_\rho$ defined by \eqref{eq:eigenvalue'} and $u=k \varphi_{\rho}$. It does not depend on $\alpha$. \end{proof} \begin{remark} \label{rmk9} %\label{Prop:EV} $(\lambda^*(\mu) - \lambda_{\rho})(\mu - \lambda_{\rho})>0$ for $\mu$ varying around $\lambda_{\rho}$. \end{remark} \begin{proof}[Proof of Theorem \ref{thm4}] Consider the equation $$ -\Delta u = \lambda \rho(x) u +f, $$ rewritten as $$ -\Delta u + \lambda \rho^-_{\alpha}(x) u = \lambda \rho^+_{\alpha}(x) u + f. $$ This equation with positive weight can be treated as \eqref{eq:1} except that the Laplacian $-\Delta$ is replaced by $L^*:=-\Delta + \lambda_{\rho} \rho^-_{\alpha}$. Obviously, its principal eigenpair is $(\lambda_{\rho}, \varphi_{\rho})$: $L^* \varphi_{\rho}= \lambda_{\rho} \rho^+_{\alpha} (x)\varphi_{\rho} $. We denote by $\sigma_2$ the second eigenvalue of this problem. As previously (proof of Theorem \ref{thm2}), we decompose $u=u^* \varphi_{\rho} + u^{\perp}$ where $\int \rho^+_{\alpha} u^{\perp} \varphi_{\rho} = 0$. Also $h =h^* \varphi_{\rho} + h^{\perp}$. We derive analogous inequalities $$ u= \frac{h^*}{\lambda_{\rho} -\lambda} \varphi_{\rho} + u^{\perp} $$ and $$ \int \rho^+_{\alpha} (u^{\perp})^2 \leq (\frac{1}{\sigma_2-\lambda})^2 \int \rho^+_{\alpha} (h^{\perp})^2. $$ For $\lambda < \lambda_{\rho} $, $\sigma_2 - \lambda > \sigma_2 - \lambda_{\rho} $. For $\lambda > \lambda_{\rho} $, choose $\lambda$ close enough to $\lambda_{\rho} $ so that $\sigma_2 - \lambda > \frac{1}{4} (\sigma_2 - \lambda_{\rho} )$. We derive as for positive weight that $|u_{\lambda}^{\perp}|$ is locally bounded and, for $|\lambda -\lambda_{\rho}|$, small enough we derive the results for one equation as for a positive weight. We conclude as in the proof of Theorems \ref{thm2} and \ref{thm3}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm5}] For a positive weight, we set $$ U=P\widetilde{U}\Leftrightarrow \widetilde{U}=P^{-1}U, \quad F=P\widetilde{F}\Leftrightarrow \widetilde{F}=P^{-1}F. $$ System $\eqref{n system}$ can be written as \begin{equation}\label{eq:8b} -\Delta \widetilde{U} = \lambda \rho(x) J\widetilde{U} +\widetilde{F}, \end{equation} where $J$ is a Jordan canonical form which has $p$ Jordan blocks $J_i$ ($1\leq j\leq p \leq n$). We are also lead to Equations \eqref{eq:9}, \eqref{eq:10} and \eqref{eq:11} in $\tilde u_j$ with indefinite weight that we can study using Theorem \ref{thm4}. Going back to $U=P\tilde{U}$, we have \eqref{eq:15}: \begin{equation*} u_i=p_{i1}\widetilde{u}_1+\sum_{j=2}^np_{ij}\widetilde{u}_j, \quad 1\leq i\leq n. \end{equation*} Finally on a ball $B_R$, we have \noindent (\textbf{I}) If $\lambda_\rho-\delta<\lambda \sigma_1<\lambda_{\rho}$ and $\lambda_\rho-\lambda\sigma_1$ small enough, we have $\widetilde{u}_1/\varphi_\rho\to +\infty$ as $\lambda\sigma_1\to\lambda_\rho$ and $\sum_{j=2}^np_{ij}\widetilde{u}_j$ is bounded by a constant times $\varphi_\rho$. \noindent(\textbf{II}) If $\lambda_{\rho}<\lambda\sigma_1<\lambda_\rho+\delta' $ and $\lambda\sigma_1-\lambda_{\rho} $ small enough, we have $\widetilde{u}_1/\varphi_\rho\to -\infty$ as $\lambda\sigma_1\to\lambda_\rho$ and $\sum_{j=2}^np_{ij}\widetilde{u}_j$ is bounded by a constant times $\varphi_\rho$. \end{proof} \subsection*{Acknowledgements} This work was partly supported by grants: 10871157 and 11001221 from the National Natural Science Foundation of China, 200806990032 from the Specialized Research Fund for the Doctoral Program of Higher Education, and 11126313 from Tianyuan Special Fund for the National Natural Science Fundation of China. It was also partly supported by Contract CTP from the Conseil R\'egional Midi-Pyr\'en\'ees. \begin{thebibliography}{00} \bibitem{Al} W. Allegretto; \emph{Principal eigenvalues for indefinite weight elliptic problems in $\mathbb{R}^n$.} Proc. Amer. Math. Soc., 116, 1992, p. 701-706. \bibitem{AH} W. Allegretto, Y. X. Huang; \emph{Eigenvalues of the indefinite-weight p-Laplacian in weighted spaces.} Funkcialaj Ekvacioj., 38, 1995, p. 233-242. \bibitem{AFL} B. Alziary, J. Fleckinger, M. H. L\'ecureux; \emph{Systems of Schr\"{o}dinger equations: positivity and negativity.