Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems

Abstract

Let us consider the following minimum problem $$ \lambda\_\alpha(p,r)=\min\_{\substack{u\in W\_{0}^{1,p}(-1,1)\u\not\equiv0}}\frac{\int\_{-1}^{1}|u'|^{p}dx+\alpha\left|\int\_{-1}^{1}|u|^{r-1}u, dx\right|^{\frac pr}}{\int\_{-1}^{1}|u|^{p}dx}, $$ where $\alpha\in\mathbb R$, $p\ge 2$ and $\frac p2 \le r \le p$. 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 $\alpha > \alpha\_{C}$.