Multiple Ordered Solutions for a Class of Problems Involving the 1-Laplacian Operator

Abstract

In this paper, we use minimax methods, comparison arguments, and an approximation result to show the existence and multiplicity of solutions for the following class of problems:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _1v =\lambda f(v)\quad \text {in}\quad \Omega \text {,}\\ v\ge 0\quad \text {in}\quad \Omega \text {,}\\ v=0\quad \text {on}\quad \partial \Omega \text {,} \end{array}\right. } \end{aligned}$$

where $\Omega$ is a bounded smooth domain of $\mathbb {R}^N,$ $N\ge 1,$ $\lambda >0$ is a parameter and the non-linearity $f:\mathbb {R}\rightarrow \mathbb {R}$ is a continuous function that can change sign and satisfies an area condition.