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 $L^{ \infty}(\partial \Omega).$
Categories: math.AP