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Prescribed mass standing waves for energy critical Hartree equations

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Abstract

In this paper, we focus on the solutions to the energy critical Hartree equations

$$\begin{aligned} -\Delta u=\lambda u+\mu (|x|^{-\alpha }*|u|^{2})u+(|x|^{-4}*|u|^{2})u,\ \ x\in \mathbb {R}^{N} \end{aligned}$$

under the normalized constraint

$$\begin{aligned} \int _{{\mathbb {R}^N}} {{u}^2}=c>0, \end{aligned}$$

where \(N\ge 5\), \(\mu \in \mathbb {R}\), \(0<\alpha <4\), and the frequency \(\lambda \in \mathbb {R}\) is a part of unknown and appears as Lagrange multiplier. Under different assumptions on c, \(\mu \) and \(\alpha \), we prove some existence, non-existence, multiplicity and asymptotic results of normalized solutions to the above problem. In addition, the stability of the corresponding standing waves to the related time-dependent problem is discussed. These results are a continuation of our previous works, Luo (J Differ Equ, 195:455–467, 2019) and Cao et al. (J Differ Equ, 276:228–263, 2021), concerning normalized solutions to Hartree equations from energy subcritical to energy critical case.

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Acknowledgements

The authors are very grateful to the referee for her/his valuable suggestions which lead to the great improvements of the present paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510520, People’s Republic of China

    Huifang Jia

  2. School of Mathematics, Hefei University of Technology, Hefei, 230009, People’s Republic of China

    Xiao Luo

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  1. Huifang Jia
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  2. Xiao Luo
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Correspondence to Xiao Luo.

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Communicated by A. Neves.

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Huifang Jia is supported by NNSF of China (No. 12001126) and GDUT grant (No. 263113459), Xiao Luo is supported by NNSF of China (No. 11901147).

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Jia, H., Luo, X. Prescribed mass standing waves for energy critical Hartree equations. Calc. Var. 62, 71 (2023). https://doi.org/10.1007/s00526-022-02416-z

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