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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 - A, 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.

[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. doi: 10.1016/j.jde.2007.05.031.

[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, ().

[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. doi: 10.1017/S0956792513000077.

[5]

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

[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. doi: 10.1016/j.na.2012.04.025.

[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. doi: 10.1017/S0956792512000435.

[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. doi: 10.1017/S0956792511000076.

[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. doi: 10.1016/j.na.2013.04.018.

[10]

C.-H. Chuang, On Exact Multiplicity and Bifurcation Diagrams of Positive Solutions of a One-Dimensional Prescribed Mean Curvature Problem,, Master Thesis, (2011).

[11]

J. Dávila and J. Wei, Point ruptures for a MEMS equation with fringing field,, Comm. Partial Differential Equations, 37 (2012), 1462. doi: 10.1080/03605302.2012.679990.

[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.

[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., (2010).

[14]

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

[15]

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

[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. doi: 10.1017/S0308210509001449.

[17]

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

[18]

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

[19]

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

[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. doi: 10.4310/MAA.2008.v15.n3.a4.

[21]

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

[22]

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

[23]

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

[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. doi: 10.1016/j.na.2011.03.020.

[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. doi: 10.1016/j.nonrwa.2012.02.012.

[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. doi: 10.1016/j.jde.2012.10.025.

[27]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS,, Chapman & Hall/CRC, (2003).

[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. doi: 10.1007/s10665-005-9013-2.

[29]

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

[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. doi: 10.1016/j.jmaa.2012.05.053.

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.

[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. doi: 10.1016/j.jde.2007.05.031.

[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, ().

[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. doi: 10.1017/S0956792513000077.

[5]

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

[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. doi: 10.1016/j.na.2012.04.025.

[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. doi: 10.1017/S0956792512000435.

[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. doi: 10.1017/S0956792511000076.

[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. doi: 10.1016/j.na.2013.04.018.

[10]

C.-H. Chuang, On Exact Multiplicity and Bifurcation Diagrams of Positive Solutions of a One-Dimensional Prescribed Mean Curvature Problem,, Master Thesis, (2011).

[11]

J. Dávila and J. Wei, Point ruptures for a MEMS equation with fringing field,, Comm. Partial Differential Equations, 37 (2012), 1462. doi: 10.1080/03605302.2012.679990.

[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.

[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., (2010).

[14]

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

[15]

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

[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. doi: 10.1017/S0308210509001449.

[17]

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

[18]

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

[19]

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

[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. doi: 10.4310/MAA.2008.v15.n3.a4.

[21]

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

[22]

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

[23]

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

[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. doi: 10.1016/j.na.2011.03.020.

[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. doi: 10.1016/j.nonrwa.2012.02.012.

[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. doi: 10.1016/j.jde.2012.10.025.

[27]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS,, Chapman & Hall/CRC, (2003).

[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. doi: 10.1007/s10665-005-9013-2.

[29]

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

[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. doi: 10.1016/j.jmaa.2012.05.053.

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