Fractional Hardy equations with critical and supercritical exponents

2021/10/27 by Mousomi Bhakta, Bhakta, Mousomi, Debdip Ganguly +3
Mathematics · #35B07 #35B65 #35R09 #35S05 #47G30 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2110.14414

openalex publication_date 2021/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the existence/nonexistence and qualitative properties of the positive solutions to the problem (-Δ)^s u -θ\fracu|x|^2samp;=u^p - u^q \quadin ℝ^N, u gt; 0 \quadin ℝ^N, u ∈ H^s(ℝ^N)∩ L^q+1(ℝ^N), where s∈ (0,1), N>2s, q>p≥(N+2s)/(N-2s), θ∈(0, Λ_N,s) and Λ_N,s is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions we mean, both the radial symmetry that is obtained by using the moving plane method in a nonlocal setting on the whole ℝ^N, and upper bound behavior of the solutions. To this last end we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space.

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