标题： 非线性非局部特征值问题的饱和现象 显示英文标题

标题： A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

摘要： 给定$1\le q \le 2$和$\alpha\in\mathbb R$，我们研究最小问题\[ \lambda(\alpha,q)=\min\left\{\dfrac{\displaystyle\int_{-1}^{1}|u'|^{2}dx+\alpha\left|\int_{-1}^{1}|u|^{q-1}u\, dx\right|^{\frac2q}}{\displaystyle\int_{-1}^{1}|u|^{2}dx}, u\in H_{0}^{1}(-1,1),\,u\not\equiv 0\right\}. \]的解的性质。特别是，根据$\alpha$和$q$，我们证明了在$\alpha=\alpha_{q}$的临界值之前，极小解具有常符号，当$\alpha>\alpha_{q}$时，极小解是奇函数。 显示英文摘要

摘要： 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.