  Given $1\le q \le 2$ and $\alpha\in\mathbb R$, we study the properties of the
solutions of the minimum problem \[
\lambda(\alpha,q)=\min\left\{\dfrac{\displaystyle\int_{-1}^{1}|u'|^{2}dx+\alpha\left|\int_{-1}^{1}|u|^{q-1}u\,
dx\right|^{\frac2q}}{\displaystyle\int_{-1}^{1}|u|^{2}dx}, u\in
H_{0}^{1}(-1,1),\,u\not\equiv 0\right\}. \] In particular, depending on
$\alpha$ and $q$, we show that the minimizers have constant sign up to a
critical value of $\alpha=\alpha_{q}$, and when $\alpha>\alpha_{q}$ the
minimizers are odd.
