where e6da6c268a9789e4eb4e64b8" title="Click to view the MathML source">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 K∈C1[r0,∞) satisfies K(r)>0 for r≥r0, 9017f1903d7a3f873dce73529f" 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">, 904" title="Click to view the MathML source">lims→∞f(s)=∞, e95f6cd2"> and is nonincreasing on [a,∞) for some 9078abf9ed957f85e9b" 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 λ≫1. We establish the uniqueness of this positive radial solution for λ≫1.