Existence of positive solutions for a class of superlinear semipositone systems

We consider an elliptic system of the form −Δu=λf(x,v)in  Ω,−Δv=λg(x,u)in  Ω,u=0=von  ∂Ω, where λ>0 is a parameter and Ω is a bounded domain in RN with C2,α boundary ∂Ω. Here the nonlinearities f,g:Ω×[0,∞)→R are Carathéodory functions that are superlinear at infinity and satisfy f(x,0)<0 and g(x,0)<0 almost everywhere in Ω. We prove that the system has a positive strong solution for λ small by using degree theory combined with re-scaling argument and a uniform L∞ apriori bound of positive strong solutions to some Lane-Emden type of systems.