Existence of positive solutions for a class of superlinear semipositone systems

We consider an elliptic system of the form − Δ u = λ f ( x , v ) in  Ω , − Δ v = λ g ( x , u ) in  Ω , u = 0 = v on  ∂ Ω , where λ > 0 is a parameter and Ω is a bounded domain in R N with C 2 , 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.