Existence of solutions for $1-$laplacian problems with singular first order terms

We prove existence of solutions to the following problem \begin{equation*} \begin{cases} -Δ_1 u +g(u)|Du|=h(u)f & $\text{in }\mathrm{\Omega }\text{,}$ \ u=0 & $\text{on }\mathrm{\partial }\mathrm{\Omega }\text{,}$ \end{cases} \end{equation*} where $\mathrm{\Omega }\subset {\mathbb{R}}^N$, with $N\ge 2$, is an open and bounded set with Lipschitz boundary, $g$ is a continuous and positive function which possibly blows up at the origin and bounded at infinity and $h$ is a continuous and nonnegative function bounded at infinity (possibly blowing up at the origin) and finally $0\le f\in L^N(\mathrm{\Omega })$. As a by-product, this paper extends the results found where $g$ is a continuous and bounded function. \We investigate the interplay between $g$ and $h$ in order to have existence of solutions.