A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

Given (1le q le 2) and (alpha in mathbb {R}), we study the properties of the solutions of the minimum problem

$$begin{aligned} lambda (alpha ,q)=min left{ dfrac{displaystyle int _{-1}^{1}|u'|^{2}dx+alpha left| int _{-1}^{1}|u|^{q-1}u, dxright| ^{frac{2}{q}}}{displaystyle int _{-1}^{1}|u|^{2}dx}, uin H_{0}^{1}(-1,1),,unot equiv 0right} . end{aligned}$$

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.