where Δpz:=div(|∇z|p−2∇z), 92874edfe89e5a4a8f8f619e47d760eb" title="Click to view the MathML source">1<p<n, λ is a positive parameter, 841b469a4" title="Click to view the MathML source">r0>0 and 92e08d603c87c379d56a30bcd8320f8" title="Click to view the MathML source">ΩE:={x∈Rn | |x|>r0}. Here the weight function 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, 84f79017f1903d7a3f873dce73529f" title="Click to view the MathML source">limr→∞K(r)=0, and the reaction term f∈C[0,∞)∩C1(0,∞) is strictly increasing and satisfies e9c4dae6a3f3497ab947229" title="Click to view the MathML source">f(0)<0 (semipositone), , e722600904" title="Click to view the MathML source">lims→∞f(s)=∞, e7940ee95f6cd2"> and is nonincreasing on [a,∞) for some bb1c45589078abf9ed957f85e9b" title="Click to view the MathML source">a>0 and 843992323ff62a" 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 84d51fa4f47191b7807ed63df861" title="Click to view the MathML source">λ≫1. We establish the uniqueness of this positive radial solution for 84d51fa4f47191b7807ed63df861" title="Click to view the MathML source">λ≫1.