Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems

Abstract

Let us consider the following minimum problem \[ λ_α(p,r)= \min_{\substack{u\in W_{0}^{1,p}(-1,1)\\ u\not\equiv0}}\dfrac{\displaystyle\int_{-1}^{1}|u'|^{p}dx+α\left|\int_{-1}^{1}|u|^{r-1}u\, dx\right|^{\frac pr}}{\displaystyle\int_{-1}^{1}|u|^{p}dx}, \] where $α\in\mathbb R$, $p\ge 2$ and $\frac p2 \le r \le p$. We show that there exists a critical value $α_C=α_C (p,r)$ such that the minimizers have constant sign up to $α=α_{C}$ and then they are odd when $α>α_{C}$.