Fractional Hardy equations with critical and supercritical exponents

Abstract

We study the existence, nonexistence and qualitative properties of the solutions to the problem $$\begin{aligned} ({\mathcal {P}}) \quad \quad \left\{ \begin{aligned} (-\Delta )^s u -\theta \frac{u}{|x|^{2s}}&=u^p - u^q \quad \text {in }\,\, {\mathbb {R}}^N\\ u&> 0 \quad \text {in }\,\, {\mathbb {R}}^N\\ u&\in {\dot{H}}^s({\mathbb {R}}^N)\cap L^{q+1}({\mathbb {R}}^N), \end{aligned} \right. \end{aligned}$$ ( P ) ( - Δ ) s u - θ u | x | 2 s = u p - u q in R N u > 0 in R N u ∈ H ˙ s ( R N ) ∩ L q + 1 ( R N ) , where $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , $$N>2s$$ N > 2 s , $$q>p\ge {(N+2s)}/{(N-2s)}$$ q > p ≥ ( N + 2 s ) / ( N - 2 s ) , $$\theta \in (0, \Lambda _{N,s})$$ θ ∈ ( 0 , Λ N , s ) and $$\Lambda _{N,s}$$ 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 $$\mathbb {R}^N$$ 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.