Sharp existence and classification results for nonlinear elliptic equations in RN ∖ { 0 } with Hardy potential

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 C¹ (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|^θu^q = 0 in R^N ∖ { 0 } with u > 0 has a C¹ (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 U₀ (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 U₀ ∪ {_{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 Wei-Du (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.