  In this paper we deal with the following boundary value problem
  \begin{equation*}
  \begin{cases}
  -\Delta_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $\Omega$,} \newline
  u\geq 0 & \text{in $\Omega$,} \newline
  u=0 & \text{on $\partial \Omega$,} \
  \end{cases}
  \end{equation*}
  in a domain $\Omega \subset \mathbb{R}^{N}$ $(N \geq 2)$, where $1\leq p<N $,
$g$ is a positive and continuous function on $[0,\infty)$, and $h$ is a
continuous function on $[0,\infty)$ (possibly blowing up at the origin). We
show how the presence of regularizing terms $h$ and $g$ allows to prove
existence of finite energy solutions for nonnegative data $f$ only belonging to
$L^1(\Omega)$.
