We prove existence of solutions to the following problem \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta _1 u +g(u)|Du|=h(u)f & \text {in } \Omega ,\\ u=0 & \text {on } \partial \Omega , \end{array}\right. \end{aligned}\) where \(\Omega \subset \mathbb {R}^N\) , 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 \(0 \le f \in L^N(\Omega )\) . 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.
