Fractional Hardy-Sobolev operator with sign-changing and singular nonlinearity. Issue 16 (9th December 2020) - Goyal, Sarika
Abstract : We study the following fractional equation with Hardy potential and singular nonlinearity ( P μ, λ ) ( − Δ ) s w − μ w | x | 2 s = a ( x ) w − q + λ b ( x ) w r in Ω w > 0 in Ω, w = 0 in R n ∖ Ω, where 0 ∈ Ω is a bounded domain in R n with smooth boundary ∂Ω, n >2 s, s ∈ ( 0, 1 ), 0 < μ < Λ n, s, the sharp constant of the fractional Hardy Sobolev inequality, λ > 0, 0 < q < 1 < r < 2 s ∗ − 1 with 2 s ∗ = 2 n / ( n − 2 s ), a : Ω ⊂ R n → R such that 0 < a ( x ) ∈ L 2 s ∗ 2 s ∗ − 1 + q ( Ω ), and b : Ω ⊂ R n → R is a sign-changing function such that b ( x ) ∈ L 2 s ∗ 2 s ∗ − 1 − r ( Ω ) . Using variational methods, we show existence, non-existence and multiplicity of positive solutions of ( P μ, λ ) with respect to the parameter λ .