Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2002; 39(3): 377-385

Printed May 1, 2002

Copyright © The Korean Mathematical Society.

A sufficient condition for the uniqueness of positive steady state to a reaction diffusion system

Joon Hyuk Kang and Yun Myung Oh

Andrews University

Abstract

In this paper, we concentrate on the uniqueness of the positive solution for the general elliptic system $$\left\{ \begin{array}{l} \left.\begin{array}{l} \Delta u + u(g_{1}(u) - g_{2}(v)) = 0\\ \Delta v + v(h_{1}(u) - h_{2}(v)) = 0 \end{array} \right.\;\;\mbox{in}\;\;R^{+} \times \Omega,\\ u|_{\partial\Omega} = v|_{\partial\Omega} = 0. \end{array} \right.$$ This system is the general model for the steady state of a competitive interacting system. The techniques used in this paper are upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.

Keywords: Lotka Voltera competition model, coexistence state

MSC numbers: 35A05, 35A07, 35G30, 35J25, 35K20

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