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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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