Existence of the minimal positive solution of some nonlinear elliptic systems when the nonlinearity is the sum of a sublinear and a superlinear term

It is shown that there exists Λ>0 such that, for every λε(0, Λ), the semilinear elliptic system: Δu = λu | u |^q-1 + u | u |^p-1 - υ, in ω, u=v=0 on ∂ ω, where ω εR ^N(N≥2) is a bounded domain with smooth boundary and 0<q<1<p, has a minimal positive solution (u_λ, v_λ). Moreover: u_λ and v_λ are strictly increasing with respect to λ.