Semipositone problems with falling zeros on exterior domains

We study boundary value problems of the form { − Δ u = λ K ( | x | ) f ( u ) , x ∈ Ω u = 0 if  | x | = r 0 u → 0 as  | x | → ∞ , where λ is a positive parameter, Δ u = div ( ∇ u ) is the Laplacian of u , Ω = { x ∈ R n ; n > 2 , | x | > r 0 } , K belongs to a class of C 1 functions such that lim r → ∞ K ( r ) = 0 , and f belongs to a class of C 1 functions which are negative at the origin and have falling zeros. We discuss the existence and uniqueness of nonnegative radial solutions when λ is large.