Abstract
We are concerned with the uniform positivity preserving property on a domain $D$ of
$\mathbb{R}^d$ ($d\geq 3$), for the noncooperative system \begin{equation}\label{sy}
\left\{ \begin{array}{cccc} -\Delta u & = & f(.,u)-\mu av & \text{in } D, \\
-\Delta v & = & bu & \text{in }D, \\ \underset{ x \rightarrow
\partial_{\infty} D }{\lim }u(x) & = & \underset{ x \rightarrow \partial_{\infty}
D}{\lim }v(x) & = 0, \end{array} \right. \end{equation} where
$\partial_{\infty}D=\left\{ \begin{array}{ccc} \partial D ,\ \ \mbox{if D is bounded},\\
\partial D\cup \{+\infty\}, \ \ \mbox{if not}. \end{array} \right.$ We give appropriate
conditions on $a$, $b$ and $f$ to get the existence and positivity of the solutions with
potential control.