Let Ω be a smooth bounded domain in RN. We prove general uniqueness results for equations of the form − Δ u = au − b(x) f(u) in Ω, subject to u = ∞ on ∂ Ω. Our uniqueness theorem is established in a setting involving Karamata's theory on regularly varying functions, which is used to relate the blow‐up behavior of u(x) with f(u) and b(x), where b ≡ 0 on ∂ Ω and a certain ratio involving b is bounded near ∂ Ω. A key step in our proof of uniqueness uses a modification of an iteration technique due to Safonov. 2000 Mathematics Subject Classification 35J25 (primary), 35B40, 35J60 (secondary).

We classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div(|∇u| p−2 ∇u) = b(x)h (u) in Ω \ {0}, where 1 < p N and Ω is an open subset of R N with 0 ∈ Ω. Our main result provides a sharp extension of a well-known theorem of Friedman and Véron for h(u) = u q and b(x) ≡ 1, and a recent result of the authors for p = 2 and b(x) ≡ 1. We assume that the function h is regularly varying at ∞ with index q (that is, lim t→∞ h(λt)/ h(t) = λ q for every λ > 0) and the weight function b(x) behaves near the origin as a function b 0 (|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.

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