Isolated singularities for weighted quasilinear elliptic equations

Abstract

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.

Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification

“…(i) When A = b = 1 and h(t) = |t| q−1 t, our Theorem 1.1(a) recovers [14,Theorem 2.1]. Moreover, Theorem 1.1(a) generalises and sharpens [10,Theorem 1.1], which analysed the case A = 1 and q < q * . Our Theorem 1.1 is also established under the optimal condition for the existence of solutions with singularities at 0 for (1.4).…”

“…Motivated by previous articles such as [3,9,10,14,27], we aim to obtain a complete understanding of the isolated singularities for nonlinear elliptic equations of the form (1.4) in the punctured unit ball B 1 \{0} in R N (N ≥ 2) under the Assumptions (A 1 )-(A 3 ) given later. A prototype model is A(|x|) = |x| ϑ , b(x) = |x| σ and h(t) = |t| q−1 t for q > p − 1 > 0 and ϑ − σ < p ≤ N + ϑ.…”

“…In Examples 1-3 above, we take α, β, γ ∈ R and ν ∈ (0, 1/2), 1 < p < N + ϑ and q + 1 > p > ϑ − σ (see Corollary 2.1 for the classification). More generally, we work in the setting of regular variation theory inspired by Cîrstea and Du [10], whose results (with A = 1) are here generalised and sharpened. The introduction of the weight function A in the operator in (1.4) adds non-trivial difficulties.…”

“…In the case of non-removable singularities, that is b(x) h(Φ) ∈ L 1 (B 1/2 ), we give a complete classification of the singularities of (1.4) in Theorem 1.1(a), accompanied by corresponding existence results in Theorem 1.2. Our analysis brings new understanding of the behaviour of the solutions to (1.4) with strong singularities at zero as the perturbation technique introduced in [10] for the subcritical case is not applicable in the critical case. The main innovation we develop is a perturbation technique which enables us to give precise explicit asymptotic formulas for the behaviour of the strong singular solutions.…”

Classification of isolated singularities for nonhomogeneous operators in divergence form

“…(1.1) the fact that it is nonhomogeneous asks for a more careful treatment and a different approach involving results coming from the theory of equations analyzed in Orlicz-Sobolev spaces. In particular, we will adapt ideas from Cîrstea and Du [4], Friedman and Véron [6] or Brandolini et al [3] to the new setting.…”