Existence and regularity results for a class of non-uniformly elliptic Robin problems

In this paper, we study the existence and the summability of solutions to a Robin boundary value problem whose prototype is the following: $$\{\begin{array}{ll}-\text{div}(b(\mathrm{\mid }u\mathrm{\mid })\mathrm{\nabla }u)=f& \text{in }\mathrm{\Omega },\\ {\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }+\beta u=0}& \text{on }\mathrm{\partial }\mathrm{\Omega }\end{array}$$ where $\mathrm{\Omega }$ 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.