  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.
