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Concentration behavior of minimizers for Kirchhoff energy functional with ellipse-shaped potential in bounded domains.

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Title: Concentration behavior of minimizers for Kirchhoff energy functional with ellipse-shaped potential in bounded domains.
Authors: Yan, Wenjing1 (AUTHOR) yanwenjing1998@126.com, Guo, Helin1 (AUTHOR) guohelin@tyut.edu.cn, Zhao, Lingling1 (AUTHOR) zhaolingling@tyut.edu.cn
Source: Zeitschrift für Angewandte Mathematik und Physik (ZAMP). Apr2026, Vol. 77 Issue 4, p1-17. 17p.
Subjects: Energy function, Concentration functions, Critical point theory, Asymptotic expansions
Abstract: We study the limiting behavior of the minimizers for the following Kirchhoff energy functional with an ellipse-shaped type trapping potential V x in a bounded domain Ω of R 2 , where the energy functional is defined by E b u = ∫ Ω ∇ u 2 d x + ∫ Ω V (x) u 2 d x + b 2 ∫ Ω ∇ u 2 d x 2 - a 2 ∫ Ω u 4 d x , u ∈ K. It has been shown that the minimizers always exist for any b > 0 . In the present paper, we consider the limiting behavior of minimizers when the endpoints of the major axis of the ellipse-shaped bottom locate at the interior or the boundary of Ω as b ↘ 0 . We first prove that the minimizers must concentrate at an inner point of Ω as b ↘ 0 if one of the endpoints of the major axis of the ellipse-shaped bottom locates at the interior of Ω. Besides, if all the endpoints of the major axis of the ellipse-shaped bottom are located at the boundary of Ω , we obtain that the minimizers must concentrate near the boundary of Ω as b ↘ 0 . [ABSTRACT FROM AUTHOR]
Copyright of Zeitschrift für Angewandte Mathematik und Physik (ZAMP) is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Concentration behavior of minimizers for Kirchhoff energy functional with ellipse-shaped potential in bounded domains.
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  Data: We study the limiting behavior of the minimizers for the following Kirchhoff energy functional with an ellipse-shaped type trapping potential V x in a bounded domain Ω of R 2 , where the energy functional is defined by E b u = ∫ Ω ∇ u 2 d x + ∫ Ω V (x) u 2 d x + b 2 ∫ Ω ∇ u 2 d x 2 - a 2 ∫ Ω u 4 d x , u ∈ K. It has been shown that the minimizers always exist for any b > 0 . In the present paper, we consider the limiting behavior of minimizers when the endpoints of the major axis of the ellipse-shaped bottom locate at the interior or the boundary of Ω as b ↘ 0 . We first prove that the minimizers must concentrate at an inner point of Ω as b ↘ 0 if one of the endpoints of the major axis of the ellipse-shaped bottom locates at the interior of Ω. Besides, if all the endpoints of the major axis of the ellipse-shaped bottom are located at the boundary of Ω , we obtain that the minimizers must concentrate near the boundary of Ω as b ↘ 0 . [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Zeitschrift für Angewandte Mathematik und Physik (ZAMP) is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s00033-026-02737-5
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      – Code: eng
        Text: English
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        PageCount: 17
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      – SubjectFull: Energy function
        Type: general
      – SubjectFull: Concentration functions
        Type: general
      – SubjectFull: Critical point theory
        Type: general
      – SubjectFull: Asymptotic expansions
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      – TitleFull: Concentration behavior of minimizers for Kirchhoff energy functional with ellipse-shaped potential in bounded domains.
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              M: 04
              Text: Apr2026
              Type: published
              Y: 2026
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