  In the present paper, we study properties of the second Dirichlet eigenvalue
of the fractional Laplacian of annuli-like domains and the corresponding
eigenfunctions. In the first part, we consider an annulus with inner radius $R$
and outer radius $R+1$. We show that for $R$ sufficiently large any
corresponding second eigenfunction of this annulus is nonradial. In the second
part, we investigate the second eigenvalue in domains of the form
$B_1(0)\setminus \overline{B_{\tau}(a)}$, where $a$ is in the unitary ball and
$0<\tau<1-|a|$. We show that this value is maximized for $a=0$, if the set
$B_1(0)\setminus \overline{B_{\tau}(0)}$ has no radial second eigenfunction. We
emphasize that the first part of our paper implies that this assumption is
indeed nonempty.
