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{equation*} \left\{ \begin{array}{ll} \displaystyle - Δ_1 u + (1 + λV(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 $Δ_1 u = \mbox{div}(\nabla u/|\nabla u|)$, $V:{\mathbb{R}}^N\to \mathbb{R}$ 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 $Ω= \mbox{int}( V^{-1}(\{0\}))$.

arXiv admin note: text overlap with arXiv:1702.06718