Positive solution of Laplacian noncooperative system with potential control

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.