A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

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.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Proposition 2.1
• proof
• Remark 2.2
• Proposition 3.1
• Proposition 3.2
• proof : Proof of Proposition \ref{['cambiosegno0']}
• Remark 3.3