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

TL;DR

This paper studies the existence and uniqueness of solutions for a class of nonlinear elliptic problems involving the p-Laplacian with singularities and Robin boundary conditions, extending understanding of such complex boundary value problems.

Contribution

It provides new results on existence and uniqueness for p-Laplacian problems with singularities and Robin boundary conditions, addressing cases with inhomogeneous boundary data.

Findings

Established conditions for existence of solutions.

Proved uniqueness under certain parameter constraints.

Extended previous results to more general 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 $L^{\infty}(\partial \Omega).$