Numerical Approach to Obtain Positive solution for Classes of Laplacian Systems

Using a numerical method based on sub-super solution, we will obtain positive solution to the coupled-system of boundary value problems of the form −∆u = λ1f(v) + μ1h(u) in Ω −∆v = λ2 g(u) + μ2γ(v) in Ω u = 0 = v on ∂Ω where −∆ is the Laplacian operator λ1, λ2, μ1, μ2 are nonnegative parameters, and Ω is a bounded region in Rn, with smooth boundary ∂Ω . We obtain numerically a large positive solution for λ1 + μ1 and λ2 + μ2 large when lim x→∞ f(M [g(x)]) x = 0 for every M > 0, limx→∞ h(x) x = 0 and limx→∞ γ(x) x = 0. In particular, we do not assume any sign conditions on f(0), g(0), h(0), or γ(0).