0) and the weight function b(x) behaves near the origin as a function b0(|x|) varying regularly at zero with index θ greater than −p. This condition includes b(x) = |x|θ and some of its perturbations, for instance, b(x) = |x|θ (−log |x|)m for any m ∈ R. Our approach makes use of the theory of regular variation and a new perturbation method for constructing sub- and super-solutions. © 2010 Elsevier Inc. All rights reserved