Anisotropic elliptic equations involving unbounded coefficients and singular nonlinearities

Abstract

In this paper, we study the existence and regularity of solutions for a class of nonlinear singular elliptic equations involving unbounded coefficients and a singular right-hand side. Specifically, we are interested to problem whose simplest model is \begin{equation*} -\sum_{j=1}^N\partial_{j}\left([1+u^{q}]\vert \partial_{j} u \vert^{p_{j}-2} \partial_{j} u\right)= \frac{f}{u^{\gamma}}\text{ in $\mathcal{D},$}\quad u>0 \text{ in $\mathcal{D},$} \quad u=0 \hbox{ on}\;\; \partial\mathcal{D}, \end{equation*} where $\mathcal{D}$ is a bounded open subset of $\mathbb{R}^{N}$ with $N>2$, $ \gamma\geq0$, $q >0 $, $p_{j}>2$ for all $j=1,...,N$ and the source term $f$ belongs to $L^1(\mathcal{D})$, with $f \geq 0$ and $f \not\equiv 0$.