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 $L^1(\Omega )$.