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

摘要

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 canvanish at infinity. The Euler-Lagrange functional is defined in a space whosedefinition resembles $BV(\mathbb{R}^N)$ and, in order to avoid working withextensions of it to some Lebesgue space, we state and prove a version of theMountain Pass Theorem without the Palais-Smale condition to Lipschitzcontinuous functionals.