Sign changing solutions of Poisson equation

Let $\Omega$ be an open, possibly unbounded, set in Euclidean space $R^m$, let $A$ be a measurable subset of $\Omega$ with measure $|A|$, and let $\gamma \in (0,1)$.
We investigate whether the solution $v_{\Omega,A,\gamma}$ of $-\Delta v=\gamma{\bf 1}_{\Omega-A}-(1-\gamma){\bf 1}_{A},\, v\in H_0^1(\Omega)$ changes sign.
Bounds are obtained for $|A|$ in terms of geometric characteristics of $\Omega$ (bottom of the spectrum of the Dirichlet Laplacian, torsion, measure, or $R$-smoothness  of the boundary) such that $v_{\Omega,A,\gamma}$ is either non-negative or is sign changing. Joint work with Dorin Bucur.