Existence and Profile of Ground-State Solutions to a 1-Laplacian Problem in RN

Abstract

In this work we prove the existence of ground state solutions for the following class of problems {-Δ1u+(1+λV(x))u|u|=f(u),x∈RN,u∈BV(RN),where λ> 0 , Δ 1 denotes the 1-Laplacian operator which is formally defined by Δ1u=div(∇u/|∇u|), V: RN→ R is a potential satisfying some conditions and f: R→ R is a subcritical nonlinearity. We prove that for λ> 0 large enough there exist ground-state solutions and, as λ→ + ∞, such solutions converges to a ground-state solution of the limit problem in Ω=int(V-1({0})).