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

Abstract

In this paper we extend the classical sub-supersolution Sattinger iteration method to $1$-Laplace type boundary value problems of the form \begin{equation*} \begin{cases} \displaystyle -\Delta_1 u = F(x,u) & \text{in}\;\Omega,\\ \newline u=0 & \text{on}\;\partial\Omega, \end{cases} \end{equation*} where $\Omega$ is an open bounded domain of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary and $F(x,s)$ is a Carathe\'{o}dory 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.