  In this work it is studied a quasilinear elliptic problem in the whole space
$\mathbb{R}^N$ involving the $1-$Laplacian operator, with potentials which can
vanish at infinity. The Euler-Lagrange functional is defined in a space whose
definition resembles $BV(\mathbb{R}^N)$ and, in order to avoid working with
extensions of it to some Lebesgue space, we state and prove a version of the
Mountain Pass Theorem without the Palais-Smale condition to Lipschitz
continuous functionals.
