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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

Hongjing Pan 1, and  Ruixiang Xing 2,

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.
Keywords: quasilinear BVPs, Mean curvature equation, time map, singular nonlinearity, splitting bifurcation, bifurcation diagram, fringing field, MEMS.
Mathematics Subject Classification: Primary: 34B18, 34C23; Secondary: 35J93, 74G3.
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
References:
[1]

D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical positive solutions of a prescribed curvature equation with singularities, Rend. Istit. Mat. Univ. Trieste, 39 (2007), 63-85.  Google Scholar

[2]

D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical solutions of a prescribed curvature equation, 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]

N. D. Brubaker and A. E. Lindsay, The onset of multivalued solutions of a prescribed mean curvature equation with singular nonlinearity, Eur. J. Appl. Math., 24 (2013), 631-656. doi: 10.1017/S0956792513000077.  Google Scholar

[5]

N. D. Brubaker and J. A. Pelesko, Non-linear effects on canonical MEMS models, Eur. J. Appl. Math., 22 (2011), 455-470. doi: 10.1017/S0956792511000180.  Google Scholar

[6]

N. D. Brubaker and J. A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal., 75 (2012), 5086-5102. doi: 10.1016/j.na.2012.04.025.  Google Scholar

[7]

N. D. Brubaker, J. I. Siddique, E. Sabo, R. Deaton and J. A. Pelesko, Refinements to the study of electrostatic deflections: theory and experiment, Eur. J. Appl. Math., 24 (2013), 343-370. doi: 10.1017/S0956792512000435.  Google Scholar

[8]

M. Burns and M. Grinfeld, Steady state solutions of a bi-stable quasi-linear equation with saturating flux, Eur. J. Appl. Math., 22 (2011), 317-331. doi: 10.1017/S0956792511000076.  Google Scholar

[9]

Y.-H. Cheng, K.-C. Hung and S.-H. Wang, Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS, Nonlinear Anal., 89 (2013), 284-298. doi: 10.1016/j.na.2013.04.018.  Google Scholar

[10]

C.-H. Chuang, On Exact Multiplicity and Bifurcation Diagrams of Positive Solutions of a One-Dimensional Prescribed Mean Curvature Problem, Master Thesis, National Tsing Hua University, Hsinchu, Taiwan, 2011, (Directed by S.-H. Wang). Google Scholar

[11]

J. Dávila and J. Wei, Point ruptures for a MEMS equation with fringing field, 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]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, vol. 20 of Courant Lect. Notes Math., Courant Inst. Math. Sci., New York., 2010.  Google Scholar

[14]

M. Fila, B. Kawohl and H. A. Levine, Quenching for quasilinear equations, Comm. Partial Differential Equations, 17 (1992), 593-614. doi: 10.1080/03605309208820855.  Google Scholar

[15]

P. Habets and P. Omari, Multiple positive solutions of a one-dimensional prescribed mean curvature problem, Commun. Contemp. Math., 9 (2007), 701-730. doi: 10.1142/S0219199707002617.  Google Scholar

[16]

P. Korman and Y. Li, Global solution curves for a class of quasilinear boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1197-1215. doi: 10.1017/S0308210509001449.  Google Scholar

[17]

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13. doi: 10.1512/iumj.1971.20.20001.  Google Scholar

[18]

H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann. Mat. Pura Appl. (4), 155 (1989), 243-260. doi: 10.1007/BF01765943.  Google Scholar

[19]

W. Li and Z. Liu, Exact number of solutions of a prescribed mean curvature equation, J. Math. Anal. Appl., 367 (2010), 486-498. doi: 10.1016/j.jmaa.2010.01.055.  Google Scholar

[20]

A. E. Lindsay and M. J. Ward, Asymptotics of some nonlinear eigenvalue problems for a MEMS capacitor. I. Fold point asymptotics, Methods Appl. Anal., 15 (2008), 297-325. doi: 10.4310/MAA.2008.v15.n3.a4.  Google Scholar

[21]

F. Obersnel, Classical and non-classical sign-changing solutions of a one-dimensional autonomous prescribed curvature equation, Adv. Nonlinear Stud., 7 (2007), 671-682.  Google Scholar

[22]

H. Pan, One-dimensional prescribed mean curvature equation with exponential nonlinearity, Nonlinear Anal., 70 (2009), 999-1010. doi: 10.1016/j.na.2008.01.027.  Google Scholar

[23]

H. Pan and R. Xing, Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations, Nonlinear Anal., 74 (2011), 1234-1260. doi: 10.1016/j.na.2010.09.063.  Google Scholar

[24]

H. Pan and R. Xing, Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. II, Nonlinear Anal., 74 (2011), 3751-3768. doi: 10.1016/j.na.2011.03.020.  Google Scholar

[25]

H. Pan and R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to MEMS models, Nonlinear Anal. Real World Appl., 13 (2012), 2432-2445. doi: 10.1016/j.nonrwa.2012.02.012.  Google Scholar

[26]

H. Pan and R. Xing, Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions, J. Differential Equations, 254 (2013), 1464-1499. doi: 10.1016/j.jde.2012.10.025.  Google Scholar

[27]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[28]

J. A. Pelesko and T. A. Driscoll, The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models, J. Engrg. Math., 53 (2005), 239-252. doi: 10.1007/s10665-005-9013-2.  Google Scholar

[29]

J. Wei and D. Ye, On MEMS equation with fringing field, Proc. Amer. Math. Soc., 138 (2010), 1693-1699. doi: 10.1090/S0002-9939-09-10226-5.  Google Scholar

[30]

