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Limiting profile of solutions for Schrödinger equations with shrinking self-focusing core

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Abstract

We consider the following equation

$$\begin{aligned} -\Delta u+u=Q_n(x)|u|^{p-2}u, \quad x\in {\mathbb {R}}^{N}, \end{aligned}$$

where \(Q_n\) are concrete bounded functions whose self-focusing core \(\text{ supp }\, Q_n^+\) shrinks to a finite set of points as \(n\rightarrow \infty \). We investigate the limiting profile of concentration for the ground state solutions and construct localized bound state solutions of concentration type.

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Acknowledgements

The authors are grateful to the referee for carefully reading the paper and for thoughtful comments which improve the presentation of the paper. The authors are partially supported by NSFC (11601057, 11771324, 11831009, 11811540026). Fang is also supported by the Fundamental Research Funds for the Central Universities (Grant. DUT18LK05).

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Xiang-Dong Fang

  2. College of Mathematics and Informatics, Fujian Normal University, Fuzhou, 350117, People’s Republic of China

    Zhi-Qiang Wang

  3. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    Zhi-Qiang Wang

Authors
  1. Xiang-Dong Fang
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  2. Zhi-Qiang Wang
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Correspondence to Zhi-Qiang Wang.

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Communicated by M. Del Pino.

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Fang, XD., Wang, ZQ. Limiting profile of solutions for Schrödinger equations with shrinking self-focusing core. Calc. Var. 59, 129 (2020). https://doi.org/10.1007/s00526-020-01799-1

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