Minimax Theory and its Applications 07 (2022), No. 2, 303--320
Copyright Heldermann Verlag 2022
Normalized Solutions for a System of Fractional Schrödinger Equations with Linear Coupling
Meiqi Liu
Dept. of Mathematical Sciences, Tsinghua University, Beijing, P. R. China
liumq20@mails.tsinghua.edu.cn
Wenming Zou
Dept. of Mathematical Sciences, Tsinghua University, Beijing, P. R. China
zou-wm@mail.tsinghua.edu.cn
[Abstract-pdf]
We study the normalized solutions of the following fractional Schr\"odinger system: \begin{equation*} \left\{\ \begin{aligned} &(-\Delta)^s u=\lambda_1 u+\mu_1|u|^{p-2}u+\beta v\quad &\hbox{in}\;\mathbb{R}^N, \\ &(-\Delta)^s v=\lambda_2 v+\mu_2|v|^{q-2}v+\beta u\quad &\hbox{in}\;\mathbb{R}^N, \end{aligned} \right. \end{equation*} with prescribed mass\ \ $\int_{\mathbb{R}^N} u^2=a$ \ and \ $\int_{\mathbb{R}^N} v^2=b$,\ \ where $s\in(0,1)$, $2 < p,q \leq2_s^*$, $\beta\in\mathbb{R}$ and $\mu_1,\mu_2,a,b$ are all positive constants. Under different assumptions on $p, q$ and $\beta\in\mathbb R$, we succeed to prove several existence and nonexistence results about the normalized solutions. Specifically, in the case of mass-subcritical nonlinear terms, we overcome the lack of compactness by establishing the least energy inequality and obtain the existence of the normalized solutions for any given $a,b > 0$ and $\beta\in\mathbb{R}$. While for the mass-supercritical case, we use the generalized Pohozaev equality to get the boundedness of the Palais-Smale sequence and obtain the positive normalized solution for any $\beta>0$. Finally, in the fractional Sobolev critical case i.e., $p=q=2_s^*$, we give a result about the nonexistence of the positive solution.
Keywords: Fractional Laplacian, Schroedinger system, normalized solutions.
MSC: 35R11, 35B09, 35B33.
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