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Normalized solutions for the fractional Choquard equations with Sobolev critical and double mass supercritical growth

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Abstract

In the present paper, we study the following Sobolev critical fractional Choquard equation

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^s u+\lambda u=\eta [|x|^{-\mu }*|u|^p]|u|^{p-2}u+|u|^{2_s^*-2}u, \quad &{}\hbox {in}\;\mathbb {R}^N,\\ \int _{\mathbb {R}^N}|u|^2dx=m^2, \end{array} \right. \end{aligned}$$

where \(m, \ \eta >0\), \(0<s<1\), \(N \ge 2\), \(0<\mu <2s\), \(2+\frac{2\,s-\mu }{N}<p\le \frac{2N-\mu +s\cdot 2_s^*}{N}<2_{\mu ,s}^*:=\frac{2N-\mu }{N-2\,s}\), \(2_s^*:=\frac{2N}{N-2s}\) is the fractional Sobolev critical exponent. By virtue of a fiber map and the concentration-compactness principle, we obtain a couple of normalized solutions to the above equation for large \(\eta \), which extends and improves the results in Li et al. (Math Methods Appl Sci 44:10331–10360, 2021) and Yang (J Math Phys 61:051505, 2020), and almost no one has studied the double mass supercritical cases.

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Correspondence to Meiqi Liu.

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This work is supported in part by the National Natural Science Foundation of China (12261031; 12161033) and the Yunnan Province Applied Basic Research for General Project (202301AT070141) and Youth Outstanding-notch Talent Support Program in Yunnan Province and the Project Funds of Xingdian Talent Support Program.

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Li, Q., Wang, W. & Liu, M. Normalized solutions for the fractional Choquard equations with Sobolev critical and double mass supercritical growth. Lett Math Phys 113, 49 (2023). https://doi.org/10.1007/s11005-023-01672-0

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  • DOI: https://doi.org/10.1007/s11005-023-01672-0

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