Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 101, pp. 1-14.

Title: Positivity and negativity of solutions to $n\times n$
       weighted systems involving the Laplace operator on $\mathbb{R}^N$

Authors: Benedicte Alziary     (Univ. de Toulouse, France)
         Jacqueline Fleckinger (Univ. de Toulouse, France)
	 Marie-Helene Lecureux (Univ. de Toulouse, France)
	 Na Wei (Northwestern Polytechnical Univ. Xi'an, China)
	 
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
 $$
 -\Delta u = \lambda \rho(x) u ,  \quad
  \lim_{|x|\to \infty} u(x)  = 0 ; \quad u(x)>0, \quad x\in \mathbb{R}^N.
 $$

Submitted February 29, 2012. Published June 15, 2012.
Math Subject Classifications: 35B50, 35J05, 35J47.
Key Words: Elliptic PDE; maximum principle; fundamental positivity;
           fundamental negativity; indefinite weight, weighted systems.