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 $\Omega = \text{ int }( V^{-1}(\{0\}))$.