In this work we prove the existence of ground state solutions for the following class of problems \begin{equation*} \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. \label{Pintro} \end{equation*} \end{abstract} where $\lambda > 0$, $\Delta_1$ denotes the $1-$Laplacian operator which is formally defined by $\Delta_1 u = \mbox{div}(\nabla u/|\nabla u|)$, $V:\mathbb{R}^N \to \mathbb{R}$ is a potential satisfying some conditions and $f:\mathbb{R} \to \mathbb{R}$ is a subcritical and superlinear nonlinearity. We prove that for $\lambda > 0$ large enough there exists ground-state solutions and, as $\lambda \to +\infty$, such solutions converges to a ground-state solution of the limit problem in $\Omega = \mbox{int}( V^{-1}(\{0\}))$.

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$$\mathbb {R}^N$$RN 中 1-Laplacian 问题的基态解的存在性和轮廓

在这项工作中，我们证明了以下类问题的基态解的存在性 \begin{equation*} \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), & \结束{数组} \right。\label{Pintro} \end{equation*} \end{abstract} 其中 $\lambda > 0$，$\Delta_1$ 表示 $1-$Laplacian 算子，其形式为 $\Delta_1 u = \mbox{div} (\nabla u/|\nabla u|)$, $V:\mathbb{R}^N \to \mathbb{R}$ 是满足某些条件的势能，$f:\mathbb{R} \to \mathbb {R}$ 是次临界和超线性非线性。我们证明，对于足够大的 $\lambda > 0$ 存在基态解，并且作为 $\lambda \to +\infty$，