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

Abstract

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.

26 pages. arXiv admin note: substantial text overlap with arXiv:1610.07369