  In this paper, we will consider the fractional Caffarelli-Kohn-Nirenberg
inequality \begin{equation*} {\Lambda} \left(\int_{\mathbb
R^n}\frac{|u(x)|^{p}}{|x|^{{\beta} {p}}}\,dx\right)^{\frac{2}{p}}\leq
\int_{\mathbb R^n}\int_{\mathbb
R^n}\frac{(u(x)-u(y))^2}{|x-y|^{n+2\gamma}|x|^{{\alpha}}|y|^{{\alpha}}}\,dy\,dx
\end{equation*} where $\gamma\in(0,1)$, $n\geq 2$, and $\alpha,\beta\in\mathbb
R$ satisfy \begin{equation*} \alpha\leq \beta\leq \alpha+\gamma, \
-2\gamma<\alpha<\frac{n-2\gamma}{2}, \end{equation*} and the exponent $p$ is
chosen to be \begin{equation*} p=\frac{2n}{n-2\gamma+2(\beta-\alpha)},
\end{equation*} such that the inequality is invariant under scaling. We first
study the existence and nonexistence of extremal solutions. Our next goal is to
show some results on the symmetry and symmetry breaking region for the
minimizers; these suggest the existence of a Felli-Schneider type curve
separating both regions but, surprisingly, we find a novel behavior as
$\alpha\to -2\gamma$. The main idea in the proofs, as in the classical case, is
to reformulate the fractional Caffarelli-Kohn-Nirenberg inequality in
cylindrical variables. Then, in order to find the radially symmetric solutions
we need to solve a non-local ODE.
  For this equation we also get uniqueness of minimizers in the radial symmetry
class; indeed, we show that the unique continuation argument of Frank-Lenzmann
(Acta'13) can be applied to more general operators with good spectral
properties. We provide, in addition, a completely new proof of non-degeneracy
which works for all critical points. It is based on the variation of constants
approach and the non-local Wronskian of
Ao-Chan-DelaTorre-Fontelos-Gonz\'alez-Wei (Duke'19).
