Anisotropic elliptic equations involving unbounded coefficients and singular nonlinearities

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^γ}$\text{ in }\mathcal{D},$\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 $N>2$, $\gamma \ge 0$, $q>0$, $p_j>2$ for all $j=1,\mathrm{.}\mathrm{.}\mathrm{.},N$ and the source term $f$ belongs to $L^1(\mathcal{D})$, with $f\ge 0$ and $f\equiv ̸0$.