Uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball

We study positive radial solutions to: −Δu=λK(|x|)f(u)−Δu=λK(|x|)f(u); x∈Ωex∈Ωe, where λ>0λ>0 is a parameter, Ωe={x∈RN:|x|>r0,r0>0,N>2}, Δ is the Laplacian operator, K∈C([r0,∞),(0,∞))K∈C([r0,∞),(0,∞)) satisfies K(r)≤1rN+μ; μ>0μ>0 for r≫1r≫1 and f∈C1([0,∞),R)f∈C1([0,∞),R) is a concave increasing function satisfying lims→∞⁡f(s)s=0 and f(0)<0f(0)<0 (semipositone). We are interested in solutions u   such that u→0u→0 as |x|→∞|x|→∞ and satisfy the nonlinear boundary condition ∂u∂η+c˜(u)u=0 if |x|=r0|x|=r0 where ∂∂η is the outward normal derivative and c˜∈C([0,∞),(0,∞)) is an increasing function. We will establish the uniqueness of positive radial solutions for large values of the parameter λ.