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Revisiting the biharmonic equation modelling electrostatic actuation in lower dimensions


Authors: Zongming Guo, Baishun Lai and Dong Ye
Journal: Proc. Amer. Math. Soc. 142 (2014), 2027-2034
MSC (2010): Primary 35J25, 35J20; Secondary 35B33, 35B40
DOI: https://doi.org/10.1090/S0002-9939-2014-11895-8
Published electronically: March 7, 2014
MathSciNet review: 3182022
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ B \subset \mathbb{R}^N$ be the unit ball. We study the structure of solutions to the semilinear biharmonic problem

$\displaystyle \left \{ \begin {array}{ll} \Delta ^2 u=\lambda (1-u)^{-p} \;\; &... ...resp.~$u = \Delta u = 0$}) \;\; & \mbox {on $\partial B$}, \end{array} \right .$

where $ p, \lambda >0$, which arises in the study of the deflection of charged plates in electrostatic actuators. We study in particular the structure of solutions for $ N=2$ or $ 3$ and show the existence of mountain-pass solutions under suitable conditions on $ p$. Our results contribute to completing the picture of solutions in previous works. Moreover, we also analyze the asymptotic behavior of the constructed mountain-pass solutions as $ \lambda \to 0$.

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Additional Information

Zongming Guo
Affiliation: Department of Mathematics, Henan Normal University, Xinxiang, 453007, People’s Republic of China
Email: gzm@htu.cn

Baishun Lai
Affiliation: School of Mathematics and Information Science, Henan University, Kaifeng 475004, People’s Republic of China
Email: laibaishun@henu.edu.cn

Dong Ye
Affiliation: Département de Mathématiques, UMR 7122, Université de Metz, Ile de Saulcy, 57045 Metz, France
Email: dong.ye@univ-metz.fr

DOI: https://doi.org/10.1090/S0002-9939-2014-11895-8
Keywords: Biharmonic equations, singular nonlinearity, asymptotic analysis
Received by editor(s): December 18, 2011
Received by editor(s) in revised form: June 28, 2012
Published electronically: March 7, 2014
Additional Notes: The first author was supported by NSFC (11171092, 10871060) and Innovation Scientists and Technicians Troop Projects of Henan Province (114200510011).
The second author was supported by the National Natural Science Foundation of China (Grants No. 11201119, 11126155, 10971061), the Natural Science Foundation of Henan Province (Grant No. 112300410054) and the Natural Science Foundation of Education Department of Henan Province (Grant No. 2011B11004).
The third author was partly supported by the French ANR project ANR-08-BLAN-0335-01.
All of the authors would like to thank the anonymous referee for a careful reading and valuable remarks.
Communicated by: Walter Craig
Article copyright: © Copyright 2014 American Mathematical Society

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