  Let $\Omega\subset\mathbb{R}^N$, $N\geq 2$, be an open bounded connected set.
We consider the fractional weighted eigenvalue problem
  $(-\Delta)^s u =\lambda \rho u$ in $\Omega$ with homogeneous Dirichlet
boundary condition,
  where $(-\Delta)^s$, $s\in (0,1)$, is the fractional Laplacian operator,
$\lambda \in \mathbb{R}$ and $ \rho\in L^\infty(\Omega)$. We study weak*
continuity, convexity and G\^ateaux differentiability of the map
$\rho\mapsto1/\lambda_1(\rho)$, where $\lambda_1(\rho)$ is the first positive
eigenvalue. Moreover, denoting by $\mathcal{G}(\rho_0)$ the class of
rearrangements of $\rho_0$, we prove the existence of a minimizer of
$\lambda_1(\rho)$ when $\rho$ varies on $\mathcal{G}(\rho_0)$. Finally, we show
that, if $\Omega$ is Steiner symmetric, then every minimizer shares the same
symmetry.
