where md5=1ea29e86e6da6c268a9789e4eb4e64b8" title="Click to view the MathML source">Δpz:=div(|∇z|p−2∇z), md5=92874edfe89e5a4a8f8f619e47d760eb" title="Click to view the MathML source">1<p<n, λ is a positive parameter, md5=68ed7b76aad25e0a2c735f8841b469a4" title="Click to view the MathML source">r0>0 and md5=a92e08d603c87c379d56a30bcd8320f8" title="Click to view the MathML source">ΩE:={x∈Rn | |x|>r0}. Here the weight function md5=b30dd4c71b5b45379a0bfa1b2c54103e" title="Click to view the MathML source">K∈C1[r0,∞) satisfies md5=139f937c8e741ece054603ce048d523b" title="Click to view the MathML source">K(r)>0 for md5=9fbc477a26e4e7bb76170b3b326cb2a7" title="Click to view the MathML source">r≥r0, md5=2284f79017f1903d7a3f873dce73529f" title="Click to view the MathML source">limr→∞K(r)=0, and the reaction term md5=46cf1ad9b16ec0f7ac0b7dd83dfae450" title="Click to view the MathML source">f∈C[0,∞)∩C1(0,∞) is strictly increasing and satisfies md5=6f41e8cf1e9c4dae6a3f3497ab947229" title="Click to view the MathML source">f(0)<0 (semipositone), md5=1bae8bdbc77500934873d946bd58fe18">, md5=2ceda53bd48c510d1b7389e722600904" title="Click to view the MathML source">lims→∞f(s)=∞, md5=d5f4bfc3f672426254e7940ee95f6cd2"> and md5=f7ee135fde423808bc7ad01fc07568a7"> is nonincreasing on md5=9913f49afa626400eb66cbdec95e43bf" title="Click to view the MathML source">[a,∞) for some md5=bf366bb1c45589078abf9ed957f85e9b" title="Click to view the MathML source">a>0 and md5=2159a9e3cd7553221b843992323ff62a" 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 md5=b42884d51fa4f47191b7807ed63df861" title="Click to view the MathML source">λ≫1. We establish the uniqueness of this positive radial solution for md5=b42884d51fa4f47191b7807ed63df861" title="Click to view the MathML source">λ≫1.