The Sattinger iteration method for 1-Laplace type problems and its application to concave-convex nonlinearities

In this paper we extend the classical sub-supersolution Sattinger iteration method to 1-Laplace type boundary value problems of the form (Formula presented.) where Ω is an open bounded domain of RN (N≥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.