A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball

We consider steady state reaction diffusion equations on the exterior of a ball, namely, boundary value problems of the form:{−Δpu=λK(|x|)f(u) in ΩE,u=0 on |x|=r0,u→0 when |x|→∞, where Δpz:=div(|∇z|p−2∇z)Δpz:=div(|∇z|p−2∇z), 10r0>0 and ΩE:={x∈Rn | |x|>r0}ΩE:={x∈Rn | |x|>r0}. Here the weight function K∈C1[r0,∞)K∈C1[r0,∞) satisfies K(r)>0K(r)>0 for r≥r0r≥r0, limr→∞⁡K(r)=0limr→∞⁡K(r)=0, and the reaction term f∈C[0,∞)∩C1(0,∞)f∈C[0,∞)∩C1(0,∞) is strictly increasing and satisfies f(0)<0f(0)<0 (semipositone), limsups→0+sf′(s)<∞, lims→∞⁡f(s)=∞lims→∞⁡f(s)=∞, lims→∞⁡f(s)sp−1=0 and f(s)sq is nonincreasing on [a,∞)[a,∞) for some a>0a>0 and q∈(0,p−1)q∈(0,p−1). For a class of such steady state equations it turns out that every nonnegative radial solution is strictly positive in the exterior of a ball, and exists for λ≫1λ≫1. We establish the uniqueness of this positive radial solution for λ≫1λ≫1.