Some elliptic equations with unbounded coefficients and singular gradient term

Abstract We are interseted in a class of nonlinear elliptic equations involving a blowing-up coefficient and a singular term whose prototype is \[\begin{aligned} \left\{ \begin{aligned}&-\mathrm{div}\Big (b(u) (1+|u|)^q \nabla u\Big )= b_1\frac{|\nabla u|^2}{|u|^\theta }+f\ \ \mathrm{in}\ \Omega, \\&u=0\ \ \mathrm{on}\ {\partial \Omega },\\ \end{aligned} \right. \end{aligned}\] where $\Omega$ is a bounded open subset of $\Real^N (N\geq 2)$, $b(s)$ is a positive continuous function which blows up for a finite value of the unknown, $b_1>0$, $q>0$, $\theta>0 $ and the nonnegative source $f$ belongs to $L^{t}(\Omega)$, $t\geq 1$.