Nonnegative solutions for nonlinear elliptic systems

In this paper, we show that the semilinear elliptic systems of the form(0.1){−Δu−μΔv=g(v),−Δv−λΔu=f(u),x∈Ω,u=v=0,x∈∂Ω, possess at least one nonnegative nontrivial solution pair (u,v)∈H01(Ω)×H01(Ω), where Ω is a smooth bounded domain in RN.