Some Results on the $1$-Laplacian Elliptic Problems with Singularities and Robin Boundary Conditions

Abstract

In this paper, we investigate the existence and uniqueness of solutions for the following model problem, involving singularities and inhomogeneous Robin boundary conditions \begin{equation*} \left\{ \begin{array}{ll} -\Delta_{p}u_{p}=\frac{f}{u_{p}^{\gamma}}& \hbox{in $\Omega,$} \frac{\partial u_{p}}{\partial \sigma}+\lambda\vert u_{p}\vert^{p-2} u_{p}+\vert u_{p}\vert^{s-1}u_{p}=\frac{g}{u_{p}^{\eta}} & \hbox{on $\partial\Omega,$} \end{array} \right. \end{equation*} where $\Omega \subset \mathbb{R}^{m}$ represents an open bounded domain, with smooth boundary, $m \geq 2$, the symbol $\sigma $ stands for the unit outward normal vector, $ \Delta_{p}u:=\mbox{div}(\vert\nabla u\vert^{p-2}\nabla u) $ is the $p-$Laplacian operator $(1\leq p<m),$ consider $0<\gamma\leq 1,$ $ \eta>0$ and $s\geq 1.$ The function $ f\in L^{\frac{m}{p}}(\Omega)$ is a nonnegative additionally $ \lambda$ and $ g$ are nonnegative functions in $L^{\infty}(\partial \Omega).$