A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

Abstract

Given $1\le q\le 2$ and $\alpha \in \mathbb{R}$, 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 $\alpha$ 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.

References in corpus (1)

• An isoperimetric inequality for a nonlinear eigenvalue problem