Low regularity results for degenerate Poisson problems

In this paper we study the Poisson problem, $$\{\begin{array}{ll}-\mathrm{d}\mathrm{i}\mathrm{v}(d^{\beta }\mathrm{\nabla }u)=f& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ u=0& \mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 2$ is a smooth bounded domain, $f$ is a continuous function, $\beta <1$, and $d(x)=dist(x,\mathrm{\partial }\mathrm{\Omega })$. We describe the behaviour of $u$ near $\mathrm{\partial }\mathrm{\Omega }$ and discuss some of its regularity properties.