Cooperative and noncooperative elliptic systems

Cooperative (+) and noncooperative (-) elliptic systems are of theform

(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 $\abla G=(G_{u},G_{v})$ . $P^{+}$ and $P^{-}$ are called cooperative and noncooperative, respectively. In addition, $\\delta$ is assumed tobe 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.