where 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 8d603c87c379d56a30bcd8320f8" title="Click to view the MathML source">ΩE:={x∈Rn | |x|>r0}. Here the weight function 9a0bfa1b2c54103e" title="Click to view the MathML source">K∈C1[r0,∞) satisfies 8d523b" title="Click to view the MathML source">K(r)>0 for r≥r0, a3f873dce73529f" 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 a3f3497ab947229" title="Click to view the MathML source">f(0)<0 (semipositone), , lims→∞f(s)=∞, and is nonincreasing on 9afa626400eb66cbdec95e43bf" title="Click to view the MathML source">[a,∞) for some abf9ed957f85e9b" 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 λ≫1. We establish the uniqueness of this positive radial solution for λ≫1.