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

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.