Existence and nonexistence of positive solutions for a class of superlinear semipositone systems

We consider an elliptic system of the form −Δu=λf(v)in Ω−Δv=λg(u)in Ωu=0=von ∂Ω,} where λ>0λ>0 is a parameter, ΩΩ is a bounded domain in RNRN with smooth boundary ∂Ω∂Ω. Here the nonlinearities f,g:[0,∞)→Rf,g:[0,∞)→R are Cloc0,σ,0<σ<1, functions that are superlinear at infinity and satisfy f(0)<0f(0)<0 and g(0)<0g(0)<0. We prove that the system has a positive solution for λλ small when ΩΩ is convex with C3C3 boundary and no positive solution for λλ large when ΩΩ is a general bounded domain with C2,βC2,β boundary. Moreover, we show that there exists a closed connected subset of positive solutions bifurcating from infinity at λ=0λ=0.