preprint

Inserted: 31 aug 2022
Last Updated: 11 sep 2022

Year: 2016

Abstract:

Given $1\le q \le 2$ and $\alpha\in\mathbb R$, we study the properties of the solutions of the minimum problem \[ \lambda(\alpha,q)=\min\left\{\dfrac{\displaystyle\int_{-1}^{1}
u'
^{2}dx+\alpha\left
\int_{-1}^{1}
u
^{q-1}u\, dx\right
^{\frac2q}}{\displaystyle\int_{-1}^{1}
u
^{2}dx}, u\in H_{0}^{1}(-1,1),\,u\not\equiv 0\right\}. \] 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.