preprint

Inserted: 18 oct 2026

Year: 2016

Abstract:

Given $1\le q \le 2$ and $α\in\mathbb R$, we study the properties of the solutions of the minimum problem \[ λ(α,q)=\min\left\{\dfrac{\displaystyle\int_{-1}^{1}
u'
^{2}dx+α\left
\int_{-1}^{1}
u
^{q-1}u\, dx\right
^{\frac2q}}{\displaystyle\int_{-1}^{1}
u
^{2}dx}, u\in H_{0}^{1}(-1,1),\,u\not\equiv 0\right\}. \] In particular, depending on $α$ and $q$, we show that the minimizers have constant sign up to a critical value of $α=α_{q}$, and when $α>α_{q}$ the minimizers are odd.