In this paper we extend the classical sub-supersolution Sattinger iteration method to 1-Laplace type boundary value problems of the form \(\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta _1 u = F(x,u) & \text {in}\;\Omega ,\\ u=0 & \text {on}\;\partial \Omega , \end{array}\right. } \end{aligned}\) where \(\Omega \) is an open bounded domain of \({\mathbb {R}}^N\) ( \(N\ge 2\) ) with Lipschitz boundary and F(x, s) is a Caratheódory function. This goal is achieved through a perturbation method that overcomes structural obstructions arising from the presence of the 1-Laplacian and by proving a weak comparison principle for these problems. As a significant application of our main result we establish existence and non-existence theorems for the so-called “concave-convex” problem involving the 1-Laplacian as leading term.
