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

TL;DR

This paper proves the existence of bounded variation solutions for a $1$-Laplacian elliptic problem in the entire space with potentials that may vanish at infinity, using a novel variational approach.

Contribution

It introduces a new variational method for $1$-Laplacian problems with vanishing potentials, avoiding the need for extensions to Lebesgue spaces.

Findings

Existence of solutions established under broad conditions.

Development of a Mountain Pass Theorem variant without Palais-Smale condition.

Application to problems with potentials vanishing at infinity.

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.