Revisiting the biharmonic equation modelling electrostatic actuation in lower dimensions

Guo, Zongming; Lai, Baishun; Ye, Dong

doi:10.1090/S0002-9939-2014-11895-8

Revisiting the biharmonic equation modelling electrostatic actuation in lower dimensions
HTML articles powered by AMS MathViewer

by Zongming Guo, Baishun Lai and Dong Ye PDF

Proc. Amer. Math. Soc. 142 (2014), 2027-2034 Request permission

Abstract:

Let $B \subset \mathbb {R}^N$ be the unit ball. We study the structure of solutions to the semilinear biharmonic problem \[ \begin {cases} \Delta ^2 u=\lambda (1-u)^{-p} & \text {in $B$},\\ 0<u<1 & \text {in $B$},\\ u=\partial _\nu =0\; (\text {resp.~$u = \Delta u = 0$}) & \text {on $\partial B$}, \end {cases} \] 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 $\lambda \to 0$.

References

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, DOI 10.1515/CRELLE.2008.052
T. Boggio, Sulle funzioni di Green d’ordine m, Rend. Circ. Mat. Palermo 20 (1905), 97-135.
C. Cowan, Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter, arXiv:1109.5206 (2011).
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, DOI 10.1007/s00205-010-0367-x
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, DOI 10.1515/ans-2009-0109
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, DOI 10.1007/s00208-009-0476-8
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, DOI 10.1090/cln/020
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, DOI 10.1016/j.na.2008.12.041
Filippo Gazzola, Hans-Christoph Grunau, and Guido Sweers, Polyharmonic boundary value problems, Lecture Notes in Mathematics, vol. 1991, Springer-Verlag, Berlin, 2010. Positivity preserving and nonlinear higher order elliptic equations in bounded domains. MR 2667016, DOI 10.1007/978-3-642-12245-3
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, DOI 10.1007/978-3-642-61798-0
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, DOI 10.1137/070703375
Zongming Guo and Juncheng Wei, Entire solutions and global bifurcations for a biharmonic equation with singular non-linearity in $\Bbb R^3$, Adv. Differential Equations 13 (2008), no. 7-8, 753–780. MR 2479029
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, DOI 10.1017/S0308210509001061
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
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, DOI 10.1098/rspa.2007.1816
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, DOI 10.1016/j.jde.2009.09.011
John A. Pelesko and David H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1955412
William C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations 42 (1981), no. 3, 400–413. MR 639230, DOI 10.1016/0022-0396(81)90113-3