论文标题
归一化溶液对非自主schrödinger方程的多样性和轨道稳定性具有混合非线性
Multiplicity and orbital stability of normalized solutions to non-autonomous Schrödinger equation with mixed nonlinearities
论文作者
论文摘要
本文研究了具有混合非线性的归一化解决方案的归一化解决方案\ begin {equation*} \ begin {cases}-ΔU=λu=λu+h（εx）| \ int _ {\ mathbb {r}^n} | u | u |^2dx = a^2，\ end {cases} \ end {equation*}其中$ a，ε，η> 0 $，$ q $ as $ l^2 $ - 2 $ -subcritical，$ l^2 $ p $是$ l^2 $ - su poy is $ - superical，$ l^2 $ - $ - $ - $ and $ ny comamit {这是Lagrange乘数，$ H $是一个积极而连续的功能。事实证明，当$ε$足够小时，归一化解决方案的数量至少是$ h $的全球最大点的数量。此外，还分析了获得的溶液的轨道稳定性。特别是，我们的结果涵盖了sobolev关键案例$ p = 2n/（n-2）$。
This paper studies the multiplicity of normalized solutions to the Schrödinger equation with mixed nonlinearities \begin{equation*} \begin{cases} -Δu=λu+h(εx)|u|^{q-2}u+η|u|^{p-2}u,\quad x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=a^2, \end{cases} \end{equation*} where $a, ε, η>0$, $q$ is $L^2$-subcritical, $p$ is $L^2$-supercritical, $λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $h$ 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. Moreover, the orbital stability of the solutions obtained is analyzed as well. In particular, our results cover the Sobolev critical case $p=2N/(N-2)$.
