A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

TL;DR

This paper investigates a nonlinear nonlocal eigenvalue problem, revealing a saturation phenomenon where minimizers change sign behavior at a critical parameter value, with solutions being constant sign below and odd above this threshold.

Contribution

It introduces a new analysis of the eigenvalue problem showing a saturation phenomenon and characterizes the sign-changing behavior of minimizers depending on parameters.

Findings

Minimizers have constant sign for \\alpha \\leq \\alpha_q.

For \\alpha > \\alpha_q, minimizers are odd functions.

The study identifies a critical value \\alpha_q for sign change behavior.

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.