In this work we study a degenerated quasilinear elliptic problem involving the \(1-\) Laplace operator and a gradient term in the whole euclidean space \( \mathbb {R}^N,\ N \ge 2. \) Our problem can be seen as a generalization to the case of \( 1-\) Laplacian of some previous results which considered a \( p-\) Laplace operator defined in the Sobolev space \( W^{1,p}( \mathbb {R}^N),\ p > 1. \) The case \( p = 1 \) is different and presents many specific features. Our problem despite having an “apparent” variational structure cannot be treated using direct tools from smooth or nonsmooth critical point theory. Actually, our existence result is established through an approximation technique, which consists in considering the problem with the \(1-\) Laplace operator as a limit of a family of problems with the \(p-\) Laplace operators when \(p \rightarrow 1^+.\) The existence of solution to the approximated problem is obtained using a perturbation argument.
