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 $L^{\infty}(\partial \Omega).$