We prove a regularity result for the unstable elliptic free boundary problem \begin{equation}\Delta u = -\chi_{\{ u\gt 0\}} \end{equation} related to traveling waves in a problem arising in solid combustion. The maximal solution and every local minimizer of the energy are regular; that is, $\{u=0\}$ is locally an analytic surface, and $u|_{\overline{\{u\gt 0\}}}, u|_{\overline{\{ u \lt 0\}}}$ are locally analytic functions. Moreover, we prove a partial regularity result for solutions that are nondegenerate of second order. Here $\{u=0\}$ is analytic up to a closed set of Hausdorff dimension $n-2$ . We discuss possible singularities