# Cooperative and noncooperative elliptic systems

Cooperative and noncooperative elliptic systems

Cooperative (+) and noncooperative (-) elliptic systems are of the form

 $(P^{\pm})\left\{\begin{array}[]{ll}-\Delta u=\lambda u{\pm}\delta v+G_{u}(x;u,% v),&\,x\in\Omega\\ -\Delta v=\delta u+\gamma v{\pm}G_{v}(x;u,v),&\,x\in\Omega\\ u=v=0\;\mbox{or}\;\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=% 0&\,x\in\partial\Omega\end{array}\right.$

where $\Omega\subset{\mathbb{R}}^{N}(N\geq 1)$ is an open bounded domain, $\lambda,\gamma,\delta$ are real parameters, $G(x;u,v)\in\mathcal{C}^{1}(\overline{\Omega}\times\mathbb{R}^{2};\mathbb{R})$ in the variables $(u,v)\in\mathbb{R}^{2}$ with $\nabla G=(G_{u},G_{v})$. $P^{+}$ and $P^{-}$ are called cooperative and noncooperative, respectively. In addition, $\delta$ is assumed to be positive for the noncooperative case. Note that systems $(P^{\pm})$ are closely related to reaction-diffusion systems arising in various chemical/physical and biological phenomena.

Title Cooperative and noncooperative elliptic systems CooperativeAndNoncooperativeEllipticSystems1 2013-03-11 19:28:25 2013-03-11 19:28:25 linor (11198) (0) 1 linor (0) Definition