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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

Wenming Zou
Dept. of Mathematical Sciences, Tsinghua University, Beijing, P. R. China


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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