where e86e6da6c268a9789e4eb4e64b8" title="Click to view the MathML source">Δpz:=div(|∇z|p−2∇z), e89e5a4a8f8f619e47d760eb" title="Click to view the MathML source">1<p<n, λ is a positive parameter, ad25e0a2c735f8841b469a4" title="Click to view the MathML source">r0>0 and ΩE:={x∈Rn | |x|>r0}. Here the weight function K∈C1[r0,∞) satisfies K(r)>0 for r≥r0, limr→∞K(r)=0, and the reaction term ad9b16ec0f7ac0b7dd83dfae450" title="Click to view the MathML source">f∈C[0,∞)∩C1(0,∞) is strictly increasing and satisfies e8cf1e9c4dae6a3f3497ab947229" title="Click to view the MathML source">f(0)<0 (semipositone), e8bdbc77500934873d946bd58fe18">, lims→∞f(s)=∞, and 808bc7ad01fc07568a7"> is nonincreasing on afa626400eb66cbdec95e43bf" title="Click to view the MathML source">[a,∞) for some 8abf9ed957f85e9b" title="Click to view the MathML source">a>0 and 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 807ed63df861" title="Click to view the MathML source">λ≫1. We establish the uniqueness of this positive radial solution for 807ed63df861" title="Click to view the MathML source">λ≫1.