In this paper, we investigate a class of problems involving the fractional \(1\) -Laplacian operator in general Lipschitz domains, addressing both singular and superlinear nonlinearities. We establish the existence of at least two distinct positive weak solutions. The singular term, combined with the norm in the fractional Sobolev space \(W^{s,1}(\Omega )\) , renders the associated energy functional non-differentiable. This non-differentiability precludes the direct application of variational methods to obtain solutions as critical points in the space \(W^{s,1}(\Omega )\) . To address this challenge, we employ an approximation scheme that involves an auxiliary problem with the fractional \(p\) -Laplacian operator, where \(p > 1\) . By deriving uniform estimates, we allow \(p\) to approach \(1\) , thereby constructing solutions for the original problem. It is worth emphasizing that the lack of a compact embedding between the relevant function spaces \(W_0^{s,p}(\Omega )\) and \(W_0^{s,q}(\Omega )\) with \(p \ne q\) introduces significant difficulties to our analysis. Additionally, the absence of the validity of Hölder and Young inequalities in this fractional context further complicates the problem. Finally, unlike the case of classical (non-fractional) Sobolev spaces, it is not known whether the Sobolev constant \(S_{s,p}\) converges to \(S_{s,1}\) as \(p \rightarrow 1^{+}\) , which necessitates working with bounding constants instead.
