Existence and Profile of Ground-State Solutions to a 1-Laplacian Problem in $$\mathbb {R}^N$$

Abstract

In this work we prove the existence of ground state solutions for the following class of problems $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle - \Delta _1 u + (1 + \lambda V(x))\frac{u}{|u|} &{} = f(u), \quad x \in \mathbb {R}^N, u \in BV(\mathbb {R}^N), &{} \end{array} \right. \end{aligned}$$ where $$\lambda > 0$$ , $$\Delta _1$$ denotes the 1-Laplacian operator which is formally defined by $$\Delta _1 u = \text{ div }(\nabla u/|\nabla u|)$$ , $$V:\mathbb {R}^N \rightarrow \mathbb {R}$$ is a potential satisfying some conditions and $$f:\mathbb {R} \rightarrow \mathbb {R}$$ is a subcritical nonlinearity. We prove that for $$\lambda > 0$$ large enough there exist ground-state solutions and, as $$\lambda \rightarrow +\infty $$ , such solutions converges to a ground-state solution of the limit problem in $$\Omega = \text{ int }( V^{-1}(\{0\}))$$ .