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: cases -div(b(|u|)∇ u)=f &in Ω,\\[.2cm] ∂ u/∂ ν+β u=0 &on ∂Ω cases where Ω is a bounded Lipschitz domain in RN, $N>2$, β>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.