Uniqueness of nonnegative solutions for semipositone problems on exterior domains

We consider the problem {−Δu=λK(|x|)f(u),x∈Ωu=0if |x|=r0u→0as |x|→∞, where λ is a positive parameter, Δu=div(∇u) is the Laplacian of u, Ω={x∈Rn;n>2,|x|>r0}, K∈C1([r0,∞),(0,∞)) is such that limr→∞K(r)=0 and f∈C1([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.