We consider the problem where λ is a positive parameter, δu=div(∇;u) is the Laplacian of u, Ω={x∈Rn;n>2,{pipe}x{pipe}>r0}, K∈C 1([r 0, ∞), (0, ∞)) is such that lim r→∞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. © 2012 Elsevier Ltd.