On existence of multiple solutions to a class of problems involving the \(1\)-Laplace operator in whole \(\mathbb{R}^N\) (Q6628893)

Impose the following conditions on a function \(f\):\N\begin{itemize}\N\item[\((f_1)\)] \(f\in C^1(\mathbb{R})\) and increasing;\N\item[\((f_2)\)] \(f(s)=o(1)\) as \(s\rightarrow 0\);\N\item[\((f_3)\)] There exist constants \(c_1,c_2>0\) and \(1<q<1^*:=\frac{N}{N-1}\) such that \(|f(s)|\leq c_1+c_2 |s|^{q-1}, \forall s\in \mathbb{R}\);\N\item[\((f_4)\)] There exists \(\theta>1\) such that \(0<\theta F(s)\leq f(s)s, \forall s\neq 0\), where \(F(s)=\int_0^s f(t)\mathrm{d}t\);\N\item[\((f_5)\)] There is \(1<q_1<1^*\) such that, for all \(p\in (1,q_1)\), \(f(s)/ |s|^{p-2}s\) is increasing on \((0,+\infty)\) and decreasing on \((-\infty,0)\),\N\end{itemize}\Nand suppose the following for \(V(x)\):\N\begin{itemize}\N\item[\((V_1)\)] \(V\in C(\mathbb{R}^N)\) and \(0<V_0:=\min_{x\in \mathbb{R}^N}V(x)\);\N\item[\((V_2)\)] \(V_\infty=\lim_{|x|\rightarrow \infty}V(x)>V_0\);\N\item[\((V_3)\)] \(V^{-1}(\{V_0\})=\{a_1,a_2,\cdots,a_\ell\}\) with \(a_1=0\) and \(a_j\neq a_s\) if \(j\neq s\).\N\end{itemize}\NThen the author uses variational methods to prove that, for \(\varepsilon>0\) small enough, the equation \[-\varepsilon \Delta_1 u+V(x)\frac{u}{|u|}=f(u)~\hbox{in}~\mathbb{R}^N, u\in BV(\mathbb{R}^N),\] possesses at least \(\ell\) different solutions. The main idea is to approximate by the \(p\)-Laplacian problem as \(p\downarrow 1\).