where e6da6c268a9789e4eb4e64b8" title="Click to view the MathML source">Δpz:=div(|∇z|p−2∇z), e5a4a8f8f619e47d760eb" title="Click to view the MathML source">1<p<n, λ is a positive parameter, 9a4" title="Click to view the MathML source">r0>0 and ΩE:={x∈Rn | |x|>r0}. Here the weight function 9a0bfa1b2c54103e" title="Click to view the MathML source">K∈C1[r0,∞) satisfies e741ece054603ce048d523b" title="Click to view the MathML source">K(r)>0 for e7bb76170b3b326cb2a7" title="Click to view the MathML source">r≥r0, e73529f" title="Click to view the MathML source">limr→∞K(r)=0, and the reaction term 9b16ec0f7ac0b7dd83dfae450" title="Click to view the MathML source">f∈C[0,∞)∩C1(0,∞) is strictly increasing and satisfies ae6a3f3497ab947229" title="Click to view the MathML source">f(0)<0 (semipositone), ae8bdbc77500934873d946bd58fe18">, e722600904" title="Click to view the MathML source">lims→∞f(s)=∞, e7940ee95f6cd2"> and is nonincreasing on 9afa626400eb66cbdec95e43bf" title="Click to view the MathML source">[a,∞) for some 9b" title="Click to view the MathML source">a>0 and 9a9e3cd7553221b843992323ff62a" title="Click to view the MathML source">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 a4f47191b7807ed63df861" title="Click to view the MathML source">λ≫1. We establish the uniqueness of this positive radial solution for a4f47191b7807ed63df861" title="Click to view the MathML source">λ≫1.