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

Abstract

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