Existence result of bounded variation solution for a perturbed \(1-\)Laplacian and \(1-\)biharmonic problem with vanishing potentials

This work focuses on the investigation of a quasilinear elliptic problem in the entire space \(\mathbb {R}^N\) ${\mathbb{R}}^N$ , which involves the 1-Laplacian and 1-biharmonic operators, as well as potentials that can vanish at infinity. This research is conducted within the space of functions with bounded variation. The main result is proven using a version of the mountain pass theorem that does not require the Palais-Smale condition.