On weak solutions to a fractional Hardy–Hénon equation, Part II: Existence. (February 2023) - Hasegawa, Shoichi; Ikoma, Norihisa; Kawakami, Tatsuki
Abstract: This paper and Hasegawa et al. (2021) treat the existence and nonexistence of stable weak solutions to a fractional Hardy–Hénon equation ( − Δ ) s u = | x | ℓ | u | p − 1 u in R N, where 0 < s < 1, ℓ > − 2 s, p > 1, N ≥ 1 and N > 2 s . In this paper, when p is critical or supercritical in the sense of the Joseph–Lundgren, we prove the existence of a family of positive radial stable solutions, which satisfies the separation property. We also show the multiple existence of the Joseph–Lundgren critical exponent for some ℓ ∈ ( 0, ∞ ) and s ∈ ( 0, 1 ), and this property does not hold in the case s = 1 .