A local symmetry result for linear elliptic problems with solutions changing sign

We prove that the only domain $\Omega$ such that there exists a solution to the following problem $\mathrm{\Delta }u+{\omega }^{2}u=-1$ in $\Omega$, $u=0$ on $\partial \Omega$, and $\frac{1}{|\partial \Omega |}{\int }_{\partial \Omega }{\partial }_{\mathbf{n}}u=c$, for a given constant $c$, is the unit ball $B_1$, if we assume that $\Omega$ lies in an appropriate class of Lipschitz domains.