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Multi-Bump Solutions for Nonlinear Choquard Equation with Potential Wells and a General Nonlinearity

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Abstract

In this article, we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity

$$-\Delta u + (\lambda a(x) + 1)u = \left(\frac{1}{|x|^\alpha}* F(u)\right) f(u) \; in \; \mathbb{R}^N,$$

where N ≥ 3, 0 < α < min{N, 4}, λ is a positive parameter and the nonnegative potential function a(x) is continuous. Using variational methods, we prove that if the potential well int(a−1(0)) consists of k disjoint components, then there exist at least 2k − 1 multi-bump solutions. The asymptotic behavior of these solutions is also analyzed as λ → +∞.

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Acknowledgements

The authors would like to thank Prof. Shuangjie Peng for stimulating discussions and helpful suggestions on this article. The first author thanks Prof. Minbo Yang very much for some useful discussions.

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Authors and Affiliations

  1. College of Science, Huazhong Agricultural University, Wuhan, 430070, China

    Lun Guo

  2. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Tingxi Hu

Authors
  1. Lun Guo
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  2. Tingxi Hu
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Corresponding author

Correspondence to Tingxi Hu.

Additional information

L. Guo is supported by the Fundamental Research Funds for the Central Universities (2662018QD039); T. Hu is supported by the Project funded by China Postdoctoral Science Foundation (2018M643389).

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Guo, L., Hu, T. Multi-Bump Solutions for Nonlinear Choquard Equation with Potential Wells and a General Nonlinearity. Acta Math Sci 40, 316–340 (2020). https://doi.org/10.1007/s10473-020-0202-x

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  • DOI: https://doi.org/10.1007/s10473-020-0202-x

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