Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in $\protect \mathbb{R}^N$

Abstract

In this paper, we consider the following 1-Laplacian problem \[ -\Delta _1 u+V(x)\frac{u}{|u|}= f(x,u),\, x\in \mathbb{R}^N,\, u>0,\ u\in BV\left(\mathbb{R}^N\right), \] where $\Delta _1 u=\mathrm{div}(\tfrac{Du}{|Du|})$, $V$ is a periodic potential and $f$ is periodic and verifies the super-primary condition at infinity. By the Nehari type manifold method, the concentration compactness principle and some analysis techniques, we show the 1-Laplacian equation has a ground state solution.