This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities\begin{equation*}\begin{cases}-\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\\int_{\mathbb{R}^N}|u|^2\,\textrm{d}x=a^2,\end{cases}\end{equation*}where $a, \epsilon, \eta \gt 0$, q is L2-subcritical, p is L2-supercritical, $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and h is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when ϵ is small enough. The solutions obtained are local minimizers and probably not ground state solutions for the lack of symmetry of the potential h. Secondly, the stability of several different sets consisting of the local minimizers is analysed. Compared with the results of the corresponding autonomous equation, the appearance of the potential h increases the number of the local minimizers and the number of the stable sets. In particular, our results cover the Sobolev critical case $p=2N/(N-2)$.

本文首先研究了具有混合非线性的非自治薛定谔方程的归一化解的多重性\begin{equation*}\begin{cases}-\Delta u=\lambda u+h(\epsilon x)|u|^{ q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\\int_{\mathbb{R}^N}|u|^2\ ,\textrm{d}x=a^2,\end{cases}\end{equation*}其中$a, \epsilon, \eta \gt 0$ , q为L 2亚临界，p为L 2超临界, $\lambda\in \mathbb{R}$是一个未知参数，表现为拉格朗日乘子，h是一个正连续函数。证明了当ε足够小时，归一化解的个数至少为h的全局极大点的个数。获得的解是局部最小化，并且可能不是由于势h缺乏对称性的基态解。其次，分析了由局部极小值组成的几个不同集合的稳定性。与相应的自治方程的结果相比，势h的出现增加了局部极小值的数量和稳定集的数量。特别是，我们的结果涵盖了 Sobolev 临界情况$p=2N/(N-2)$。