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

In this work we use variational methods to prove the existence of multiple solutions for the following class of problem\[-\epsilon \Delta_1 u + V(x) \frac{u}{\lvert u \rvert} = f(u) \quad \textrm{in} \quad \mathbb{R}^n \,\textrm{,} \quad u \in BV (\mathbb{R}^N)\, \textrm{,}\]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.

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C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7.

Received 9 September 2020

Accepted 25 May 2021

Published 9 August 2024