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August  2015, 35(8): 3627-3682. doi: 10.3934/dcds.2015.35.3627

On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

2. 

School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China

Received  January 2014 Revised  December 2014 Published  February 2015

Motivated by some nonlinear models recently arising in Micro-Electro-Mechanical System (MEMS) and new progress on one-dimensional mean curvature type problems, we investigate the existence and exact numbers of positive solutions for a class of boundary value problems with $\varphi$-Laplacian $$ -(\varphi(u'))'=\lambda f(u)\; on (-L, L),\quad u(-L)=u(L)=0, $$ when the parameters $\lambda$ and $L$ vary. Various exact multiplicity results as well as global bifurcation diagrams are obtained. These results include the applications to one-dimensional MEMS equations with fringing field as well as mean curvature type problems. We also extend and improve one of the main results of Korman and Li [Proc. Roy. Soc. Edinburgh Sect. A, 140(6):1197--1215, 2010] (Theorem 3.4). With the aid of numerical simulations, we find many interesting new examples, which reveal the striking complexity of bifurcation patterns for the problem.
Citation: Hongjing Pan, Ruixiang Xing. On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3627-3682. doi: 10.3934/dcds.2015.35.3627
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show all references

References:
[1]

Rend. Istit. Mat. Univ. Trieste, 39 (2007), 63-85.  Google Scholar

[2]

J. Differential Equations, 243 (2007), 208-237. doi: 10.1016/j.jde.2007.05.031.  Google Scholar

[3]

N. D. Brubaker and A. E. Lindsay, Analysis of the singular solution branch of a prescribed mean curvature equation with singular nonlinearity modeling a MEMS capacitor,, Preprint, ().   Google Scholar

[4]

Eur. J. Appl. Math., 24 (2013), 631-656. doi: 10.1017/S0956792513000077.  Google Scholar

[5]

Eur. J. Appl. Math., 22 (2011), 455-470. doi: 10.1017/S0956792511000180.  Google Scholar

[6]

Nonlinear Anal., 75 (2012), 5086-5102. doi: 10.1016/j.na.2012.04.025.  Google Scholar

[7]

Eur. J. Appl. Math., 24 (2013), 343-370. doi: 10.1017/S0956792512000435.  Google Scholar

[8]

Eur. J. Appl. Math., 22 (2011), 317-331. doi: 10.1017/S0956792511000076.  Google Scholar

[9]

Nonlinear Anal., 89 (2013), 284-298. doi: 10.1016/j.na.2013.04.018.  Google Scholar

[10]

Master Thesis, National Tsing Hua University, Hsinchu, Taiwan, 2011, (Directed by S.-H. Wang). Google Scholar

[11]

Comm. Partial Differential Equations, 37 (2012), 1462-1493. doi: 10.1080/03605302.2012.679990.  Google Scholar

[12]

J. Escher, P. Laurencot and C. Walker, Dynamics of a free boundary problem with curvature modeling electrostatic MEMS,, Trans. Amer. Math. Soc., ().  doi: 10.1090/S0002-9947-2014-06320-4.  Google Scholar

[13]

Courant Inst. Math. Sci., New York., 2010.  Google Scholar

[14]

Comm. Partial Differential Equations, 17 (1992), 593-614. doi: 10.1080/03605309208820855.  Google Scholar

[15]

Commun. Contemp. Math., 9 (2007), 701-730. doi: 10.1142/S0219199707002617.  Google Scholar

[16]

Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1197-1215. doi: 10.1017/S0308210509001449.  Google Scholar

[17]

Indiana Univ. Math. J., 20 (1970), 1-13. doi: 10.1512/iumj.1971.20.20001.  Google Scholar

[18]

Ann. Mat. Pura Appl. (4), 155 (1989), 243-260. doi: 10.1007/BF01765943.  Google Scholar

[19]

J. Math. Anal. Appl., 367 (2010), 486-498. doi: 10.1016/j.jmaa.2010.01.055.  Google Scholar

[20]

Methods Appl. Anal., 15 (2008), 297-325. doi: 10.4310/MAA.2008.v15.n3.a4.  Google Scholar

[21]

Adv. Nonlinear Stud., 7 (2007), 671-682.  Google Scholar

[22]

Nonlinear Anal., 70 (2009), 999-1010. doi: 10.1016/j.na.2008.01.027.  Google Scholar

[23]

Nonlinear Anal., 74 (2011), 1234-1260. doi: 10.1016/j.na.2010.09.063.  Google Scholar

[24]

Nonlinear Anal., 74 (2011), 3751-3768. doi: 10.1016/j.na.2011.03.020.  Google Scholar

[25]

Nonlinear Anal. Real World Appl., 13 (2012), 2432-2445. doi: 10.1016/j.nonrwa.2012.02.012.  Google Scholar

[26]

J. Differential Equations, 254 (2013), 1464-1499. doi: 10.1016/j.jde.2012.10.025.  Google Scholar

[27]

Chapman & Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[28]

J. Engrg. Math., 53 (2005), 239-252. doi: 10.1007/s10665-005-9013-2.  Google Scholar

[29]

Proc. Amer. Math. Soc., 138 (2010), 1693-1699. doi: 10.1090/S0002-9939-09-10226-5.  Google Scholar

[30]

J. Math. Anal. Appl., 395 (2013), 393-402. doi: 10.1016/j.jmaa.2012.05.053.  Google Scholar

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