Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms

In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -Δ_{p}u + g(u) | \nabla u|^{p} = h(u)f & $\text{in }\mathrm{\Omega }\text{,}$ \newline u\geq 0 & $\text{in }\mathrm{\Omega }\text{,}$ \newline u=0 & $\text{on }\mathrm{\partial }\mathrm{\Omega }\text{,}$ \ \end{cases} \end{equation*} in a domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ $(N\ge 2)$, where $1\le p<N$, $g$ is a positive and continuous function on $[0,\mathrm{\infty })$, and $h$ is a continuous function on $[0,\mathrm{\infty })$ (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 $L^1(\mathrm{\Omega })$.