Existence and uniqueness results for an elliptic equation with blowing-up coefficient and singular lower order term

In this paper, we study a class of nonlinear elliptic problems whose model is the following \(\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&-\textrm{div}\Big (b(u) |\nabla u|^{p-2}\nabla u\Big )=f\Big (1+\frac{1}{|u|^\gamma }\Big )\ \ \textrm{in}\ \Omega , \\ {}&u=0\ \ \textrm{on}\ {\partial \Omega },\\ \end{aligned} \right. \end{aligned} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& -{\text{div}(b(u)|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)=f(1+\frac{1}{{|u|}^{\gamma }})\text{in}\mathrm{\Omega },\hfill \\ \hfill & u=0\text{on}\partial \mathrm{\Omega },\hfill \\ \hfill \end{array}\right)\end{array}\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded open subset of \({\mathbb {R}}^N (N\ge 2)\) ${\mathbb{R}}^N(N\ge 2)$ , \(\gamma > 0\) $\gamma >0$ , b is a positive continuous function which blows up for a finite value of the unknown u. We will prove existence and uniqueness of a renormalized nonnegative solution in the case where the nonnegative source f belongs to \(L^1(\Omega )\) $L^1(\mathrm{\Omega })$ .