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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]
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Database: Engineering Source
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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]
ISSN:00442275
DOI:10.1007/s00033-026-02737-5