This paper deals with the Dirichlet problem for the 1-Laplacian \(\Delta_1 u= \text{div}(\frac{Du}{|Du|})\) and having singular lower order term \(f(x)/u^\gamma\), \(f\in L^N(\Omega)\), \(f\ge 0\) a.e., \(0<\gamma\le 1\). \(\Omega\subset \mathbb{R}^N\) is a bounded open domain with Lipschitz boundary \(\partial\Omega\). The considerations are in \(BV(\Omega)\), the space of all functions \(u\in L^1(\Omega)\) whose distributional derivatives \(Du\) are Radon measures with finite total variation. The notion of weak solution \(u\ge 0\) a.e., \(u\in BV(\Omega)\cap L^\infty(\Omega)\) of the above-mentioned Dirichlet problem with \(u=0\) on \(\partial\Omega\) is given by Def. 4.1. The solution is obtained in Th. 4.5 as the limit of a singular bvp with principal part the \(p\)-Laplacian \(\Delta_p\), \(p>1\) as \(p\to 1\). The latter Dirichlet problems possess unique bounded weak solutions \(u_p\) for fixed \(p>1\) (Theorem 3.4).
In the special case \(f>0\) a.e., \(u\) has several interesting properties and among them \(u>0\) a.e., and \(u\) is unique. At the end of the paper explicit examples of 1-dimensional solutions for \(f\ge 0\), \(f\not\equiv 0\) are given. The authors show that uniqueness does not hold in general.