Semipositone problems with falling zeros on exterior domains

We study boundary value problems of the form {−Δu=λK(|x|)f(u),x∈Ωu=0if |x|=r0u→0as |x|→∞, where λλ is a positive parameter, Δu=div(∇u)Δu=div(∇u) is the Laplacian of uu, Ω={x∈Rn;n>2,|x|>r0}, KK belongs to a class of C1C1 functions such that limr→∞K(r)=0limr→∞K(r)=0, and ff belongs to a class of C1C1 functions which are negative at the origin and have falling zeros. We discuss the existence and uniqueness of nonnegative radial solutions when λλ is large.