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Multiplicity of Concentrating Solutions for Choquard Equation with Critical Growth

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Abstract

In this paper, we consider the multiplicity and concentration phenomenon of positive solutions to the following Choquard equation

$$\begin{aligned} -\varepsilon ^2\Delta u+V(x)u=\varepsilon ^{-\alpha }Q(x)(I_\alpha *|u|^{2^{*}_{\alpha }}) |u|^{2^{*}_{\alpha }-2}u+ f(u)\quad \text{ in }\;\mathbb {R}^N, \end{aligned}$$

where \(N\ge 3\), \((N-4)_+<\alpha < N\), \(I_\alpha \) is the Riesz potential, \(\varepsilon \) is a small parameter, \(V(x)\in C(\mathbb {R}^N)\cap L^\infty (\mathbb {R}^N)\) is a positive potential, \(f\in C^1(\mathbb {R}^+,\mathbb {R})\) is a subcritical nonlinear term and \( 2^{*}_{\alpha }=\frac{N+\alpha }{N-2}\) is the upper-critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. By means of variational methods and delicate energy estimates, we establish the relationship between the number of solutions and the profiles of potentials V and Q, and the concentration behavior of positive solutions is also obtained for \(\varepsilon >0\) small.

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Acknowledgements

The authors are very grateful for the anonymous reviewers for their careful reading the manuscript and valuable comments. Y. Meng is supported by Beijing Natural Science Foundation (1222017) and National Natural Science Foundation of China (12271028) and X. He is supported by National Natural Science Foundation of China (12171497,11771468, 11971027).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Yuxi Meng

  2. College of Science, Minzu University of China, Beijing, 100081, China

    Xiaoming He

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  1. Yuxi Meng
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Correspondence to Xiaoming He.

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Meng, Y., He, X. Multiplicity of Concentrating Solutions for Choquard Equation with Critical Growth. J Geom Anal 33, 78 (2023). https://doi.org/10.1007/s12220-022-01129-1

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