混合非线性非自治Schrödinger方程归一化解的多重性与稳定性

IF 0.7 3区 数学 Q2 MATHEMATICS
Xinfu Li, Li Xu, Meiling Zhu
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引用次数: 0

摘要

摘要本文首先研究了一类混合非线性方程\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*}的归一化解的多重性,其中$a, \epsilon, \eta \gt 0$, q是l2 -次临界,p是l2 -超临界,$\lambda\in \mathbb{R}$是一个以拉格朗日乘子形式出现的未知参数,h是一个正连续函数。证明了当λ足够小时,归一化解的个数至少是h的全局最大值点的个数。得到的解是局部极小值,可能不是基态解,因为势h缺乏对称性。其次,分析了由局部最小值组成的不同集合的稳定性。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。特别地,我们的结果涵盖了Sobolev临界情况$p=2N/(N-2)$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Multiplicity and Stability of Normalized Solutions to Non-autonomous Schrödinger Equation with Mixed Non-linearities
Abstract 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 L 2 -subcritical, p is L 2 -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 $p=2N/(N-2)$ .
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来源期刊
CiteScore
1.10
自引率
0.00%
发文量
49
审稿时长
6 months
期刊介绍: The Edinburgh Mathematical Society was founded in 1883 and over the years, has evolved into the principal society for the promotion of mathematics research in Scotland. The Society has published its Proceedings since 1884. This journal contains research papers on topics in a broad range of pure and applied mathematics, together with a number of topical book reviews.
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