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} -\Delta_1 u +g(u)|Du|=h(u)f & \text{in $\Omega$,} \\ u=0 & \text{on $\partial\Omega$,} \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^N$, with $N\ge2$, 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 $0 \le f \in L^N(\Omega)$. 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.