Uniqueness of nonnegative solutions for semipositone problems on exterior domains

We consider the problem { − 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 ∈ C 1 ( [ r 0 , ∞ ) , ( 0 , ∞ ) ) is such that lim r → ∞ K ( r ) = 0 and f ∈ C 1 ( [ 0 , ∞ ) , R ) is a concave function which is sublinear at ∞ and f ( 0 ) < 0 . We establish the uniqueness of nonnegative radial solutions when λ is large.