  We deal with existence and uniqueness of nonnegative solutions to
  \begin{equation*} \left\{
  \begin{array}{l}
  -\Delta u = f(x) \text{ in }\Omega,
  \frac{\partial u}{\partial \nu} + \lambda(x) u = \frac{g(x)}{u^\eta} \text{
on } \partial\Omega,
  \end{array} \right.
  \end{equation*}
  where $\eta\ge 0$ and $f,\lambda$ and $g$ are nonnegative integrable
functions. The set $\Omega\subset\mathbb{R}^N (N> 2)$ is open and bounded with
smooth boundary and $\nu$ denotes its unit outward normal vector.
  More generally, we handle equations driven by monotone operators of
$p$-Laplacian type jointly with nonlinear boundary conditions. We prove
existence of an entropy solution and check that this solution is unique under
natural assumptions. Among other features, we study the regularizing effect
given to the solution by both the absorption and the nonlinear boundary term.
