Existence and profile of ground-state solutions to a $1-$Laplacian problem in $\mathbb{R}^N$

TL;DR

This paper proves the existence of ground-state solutions for a class of $1$-Laplacian problems in $ ^N$, showing solutions exist for large parameters and converge to a limit problem as the parameter grows.

Contribution

It establishes the existence and convergence of ground-state solutions for a $1$-Laplacian problem with potential in unbounded domains, a novel result in this context.

Findings

Existence of ground-state solutions for large $\lambda$

Solutions converge to a limit problem as $\lambda o + abla$

Provides conditions on potential $V$ and nonlinearity $f$

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