In this paper, we study the existence of solutions for the equation \((-\Delta )_1^s u=f\) in a bounded open set with Lipschitz boundary \(\Omega \subset \mathbb {R}^n\) , vanishing on \(\mathbb {R}^n\setminus \Omega \) , given some \(s\in (0,1)\) . Contextually, we obtain that the sequence of solutions for \((-\Delta )_p^s u=f\) convergences to a solution of \((-\Delta )_1^s u=f\) when \(p\rightarrow 1\) . We obtain our existence and convergence results by comparing the \(L^{\frac{n}{s}}\) norm of f to \((2S_{n,s})^{-1}\) , where \(S_{n,s}\) is the sharp fractional Sobolev constant, or, when f is non-negative, a weighted version of the fractional Cheegar constant to 1, and in this case, the results are sharp. We further prove that solutions are “flat” on sets of positive Lebesgue measure.
