Positive solutions for a class of superlinear semipositone systems on exterior domains

We study the existence of a positive radial solution to the nonlinear eigenvalue problem -Delta u = lambda K-1 (vertical bar x vertical bar)f (v) in Omega(e), -Delta v = lambda K-2(vertical bar x vertical bar)g(u) in Omega(e), u(x) = v(x) = 0 if vertical bar x vertical bar = r(0) (> 0), u(x) -> 0, v(x) -> infinity as lx1 infinity, where lambda > 0 is a parameter, Delta u = div(del u) is the Laplace operator, Omega(e), = {x is an element of R-n vertical bar vertical bar x vertical bar > r(0), n > 2}, and K E Ci (r(0), infinity), (0, infinity)); i = 1,2 are such that K-i(x) -> 0 as vertical bar x vertical bar infinity. Here f,g : [0 infinity) -> R are C-1 functions such that they are negative at the origin (semipositone) and superlinear at infinity. We establish the existence of a positive solution for A small via degree theory and rescaling arguments. We also discuss a non-existence result for lambda >> 1 for the single equations case.