Strauss' and Lions' type results in $BV(\mathbb{R}^N)$ with an application to $1-$Laplacian problem

In this work we state and prove versions of some classical results, in the framework of functionals defined in the space of functions of bounded variation in $\mathbb{R}^N$. More precisely, we present versions of the Radial Lemma of Strauss, the compactness of the embeddings of the space of radially symmetric functions of $BV(\mathbb{R}^N)$ in some Lebesgue spaces and also a version of the Lions Lemma, proved in his celebrated paper of 1984. As an application, we state and prove a version of the Mountain Pass Theorem without the Palais-Smale condition in order to get existence of a ground-state bounded variation solution of a quasilinear elliptic problem involving the $1-$Laplacian operator in $\mathbb{R}^N$. This seems to be the very first work dealing with stationary problems involving this operator in the whole space.