A generalized porous medium equation related to some singular quasilinear problems

In this paper we study existence and nonexistence of solutions for a Dirichlet boundary value problem whose model is $$\begin{cases} -\sum_{m=1}^{\infty} a_m \Delta u^m= f&\text{in}\ \Omega \newline u=0 & \text{on}\ \partial\Omega\,, \end{cases}$$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $a_m$ is a sequence of nonnegative real numbers, and $f$ is in $L^q(\Omega)$, $q>\frac{N}{2}$.