} Monograf\'{\i}as del Sem. Matem\'{a}tico Garc\'{\i}a de Galdeano, 33, 2006, p. 19-26. \bibitem{AFT} B. Alziary, J. Fleckinger, P. Tak\'a\v{c}; \emph{Positivity and negativity of solutions to a Schr\"{o}dinger equation in $\mathbb{R}^N$.} Positivity, 5, 4, 2001, p. 359-382. \bibitem{AFT2} B. Alziary, J. Fleckinger, P. Tak\'{a}\v{c}; \emph{An extension of maximum and anti-maximum principles to a Schr\"{o}dinger equation in $\mathbb{R}^2$.} J. Diff. Equ., 156, 1999, p. 122-152. \bibitem{AT1} B. Alziary, P. Tak\'{a}\v{c}; \emph{A pointwise lower bound for positive solutions of a Schr\"{o}dinger equation in $\mathbb{R}^N$.} J. Diff. Equ., 133, 2, 1997, p. 280-295. \bibitem{ATa} B. Alziary, P. Tak\'{a}\v{c}; \emph{Compactness for a Schr\"{o}dinger operator in the ground-state space over $\mathbb{R}^N$.} Proceedings of the 2006 International Conference in honor of Jacqueline Fleckinger, Electron. J. Differ. Equ. Conf., 16, Texas State Univ. -San Marcos, Dept. Math., San Marcos, TX, 2007, p. 35-58. \bibitem{BePl} A. Bermann, R. J. Plemmons; \emph{Nonnegative matrices in the Mathematical Sciences}, Academic Press, New York, 1979. \bibitem{BCF} K. J. Brown, C. Cosner, J. Fleckinger; \emph{Principal eigenvalues for problems with indefinite weight function on $\mathbb{R}^N$.} Proc. Amer. Math. Soc., 109, 1, 1990, p. 147-155. \bibitem{BTe} K. J. Brown, A. Tertikas; \emph{The existence of principal eigenvalues for problems with indefinite weight function in $\mathbb{R}^n$.} Proc. Royal Soc. Edinburgh, 123A, 1993, p. 561-569. \bibitem{ClPe} P. Cl\'ement, L. Peletier; \emph{An anti-maximum principle for second order elliptic operators.} J. Diff. Equ., 34, 1979, p. 218-229. \bibitem{FiMi} D. de Figueiredo, E. Mitidieri; \emph{Maximum principles for cooperative elliptic systems.} C. R. Acad. Sci. Paris 310, Serie I, 1990, p. 49-52. \bibitem{FGoTaT} J. Fleckinger, J. P. Gossez, P. Tak\'{a}\v{c}, F. de Th\'elin; \emph{Existence, nonexistence et principe de l'antimaximum pour le p-laplacien.} C. R. Acad. Sci. Paris 321, 1995, p. 731-734. \bibitem{FGoT04} J. Fleckinger, J. P. Gossez, F. de Th\'elin; \emph{Antimaximum principle in $\mathbb{R}^N$: local versus global.} J. Diff. Equ., 8, 7B, 2004, p. 159-188. \bibitem{FHeT1995} J. Fleckinger, J. Hernandez, F. de Th\'elin; \emph{On Maximum principles and existence of positive solutions for some cooperative elliptic systems}, Diff. Int. Eq., 8, 12, 1995, p. 69-85. \bibitem{FHeT2004} J. Fleckinger, J. Hernandez, F. de Th\'elin; \emph{Existence of multiple principal eigenvalues for some indefinite linear eigenvalue problems.}, Boll. U.M.I., 8, 7B, 2004, p. 159-188. \bibitem{FHeT2011} J. Fleckinger, J. Hernandez, F. de Th\'elin; \emph{Estimate of the validity interval for the antimaximum principle } In preparation. \bibitem{FMST} J. Fleckinger, R. Manasevich, N. Stavrakakis, F. de Th\'elin; \emph{Principal eigenvalues for some quasilinear elliptic systems on $\mathbb{R}^n$.} Advances in Diff. Eq., 2, 6, 1997, p. 981-1003. \bibitem{FM} J. Fleckinger, A. B. Mingarelli; \emph{On the eigenvalues of non-definite elliptic operators}, Differential equations, North Holland, 1984, p. 219-222. \bibitem{FS} J. Fleckinger-Pell\'e, H. Serag; \emph{Semilinear cooperative elliptic systems on $\mathbb{R}^n$}, Rend. Mat., S.VII, V15, 1995, p. 89-108. \bibitem{FW} J. Fleckinger, N. Wei; \emph{Estimates of solutions to some weighted systems defined on $\mathbb{R}^N$.} Operator theory and its applications, Amer. Math. Soc. Trans. Series 2, 231, 2010, p. 89-98. \bibitem{Gan} F.R. Gantmacher; \emph{The Theory of Matrices} vol. 2, 2nd ed., Chelsea, New York, 1964. \bibitem{Hess} P. Hess; \emph{On the eigenvalue problem for weakly coupled elliptic systems.} Arch. Rat. Mech. Anal., 81, 1983, p. 151-159. \bibitem{KS91} H. Kozono, and H. Sohr, \emph{New A Priori estimates for the Stokes Equations in Exterior domains.}, Indiana Univ. Math. Journal 40 (1991), 1-27. \bibitem{Lec} M. H. L\'ecureux; \emph{Comparison with groundstate for solutions of noncooperative systems of Schr\" odinger operator on $\mathbb{R}^N$.} Rostocker Math. Koll 65, 2010, p. 51-69. \bibitem{StT} N. Stavrakakis, F. de Th\'elin; \emph{Principal eigenvalues and anti-maximum principle for some quasilinear elliptic equations on $\mathbb{R}^n$.} Math. Nachr., 212, 1, 2000, p. 155-171. \end{thebibliography} \end{document}