Uniqueness for an overdetermined boundary value problem for the p-Laplacian

For $p>1$ set $\Delta _p u = {\mathrm{div}}(|\nabla u|^{p-2}\nabla u)$, and let $\mu$ be a measure with compact support. Suppose, for $j=1,2$, there are functions $u_j \in W^{1,p}$ and (bounded) domains $\Omega _j$, both containing the support of $\mu$ with the property that $\Delta _p u_j =\chi _{\Omega _j} - \mu$ in $\mathbf{R}^N$ (weakly) and $u_j=0$ in the complement of $\Omega _j$. If in addition $\Omega _1 \cap \Omega _2$ is convex, then $\Omega _1 \equiv \Omega _2$ and $u_1\equiv u_2$.