Existence of bounded variation solutions for a 1-Laplacian problem with vanishing potentials

In this work it is studied a quasilinear elliptic problem in the whole space RN 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(RN). It is proved the existence of a nonnegative nontrivial bounded variation solution and the proof relies on a version of the Mountain Pass Theorem without the Palais-Smale condition to Lipschitz continuous functionals.