Quasilinear Elliptic Problem with Singular Lower Order Term and $$L^1$$ Data

Abstract

In this paper, we are interested in the existence result of solutions for the nonlinear Dirichlet problem of the type: $$\begin{aligned} \left\{ \begin{aligned}&-\mathrm{div} (M(x) \nabla u )+ \gamma u^p= B \frac{|\nabla u|^q}{u^\theta }+f\ \ \mathrm{in}\ \Omega ,\\&u> 0\ \ \mathrm{in}\ \Omega ,\\&u=0\ \ \mathrm{on}\ {\partial \Omega },\\ \end{aligned} \right. \end{aligned}$$ where $$\Omega $$ is a bounded open subset of $$\mathbb {R}^N$$ , $$N>2$$ , M(x) is a uniformly elliptic and bounded matrix, $$\gamma > 0$$ , $$B> 0$$ , $$1\le q<2$$ , $$0<\theta \le 1$$ , and the source f is a nonnegative (not identically zero) function belonging to $$L^1(\Omega )$$ .