Positivity and negativity of solutions to nXn weighted systems involving the Laplace operator on R^N

We consider the sign of the solutions of a $n imes n$ system defined on the whole space $mathbb{R}^N$, $Ngeq 3$ and a weight function $ ho$ with a positive part decreasing fast enough, $$ -Delta U = lambda ho(x) MU +F, $$ where $F$ is a vector of functions, $M$ is a $n imes n$ matrix with constant coefficients, not necessarily cooperative, and the weight function $ ho$ is allowed to change sign. We prove that the solutions of the $n imes n$ system exist and then we prove the local fundamental positivity and local fundamental negativity of the solutions when $|lambdasigma_1-lambda_ ho|$ is small enough, where $sigma_1$ is the largest eigenvalue of the constant matrix $M$ and $lambda_ ho$ is the "principal" eigenvalue of $$ -Delta u = lambda ho(x) u , quad lim_{|x| o infty} u(x) = 0 ; quad u(x)>0, quad xin mathbb{R}^N. $$