On bounded variation solutions of quasi-linear 1-Laplacian problems with periodic potential in [formula omitted

Abstract

This paper considers the following 1-Laplacian problem{−Δ1u+V(x)u|u|=f(x,u),x∈RN,u∈BV(RN), where Δ1u=div(Du|Du|), V is a periodic and bounded potential and f is periodic in x and satisfies some super-linear conditions. By applying Mountain Pass Theorem with Cerami condition to non-smooth functionals and Concentration-Compactness Lemma to the Cerami sequence obtained from Mountain Pass Theorem together with some analytic techniques, it is shown that there exist nontrivial bounded variation solutions of this problem for some general f(x,s). Moreover, under extra conditions on f(x,s), a nontrivial bounded variation ground state solution can be obtained.