Some remarks on a shape optimization problem

Given $\Omega$ bounded open set of $\mathbb R^{n}$ and $\alpha\in \mathbb R$, let us consider $$\mu(\Omega,\alpha)=\min_{\substack{v\in W_{0}^{1,2}(\Omega)\\v\not\equiv 0}} \frac{\displaystyle\int_{\Omega} |\nabla v|^{2}dx+\alpha \left|\displaystyle\int_{\Omega}|v|v\,dx \right|}{\displaystyle\int_{\Omega} |v|^{2}dx}.$$ We study some properties of $\mu(\Omega,\alpha)$ and of its minimizers, and, depending on $\alpha$, we determine the set $\Omega_{\alpha}$ among those of fixed measure such that $\mu(\Omega_{\alpha},\alpha)$ is the smallest possible.