On existence and concentration of solutions to a class of quasilinear problems involving the $1-$Laplace operator

Abstract

In this work we use variational methods to prove results on existence and concentration of solutions to a problem in ${\mathbb{R}}^N$ involving the $1-$Laplacian operator. A thorough analysis on the energy functional defined in the space of functions of bounded variation $BV({\mathbb{R}}^N)$ is necessary, where the lack of compactness is overcome by using the Concentration of Compactness Principle of Lions.