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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
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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  • [1] Elvise Berchio, Filippo Gazzola, and Tobias Weth, Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math. 620 (2008), 165-183. MR 2427979 (2009i:35069),
  • [2] T. Boggio, Sulle funzioni di Green d'ordine m, Rend. Circ. Mat. Palermo 20 (1905), 97-135.
  • [3] C. Cowan, Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter, arXiv:1109.5206 (2011).
  • [4] Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub, and Amir Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal. 198 (2010), no. 3, 763-787. MR 2729319 (2012a:35057),
  • [5] Daniele Cassani, João Marcos do Ó, and Nassif Ghoussoub, On a fourth order elliptic problem with a singular nonlinearity, Adv. Nonlinear Stud. 9 (2009), no. 1, 177-197. MR 2473155 (2010c:35044)
  • [6] Juan Dávila, Isabel Flores, and Ignacio Guerra, Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann. 348 (2010), no. 1, 143-193. MR 2657438 (2011m:35075),
  • [7] Pierpaolo Esposito, Nassif Ghoussoub, and Yujin Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lecture Notes in Mathematics, vol. 20, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010. MR 2604963 (2011c:35005)
  • [8] Alberto Ferrero and Guillaume Warnault, On solutions of second and fourth order elliptic equations with power-type nonlinearities, Nonlinear Anal. 70 (2009), no. 8, 2889-2902. MR 2509377 (2010f:35063),
  • [9] Filippo Gazzola, Hans-Christoph Grunau, and Guido Sweers, Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains, Lecture Notes in Mathematics, vol. 1991, Springer-Verlag, Berlin, 2010. MR 2667016 (2011h:35001)
  • [10] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
  • [11] Zongming Guo and Juncheng Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal. 40 (2008/09), no. 5, 2034-2054. MR 2471911 (2010b:35125),
  • [12] Zongming Guo and Juncheng Wei, Entire solutions and global bifurcations for a biharmonic equation with singular non-linearity in $ \mathbb{R}^3$, Adv. Differential Equations 13 (2008), no. 7-8, 753-780. MR 2479029 (2009m:35091)
  • [13] Zongming Guo and Zhongyuan Liu, Further study of a fourth-order elliptic equation with negative exponent, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 3, 537-549. MR 2805617 (2012f:35092),
  • [14] Saïma Khenissy, Nonexistence and uniqueness for biharmonic problems with supercritical growth and domain geometry, Differential Integral Equations 24 (2011), no. 11-12, 1093-1106. MR 2866013 (2012k:35103)
  • [15] Fanghua Lin and Yisong Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2081, 1323-1337. MR 2313813 (2008b:78005),
  • [16] Amir Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. Differential Equations 248 (2010), no. 3, 594-616. MR 2557908 (2010j:35171),
  • [17] J.A. Pelesko and D.H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1955412 (2003m:74004)
  • [18] William C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations 42 (1981), no. 3, 400-413. MR 639230 (83b:35051),

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Additional Information

Zongming Guo
Affiliation: Department of Mathematics, Henan Normal University, Xinxiang, 453007, People’s Republic of China

Baishun Lai
Affiliation: School of Mathematics and Information Science, Henan University, Kaifeng 475004, People’s Republic of China

Dong Ye
Affiliation: Département de Mathématiques, UMR 7122, Université de Metz, Ile de Saulcy, 57045 Metz, France

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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