On Some Weighted 1-Laplacian Problem on RN with Singular Behavior at the Origin

In this work, we prove the existence of a nontrivial solution to a quasilinear elliptic problem defined on the whole Euclidean space R-N, N >= 2, and involving a weighted 1-Laplacian operator. The nonlinear term has a singular behavior at the origin. This solution is obtained through an approximation technique, which consists in considering the problem with the 1-Laplacian operator as a limit of a family of problems with the p-Laplacian operators when p -> 1(+). For that aim, a new version of Anzellotti's L-infinity-divergence-measure pairing theory is established and new arguments are used.