  In this paper, we study the existence and the summability of solutions to a
Robin boundary value problem whose prototype is the following: $$ \begin{cases}
-\text{div}(b(|u|)\nabla u)=f &\text{in }\Omega,\\[.2cm]
\displaystyle\frac{\partial u}{\partial \nu}+\beta u=0 &\text{on
}\partial\Omega \end{cases} $$ where $\Omega$ is a bounded Lipschitz domain in
$\mathbb R^N$, $N>2$, $\beta>0$, $b(s)$ is a positive function which may vanish
at infinity and $f$ belongs to a suitable Lebesgue space. The presence of such
a function $b$ in the principal part of the operator prevents it from being
uniformly elliptic when $u$ is large.
