Anisotropic elliptic problem involving a singularity and a Radon measure

Abstract

In this study, we demonstrate the existence of solutions to an anisotropic elliptic problem featuring a singularity, where the non-homogeneous term is characterized by a non-negative Radon measure $\mu$. The model problem is\begin{equation*}\left\{\begin{array}{ll}-\sum\limits_{i=1}^{N} \partial_{i}\left(\vert \partial_{i}u\vert^{p_{i}-2}\partial_{i}u\right)=\frac{f}{\left(e^u-1\right)^{\gamma}}+\mu & \hbox{in}\;\;\Omega, \\ u =0 & \hbox{on}\;\; \partial\Omega, \\ u >0 & \hbox{in}\;\; \Omega, \end{array} \right.\end{equation*}where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $\gamma>0$, $f\in L^{1}(\Omega)$ and $2< p_{1}\leq p_{2}\leq \ldots \leq p_{N}$. The primary goal of this work is to establish the existence of solutions based on the values of $\gamma$.