Sharp existence and classification results for nonlinear elliptic equations in R^{N} ∖ { 0 } with Hardy potential
Abstract
For N ≥ 3, by the seminal paper of Brezis and Véron (1980/81), no positive solutions of  Δu +u^{q} = 0 in R^{N} ∖ { 0 } exist if q ≥ N / (N  2). In turn, for 1 < q < N / (N  2) the existence and profiles near zero of all positive C^{1} (R^{N} ∖ { 0 }) solutions are given by Friedman and Véron (1986).
In this paper, for every q > 1 and θ ∈ R, we prove that the elliptic problem (⋆)  Δu  λ  x^{2} u +  x^{θ}u^{q} = 0 in R^{N} ∖ { 0 } with u > 0 has a C^{1} (R^{N} ∖ { 0 }) solution if and only if λ >λ^{⁎}, where λ^{⁎} = Θ (N  2  Θ) with Θ = (θ + 2) / (q  1). We show that (a) if λ >^{(N  2) 2} / 4, then U_{0} (x) =^{(λ λ⁎) 1 / (q  1)}  x^{Θ} is the only solution of (⋆) and (b) if λ^{⁎} < λ ≤^{(N  2) 2} / 4, then all solutions of (⋆) are radially symmetric and their total set is U_{0} ∪ {_{U γ , q , λ} : γ ∈ (0 , ∞) }. We give the precise behavior of _{U γ , q , λ} near zero and at infinity, distinguishing between 1 < q <q_{N,θ} and q > max {q_{N,θ} , 1 }, where q_{N,θ} = (N + 2 θ + 2) / (N  2).
In addition, we answer open questions arising from the works of Cîrstea (2014) and WeiDu (2017) for θ ≤  2 by settling the existence and sharp profiles near zero of all positive solutions of (⋆) in Ω ∖ { 0 }, subject to u_{∂Ω} = 0, where Ω is a smooth bounded domain containing zero.
 Publication:

Journal of Differential Equations
 Pub Date:
 August 2021
 DOI:
 10.1016/j.jde.2021.05.005
 arXiv:
 arXiv:2009.00157
 Bibcode:
 2021JDE...292..461C
 Keywords:

 35J60;
 35B40;
 35J25;
 35B33;
 Mathematics  Analysis of PDEs;
 35J60 (Primary) 35B40;
 35J25 (Secondary)
 EPrint:
 32 pages