  In this paper, we study the existence of solutions of the equation
$(-\Delta)_1^s u=f$ in a bounded open set with Lipschitz boundary
$\Omega\subset \Rn$, vanishing on $\Co \Omega$, for some given $s\in (0,1)$,
and asymptotics as $p\to 1$ of solutions of $(-\Delta)_p^s u=f$. We obtain
existence and convergence by comparing the $L^{\frac{n}{s}}$ norm of $f$ to the
sharp fractional Sobolev constant, or, when $f$ is non-negative, the weighted
fractional Cheegar constant to $1$ -- in this case, the results are sharp. We
further prove that solutions are "flat" on sets of positive Lebesgue measure.
