  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.
