## Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems

created by piscitelli on 31 Aug 2022
modified on 11 Sep 2022

[BibTeX]

preprint

Inserted: 31 aug 2022
Last Updated: 11 sep 2022

Year: 2019

ArXiv: 1902.04578 PDF

Abstract:

Let us consider the following minimum problem $\lambda_\alpha(p,r)= \min_{\substack{u\in W_{0}^{1,p}(-1,1)\\ u\not\equiv0}}\dfrac{\displaystyle\int_{-1}^{1} u' ^{p}dx+\alpha\left \int_{-1}^{1} u ^{r-1}u\, dx\right ^{\frac pr}}{\displaystyle\int_{-1}^{1} u ^{p}dx},$ 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 $\alpha>\alpha_{C}$.