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^{ \theta}}=f& \hbox{ in}\;\Omega, \\ u>0& \hbox{ in}\;\Omega, u =0 & \hbox{ on}\; \partial\Omega, \end{ array} \right. \end{ equation*} $\Omega$ is a bounded domain in $\mathbb{ R}^{ N}$, $j\in J=\{ 1,2,\ldots,N\},$ $q>0$, $0< \theta<1$, $2\leq p_{ 1}\leq p_{ 2}\leq... \leq p_{ N}$ and $f\in L^{ 1}(\Omega)$. Our study's conclusions will depend on the values of $q$ and $\theta$.
Categories: math.AP