# Cooperative and noncooperative elliptic systems

Cooperative and noncooperative elliptic systems

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

$$({P}^{\pm})\{\begin{array}{cc}-\mathrm{\Delta}u=\lambda u\pm \delta v+{G}_{u}(x;u,v),\hfill & x\in \mathrm{\Omega}\hfill \\ -\mathrm{\Delta}v=\delta u+\gamma v\pm {G}_{v}(x;u,v),\hfill & x\in \mathrm{\Omega}\hfill \\ u=v=0\text{or}\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0\hfill & x\in \partial \mathrm{\Omega}\hfill \end{array}$$ |

where $\mathrm{\Omega}\subset {\mathbb{R}}^{N}(N\ge 1)$ is an open bounded domain, $\lambda ,\gamma ,\delta $ are real parameters, $G(x;u,v)\in {\mathcal{C}}^{1}(\overline{\mathrm{\Omega}}\times {\mathbb{R}}^{2};\mathbb{R})$ 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 |
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Canonical name | CooperativeAndNoncooperativeEllipticSystems1 |

Date of creation | 2013-03-11 19:28:25 |

Last modified on | 2013-03-11 19:28:25 |

Owner | linor (11198) |

Last modified by | (0) |

Numerical id | 1 |

Author | linor (0) |

Entry type | Definition |