where Δpz:=div(|∇z|p−2∇z), 874edfe89e5a4a8f8f619e47d760eb" title="Click to view the MathML source">1<p<n, λ is a positive parameter, r0>0 and 87c379d56a30bcd8320f8" title="Click to view the MathML source">ΩE:={x∈Rn | |x|>r0}. Here the weight function bfa1b2c54103e" title="Click to view the MathML source">K∈C1[r0,∞) satisfies 9f937c8e741ece054603ce048d523b" title="Click to view the MathML source">K(r)>0 for 9fbc477a26e4e7bb76170b3b326cb2a7" title="Click to view the MathML source">r≥r0, 873dce73529f" 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), 873d946bd58fe18">, e722600904" title="Click to view the MathML source">lims→∞f(s)=∞, bfc3f672426254e7940ee95f6cd2"> and is nonincreasing on 913f49afa626400eb66cbdec95e43bf" title="Click to view the MathML source">[a,∞) for some bf366bb1c45589078abf9ed957f85e9b" 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 91b7807ed63df861" title="Click to view the MathML source">λ≫1. We establish the uniqueness of this positive radial solution for 91b7807ed63df861" title="Click to view the MathML source">λ≫1.