Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (3): 997-1019.

### MULTIPLICITY OF NORMALIZED SOLUTIONS FOR THE FRACTIONAL SCHRÖDINGER-POISSON SYSTEM WITH DOUBLY CRITICAL GROWTH

Yuxi Meng1,*, Xiaoming He2

1. 1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;
2. College of Science, Minzu University of China, Beijing 100081, China
• Received:2022-09-05 Revised:2023-04-16 Online:2024-06-25 Published:2024-05-21
• Contact: *Yuxi Meng, E-mail:yxmeng125@163.com
• About author:Xiaoming He, E-mail:xmhe923@muc.edu.cn
• Supported by:
BIT Research and Innovation Promoting Project (2023YCXY046), the NSFC (11771468, 11971027, 11971061, 12171497 and 12271028), the BNSF (1222017) and the Fundamental Research Funds for the Central Universities.

Abstract: In this paper, we are concerned with solutions to the fractional Schrödinger-Poisson system
$\begin{cases}(-\Delta)^su-\phi|u|^{2_s^*-3}u=\lambda u+\mu|u|^{q-2}u+|u|^{2_s^*-2}u,&x\in\mathbb R^3,\\(-\Delta)^s\phi=|u|^{2_s^*-1},&x\in\mathbb R^3,\end{cases}$
with prescribed mass $\int_{\mathbb{R}^3}|u|^2\mathrm{d}x = a^2,$ where $a>0$ is a prescribed number, $\mu>0$ is a paremeter, $s \in (0,1), 2<q<2^*_s$, and $2^*_s = \frac{6}{3-2s}$ is the fractional critical Sobolev exponent. In the $L^2$-subcritical case, we show the existence of multiple normalized solutions by using the genus theory and the truncation technique; in the $L^2$-supercritical case, we obtain a couple of normalized solutions by developing a fiber map. Under both cases, to recover the loss of compactness of the energy functional caused by the doubly critical growth, we need to adopt the concentration-compactness principle. Our results complement and improve upon some existing studies on the fractional Schrödinger-Poisson system with a nonlocal critical term.

CLC Number:

• 35A15
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