40A05-CooperativeAndNoncooperativeEllipticSystems1.tex

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\pmtitle{Cooperative and noncooperative elliptic systems}
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Cooperative (+) and noncooperative (-) elliptic systems are of the
form
    $$ 
       (P^{\pm}) \left \{ \begin{array}{ll}
              -\Delta u=\lambda u { \pm} \delta v + G_u(x; u,v),  &\, x\in \Omega \nonumber \\
              -\Delta v= \delta u + \gamma v {\pm} G_v(x; u,v),  &\, x\in \Omega\\
                u=v=0 \; \mbox{or} \;
           \frac{\partial u}{\partial n}= \frac{\partial v}{\partial n}=0
           &\, x\in \partial \Omega
         \end{array}  \right.
   $$
where $\Omega\subset {\mathbb R}^N (N\ge 1)$ 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.



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