Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems

Let us consider the following minimum problem $${\lambda }_{\alpha }(p,r)=\underset{\begin{array}{c}u\in W_0^{1,p}(-1,1)\\ u\equiv ̸0\end{array}}{\min }{\displaystyle \frac{{\displaystyle {\int }_{-1}^1\mathrm{\mid }{u}^{\mathrm{\prime }}{\mathrm{\mid }}^{p}dx+\alpha {\mid {\int }_{-1}^1\mathrm{\mid }u{\mathrm{\mid }}^{r-1}udx\mid }^{\frac{p}{r}}}}{{\displaystyle {\int }_{-1}^1\mathrm{\mid }u{\mathrm{\mid }}^{p}dx}}},$$ where $\alpha \in \mathbb{R}$, $p\ge 2$ and $\frac{p}{2}\le r\le p$. We show that there exists a critical value ${\alpha }_C={\alpha }_C(p,r)$ such that the minimizers have constant sign up to $\alpha ={\alpha }_C$ and then they are odd when $\alpha >{\alpha }_C$.