A Sufficient Condition for the Uniqueness of Positive Steady State to a Reaction Diffusion System

In this paper, we concentrate on the uniqueness of the positive solution for the general elliptic system { Δu + u(g1(u) - g 2(v)) = 0 in R+ × Ω, Δν + v(h 1(u) - h2(v)) = 0 u|∂Ω = v| ∂Ω = 0. 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.