## Differential and Integral Equations

### 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 $$\label{sy} \left\{ \begin{array}{cccc} -\Delta u & = & f(.,u)-\mu av & \text{in } D, \\ -\Delta v & = & bu & \text{in }D, \\ \underset{ x \rightarrow \partial_{\infty} D }{\lim }u(x) & = & \underset{ x \rightarrow \partial_{\infty} D}{\lim }v(x) & = 0, \end{array} \right.$$ 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.

#### Article information

Source
Differential Integral Equations, Volume 19, Number 9 (2006), 1019-1034.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356050330

Mathematical Reviews number (MathSciNet)
MR2262094

Zentralblatt MATH identifier
1210.35077

Subjects
Primary: 35J55
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc.

#### Citation

Bezzarga, M.; Kefi, Khaled. Positive solution of Laplacian noncooperative system with potential control. Differential Integral Equations 19 (2006), no. 9, 1019--1034. https://projecteuclid.org/euclid.die/1356050330