Fractional Hardy equations with critical and supercritical exponents

Abstract

We study the existence, nonexistence and qualitative properties of the solutions to the problem (P)(-Δ)su-θu|x|2s=up-uqinRNu>0inRNu∈H˙s(RN)∩Lq+1(RN),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} ({\\mathcal {P}}) \\quad \\quad \\left\\{ \\begin{aligned} (-\\Delta )^s u -\ heta \\frac{u}{|x|^{2s}}&=u^p - u^q \\quad \ ext {in }\\,\\, {\\mathbb {R}}^N\\\\ u&> 0 \\quad \ ext {in }\\,\\, {\\mathbb {R}}^N\\\\ u&\\in {\\dot{H}}^s({\\mathbb {R}}^N)\\cap L^{q+1}({\\mathbb {R}}^N), \\end{aligned} \\right. \\end{aligned}$$\\end{document}where sin (0,1), N>2s, q>pge {(N+2s)}/{(N-2s)}, theta in (0, Lambda _{N,s}) and Lambda _{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 mathbb {R}^N, and a suitable 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.