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

Abstract

We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations of the formwhere B r denotes the open ball with radius r > 0 centred at 0 in R N (N ≥ 2). We assume thatare positive functions associated with regularly varying functions of index ϑ, σ and q at 0, 0 and ∞ respectively, satisfying q > p − 1 > 0 and ϑ − σ < p < N + ϑ. We prove that the condition b(x) h(Φ) L 1 (B 1/2 ) is sharp for the removability of all singularities at 0 for the positive solutions of (0.1), where Φ denotes the "fundamental solution" of −div (A(|x|) |∇u| p−2 ∇u) = δ 0 (the Dirac mass at 0) in B 1 , subject to Φ| ∂B 1 = 0. If b(x) h(Φ) ∈ L 1 (B 1/2 ), we show that any non-removable singularity at 0 for a positive solution of (0.1) is either weak (i.e., lim |x|→0 u(x)/Φ(|x|) ∈ (0, ∞)) or strong (lim |x|→0 u(x)/Φ(|x|) = ∞). The main difficulty and novelty of this paper, for which we develop new techniques, come from the explicit asymptotic behaviour of the strong singularity solutions in the critical case, which had previously remained open even for A = 1. We also study the existence and uniqueness of the positive solution of (0.1) with a prescribed admissible behaviour at 0 and a Dirichlet condition on ∂B 1 .

Sharp existence and classification results for nonlinear elliptic equations in $\mathbb R^N\setminus\{0\}$ with Hardy potential

“…Their results were generalized to p-Laplacian type equations with 1 < p < N by Friedman and Véron [23] for p − 1 < q < N (p − 1)/(N − p) and by Vázquez and Véron [37] for q ≥ N (p − 1)/(N − p). More recent generalizations exist in various directions, but without a Hardy potential [4,10,12,14,15,35].…”

“…In fact, more general weights than |x| −2a were considered in [4] using the framework of regular variation theory. For recent generalizations of these local existence and classification results to weighted quasilinear elliptic equations, see [10,35].…”

“…Construction and motivation of our refined sub/super-solutions. The idea of constructing a suitable family of sub-solutions and super-solutions to obtain more precise upper and lower bound estimates near zero has been used successfully for various nonlinear elliptic equations without a Hardy potential, see for example [10,12,14,15]. In our situation, the introduction of the Hardy potential in Case (U) and Case (M 1 ) poses an extra difficulty when comparing an arbitrary solution u of (1.2) with a super-solution (or subsolution).…”

Existence of sharp asymptotic profiles of singular solutions to an elliptic equation with a sign-changing non-linearity

“…The isolated singularity problem has been studied extensively, see Véron's monograph [21]. Recent works of the first author and her collaborators such as [4,10,11] give a full classification of the isolated singularities for various classes of elliptic equations.…”

“…Let q ∈ (1, 2 ⋆ − 1). Suppose that (5) admits a positive smooth solution u satisfying (4). From u = 0 on ∂ Ω, we have ∇u = (∂ ν u) ν for x ∈ ∂ Ω, where ν denotes the unit outward normal at ∂ Ω.…”