On existence of multiple solutions to a class of problems involving the 1-Laplace operator in whole $\mathbb{R}^N$

Abstract

In this work we use variational methods to prove the existence of multiple solutions for the following class of problem $$- \epsilon \Delta_1 u + V(x)\frac{u}{|u|} = f(u) \quad \mbox{in} \quad \mathbb{R}^N, \quad u \in BV(\mathbb{R}^N), $$ where $\Delta_1$ is the $1-$Laplacian operator and $\epsilon$ is a positive parameter. It is proved that the numbers of solutions is at least the numbers of global minimum points of $V$ when $\epsilon$ is small enough.