Fractional Hardy equations with critical and supercritical exponents

TL;DR

This paper investigates the existence, nonexistence, and qualitative properties of positive solutions to fractional Hardy equations with critical and supercritical exponents, using advanced nonlocal analysis techniques.

Contribution

It introduces new methods to analyze fractional Hardy equations with critical and supercritical exponents, including a nonlocal moving plane method and a transformation into weighted fractional spaces.

Findings

Established conditions for existence and nonexistence of solutions.

Proved radial symmetry of solutions using nonlocal moving plane method.

Derived upper bounds for solutions through a novel representation approach.

Abstract

We study the existence/nonexistence and qualitative properties of the positive solutions to the problem \begin{align*} (-\Delta)^s u -\theta\frac{u}{|x|^{2s}}&=u^p - u^q \quad\text{in }\,\, \mathbb{R}^N,\quad u > 0 \quad\text{in }\,\, \mathbb{R}^N, \quad u \in \dot{H}^s(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N), \end{align*} where $s\in (0,1)$, $N>2s$, $q>p\geq{(N+2s)}/{(N-2s)}$, $\theta\in(0, \Lambda_{N,s})$ 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 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.