We study positive radial solutions to the problem {equation presented} where λ is a positive parameter, Δu = div (δu) is the Laplacian of u, ω = {x ∈ Rn, n > 2 : |x| > r 0} is an exterior domain and f : (0.∞) → R belongs to a class of sublinear functions at ∞ such that they are continuous and f(0+) = lims→0+ f(s) < 0. In particular we also study infinite semipositone problems where lims→0+ f(s) = -∞. Here K : [r0,∞) → (0,∞) belongs to a class of continuous functions such that limr→∞ K(r) = 0. We establish various existence results for such boundary value problems and also extend our results to classes of systems. We prove our results by the method of sub-/supersolutions.