Existence Results for Semipositone Systems

We study existence of positive solutions to the coupled-system of boundary value problems of the form -Δu(x) = λf(x,u,v); x ∈ Ω -Δv(x) = λg(x,u,v); x ∈ Ω u(x) = 0 = v(x); x ∈ ∂Ω where λ \u3e 0 is a parameter, Ω is a bounded domain in R^N; N ≥ 1 with a smooth boundary ∂Ω and f,g are C^1 function with at least one of f(x_0,0,0) or g(x_0,0,0) being negative for some x_0 ∈ Ω (semipositone). We establish our existence results using the method of sub-super solutions. We also discuss non-existence results for λ small