X. Zhang and M. Feng, Exact number of solutions of a one-dimensional prescribed mean curvature equation with concave-convex nonlinearities, J. Math. Anal. Appl., 395 (2013), 393-402. doi: 10.1016/j.jmaa.2012.05.053.  Google Scholar

show all references

References:
[1]

D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical positive solutions of a prescribed curvature equation with singularities, Rend. Istit. Mat. Univ. Trieste, 39 (2007), 63-85.  Google Scholar

[2]

D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical solutions of a prescribed curvature equation, 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]

N. D. Brubaker and A. E. Lindsay, The onset of multivalued solutions of a prescribed mean curvature equation with singular nonlinearity, Eur. J. Appl. Math., 24 (2013), 631-656. doi: 10.1017/S0956792513000077.  Google Scholar

[5]

N. D. Brubaker and J. A. Pelesko, Non-linear effects on canonical MEMS models, Eur. J. Appl. Math., 22 (2011), 455-470. doi: 10.1017/S0956792511000180.  Google Scholar

[6]

N. D. Brubaker and J. A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal., 75 (2012), 5086-5102. doi: 10.1016/j.na.2012.04.025.  Google Scholar

[7]

N. D. Brubaker, J. I. Siddique, E. Sabo, R. Deaton and J. A. Pelesko, Refinements to the study of electrostatic deflections: theory and experiment, Eur. J. Appl. Math., 24 (2013), 343-370. doi: 10.1017/S0956792512000435.  Google Scholar

[8]

M. Burns and M. Grinfeld, Steady state solutions of a bi-stable quasi-linear equation with saturating flux, Eur. J. Appl. Math., 22 (2011), 317-331. doi: 10.1017/S0956792511000076.  Google Scholar

[9]

Y.-H. Cheng, K.-C. Hung and S.-H. Wang, Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS, Nonlinear Anal., 89 (2013), 284-298. doi: 10.1016/j.na.2013.04.018.  Google Scholar

[10]

C.-H. Chuang, On Exact Multiplicity and Bifurcation Diagrams of Positive Solutions of a One-Dimensional Prescribed Mean Curvature Problem, Master Thesis, National Tsing Hua University, Hsinchu, Taiwan, 2011, (Directed by S.-H. Wang). Google Scholar

[11]

J. Dávila and J. Wei, Point ruptures for a MEMS equation with fringing field, 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]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, vol. 20 of Courant Lect. Notes Math., Courant Inst. Math. Sci., New York., 2010.  Google Scholar

[14]

M. Fila, B. Kawohl and H. A. Levine, Quenching for quasilinear equations, Comm. Partial Differential Equations, 17 (1992), 593-614. doi: 10.1080/03605309208820855.  Google Scholar

[15]

P. Habets and P. Omari, Multiple positive solutions of a one-dimensional prescribed mean curvature problem, Commun. Contemp. Math., 9 (2007), 701-730. doi: 10.1142/S0219199707002617.  Google Scholar

[16]

P. Korman and Y. Li, Global solution curves for a class of quasilinear boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1197-1215. doi: 10.1017/S0308210509001449.  Google Scholar

[17]

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13. doi: 10.1512/iumj.1971.20.20001.  Google Scholar

[18]

H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann. Mat. Pura Appl. (4), 155 (1989), 243-260. doi: 10.1007/BF01765943.  Google Scholar

[19]

W. Li and Z. Liu, Exact number of solutions of a prescribed mean curvature equation, J. Math. Anal. Appl., 367 (2010), 486-498. doi: 10.1016/j.jmaa.2010.01.055.  Google Scholar

[20]

A. E. Lindsay and M. J. Ward, Asymptotics of some nonlinear eigenvalue problems for a MEMS capacitor. I. Fold point asymptotics, Methods Appl. Anal., 15 (2008), 297-325. doi: 10.4310/MAA.2008.v15.n3.a4.  Google Scholar

[21]

F. Obersnel, Classical and non-classical sign-changing solutions of a one-dimensional autonomous prescribed curvature equation, Adv. Nonlinear Stud., 7 (2007), 671-682.  Google Scholar

[22]

H. Pan, One-dimensional prescribed mean curvature equation with exponential nonlinearity, Nonlinear Anal., 70 (2009), 999-1010. doi: 10.1016/j.na.2008.01.027.  Google Scholar

[23]

H. Pan and R. Xing, Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations, Nonlinear Anal., 74 (2011), 1234-1260. doi: 10.1016/j.na.2010.09.063.  Google Scholar

[24]

H. Pan and R. Xing, Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. II, Nonlinear Anal., 74 (2011), 3751-3768. doi: 10.1016/j.na.2011.03.020.  Google Scholar

[25]

H. Pan and R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to MEMS models, Nonlinear Anal. Real World Appl., 13 (2012), 2432-2445. doi: 10.1016/j.nonrwa.2012.02.012.  Google Scholar

[26]

H. Pan and R. Xing, Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions, J. Differential Equations, 254 (2013), 1464-1499. doi: 10.1016/j.jde.2012.10.025.  Google Scholar

[27]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[28]

J. A. Pelesko and T. A. Driscoll, The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models, J. Engrg. Math., 53 (2005), 239-252. doi: 10.1007/s10665-005-9013-2.  Google Scholar

[29]

J. Wei and D. Ye, On MEMS equation with fringing field, Proc. Amer. Math. Soc., 138 (2010), 1693-1699. doi: 10.1090/S0002-9939-09-10226-5.  Google Scholar

[30]

X. Zhang and M. Feng, Exact number of solutions of a one-dimensional prescribed mean curvature equation with concave-convex nonlinearities, J. Math. Anal. Appl., 395 (2013), 393-402. doi: 10.1016/j.jmaa.2012.05.053.  Google Scholar

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