Abstract
In this paper, we study the critical fractional Choquard equation with a local perturbation \((-\Delta )^su=\lambda u+\mu |u|^{q-2}u+(I_{\alpha }*|u|^{2^*_{\alpha ,s}})|u|^{2^*_{\alpha ,s}-2}u,~~x\in {\mathbb {R}}^N,\) having prescribed mass \(\int _{{\mathbb {R}}^N}u^2\text {d}x=a^2,\) where \(I_{\alpha }(x)\) is the Riesz potential, \(s\in (0,1), N>2s, 0<\alpha<\min \{N,4s\}, 2<q<2^*_s=\frac{2N^{}}{N-2s}\) is the fractional critical Sobolev exponent, and \(2^*_{\alpha ,s}=\frac{2N-\alpha }{N-2s}\) is the fractional Hardy–Littlewood–Sobolev critical exponent, \(a>0,\ \mu \in {\mathbb {R}}.\). Under some \(L^2\)-subcritical, \(L^2\)-critical and \(L^2\)-supercritical perturbation \(\mu |u|^{q-2}u\), respectively, we prove several existence and non-existence results. The qualitative behavior of the ground states as \(\mu \rightarrow 0^+\) is also studied. The mathematical analysis carried out in this paper can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions for fractional Choquard equation. In this framework, several related results are extended and improved.
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Acknowledgements
The authors would like to thank two anonymous reviewers for the careful reading of the manuscript and for their valuable comments. Xiaoming He is supported by the National Natural Science Foundation of China (11771468, 11971027, 12171497), while Wenming Zou is supported by the National Natural Science Foundation of China (11771234, 11926323). The research of Vicenţiu D. Rădulescu is supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI-UEFISCDI, project number PCE 137/2021, within PNCDI III.
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He, X., Rădulescu, V.D. & Zou, W. Normalized Ground States for the Critical Fractional Choquard Equation with a Local Perturbation. J Geom Anal 32, 252 (2022). https://doi.org/10.1007/s12220-022-00980-6
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DOI: https://doi.org/10.1007/s12220-022-00980-6