# On existence and concentration of solutions to a class of quasilinear problems involving the $1-$Laplace operator

## TL;DR

This paper investigates the existence and concentration phenomena of solutions to a class of quasilinear problems involving the 1-Laplacian operator in b(R^N), employing variational methods and the Concentration of Compactness Principle.

## Contribution

It provides new existence and concentration results for solutions involving the 1-Laplacian, addressing challenges due to lack of compactness in BV spaces.

## Key findings

- Existence of solutions established using variational methods.
- Concentration phenomena analyzed via Lions' principle.
- Overcoming lack of compactness in BV space.

## Abstract

In this work we use variational methods to prove results on existence and concentration of solutions to a problem in $\mathbb{R}^N$ involving the $1-$Laplacian operator. A thorough analysis on the energy functional defined in the space of functions of bounded variation $BV(\mathbb{R}^N)$ is necessary, where the lack of compactness is overcome by using the Concentration of Compactness Principle of Lions.