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.