A Class of Non-linear Anisotropic Elliptic problems with Unbounded Coefficients and Singular Quadratic Lower Order Terms

In this work, we study the existence and regularity results of anisotropic elliptic equations with a singular lower order term that grows naturally with respect to the gradient and unbounded coefficients. We take up the following model problem \begin{equation*} \left{\begin{array}{ll}-\displaystyle\sum\limits_{j\in J} D_{j}\left(\left[ 1+ u^{q}\right]\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+\sum\limits_{j\in J}\frac{\vert D_{j}u\vert^{p_{j}}}{ u^θ}=f& \hbox{in};Ω, \ u>0& \hbox{in};Ω, u =0 & \hbox{on}; \partialΩ, \end{array} \right. \end{equation*} $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$, $j\in J=\{1,2,\dots ,N\},$ $q>0$, $0<\theta <1$, $2\le p_1\le p_2\le \mathrm{.}\mathrm{.}\mathrm{.}\le p_N$ and $f\in L^1(\mathrm{\Omega })$. Our study’s conclusions will depend on the values of $q$ and $\theta$.