Uniqueness of Positive Radial Solutions for a Class of Semipositone Systems on the Exterior of a Ball

In this paper, we study the positive radial solutions for elliptic systems to the nonlinear BVP:
, where Δu = div (∇u) and Δv = div (∇v) are the Laplacian of u, λ is a positive parameter, Ω = {x ∈ Rn : N > 2, |x| > r0, r0 > 0}, let i = [1,2] then Ki :[r0,∞] → (0,∞) is a continuous function such that limr→∞ ki(r) = 0 and is The external natural derivative, and : [0, ∞) → (0, ∞) is a continuous function. We discuss existence and multiplicity results for classes of f with a) fi > 0, b) fi < 0, and c) fi = 0. We base our presence and multiple outcomes via the Sub-solutions method. We also discuss some unique findings.