  In the present paper we prove existence results for solutions to nonlinear
elliptic Neumann problems whose prototype is \begin{equation*}
  \begin{cases}
  -\Delta_{p} u -\text{div} (c(x)|u|^{p-2}u)) =f & \text{in}\ \Omega, \\ \left(
|\nabla u|^{p-2}\nabla u+ c(x)|u|^{p-2}u \right)\cdot\underline n=0 &
\text{on}\ \partial \Omega \,, \end{cases} \end{equation*} when $f$ is just a
summable function. Our approach allows also to deduce a stability result for
renormalized solutions and an existence result for operator with a zero order
term.
