
Minimax Theory and its Applications 07 (2022), No. 2, 303320 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 zouwm@mail.tsinghua.edu.cn [Abstractpdf] We study the normalized solutions of the following fractional Schr\"odinger system: \begin{equation*} \left\{\ \begin{aligned} &(\Delta)^s u=\lambda_1 u+\mu_1u^{p2}u+\beta v\quad &\hbox{in}\;\mathbb{R}^N, \\ &(\Delta)^s v=\lambda_2 v+\mu_2v^{q2}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 masssubcritical 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 masssupercritical case, we use the generalized Pohozaev equality to get the boundedness of the PalaisSmale 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. [ Fulltextpdf (156 KB)] for subscribers only. 