Sobolev gradients of viscosity supersolutions

We investigate which elliptic PDEs that have the property that every viscosity supersolution is $W_{loc}^{1,q}(\mathrm{\Omega })$, $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$. The asymptotic cone of the operator’s sublevel set seems to be essential. It turns out that much can be said if we know how this cone compares to the sublevel set of a certain minimal operator associated with the exponent $q$.