December 2018, 23(10): 4187-4205. doi: 10.3934/dcdsb.2018132

A perturbed fourth order elliptic equation with negative exponent

1. 

Department of Mathematics, Henan Normal University, Xinxiang 453007, China

2. 

Department of Mathematics, Hangzhou Dianzi University, Zhejiang 310018, China

* Corresponding author

Received  August 2017 Revised  December 2017 Published  April 2018

Fund Project: The research of the first author is supported by NSFC (11571093)

By a new type of comparison principle for a fourth order elliptic problem in general domains, we investigate the structure of positive solutions to Navier boundary value problems of a perturbed fourth order elliptic equation with negative exponent, which arises in the study of the deflection of charged plates in electrostatic actuators in the modeling of electrostatic micro-electromechanical systems (MEMS). It is seen that the structure of solutions relies on the boundary values. The global branches of solutions to the Navier boundary value problems are established. We also show that the behaviors of these branches are relatively "stable" with respect to the Navier boundary values.

Citation: Zongming Guo, Long Wei. A perturbed fourth order elliptic equation with negative exponent. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4187-4205. doi: 10.3934/dcdsb.2018132
References:
[1]

C. CowanP. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, Discrete Contin. Dyn. Syst., 28 (2010), 1033-1050. doi: 10.3934/dcds.2010.28.1033.

[2]

C. CowanP. EspositoN. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., 198 (2010), 763-787. doi: 10.1007/s00205-010-0367-x.

[3]

J. DávilaI Flores and I. Guerra, Multiplicity of solutions for a fourth order problem with power-type nonlinearity, Math. Ann., 348 (2010), 143-193. doi: 10.1007/s00208-009-0476-8.

[4]

J. Dávila and D. Ye, On finite Morse index solutions of two equations with negative exponent, Proc. R. Soc. Edinb., 143 (2013), 121-128. doi: 10.1017/S0308210511001144.

[5]

J. D. DiazJ. M. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations, 12 (1987), 1333-1344. doi: 10.1080/03605308708820531.

[6]

P. Esposito, Compactness of a nonlinear eigenvalue problem with a singular nonlinearity, Comm. Contemp. Math., 10 (2008), 17-45. doi: 10.1142/S0219199708002697.

[7]

P. EspositoN. Ghoussoub and Y. Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2007), 1731-1768. doi: 10.1002/cpa.20189.

[8]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differetial Equations Modeling Electrostatic MEMS, Courant Lecture Notes in Mathematics, 20 (2010), Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010.

[9]

G. FloresG. A. Mercado and J. A. Pelesko, Dynamics and touchdown in electrostatic MEMS, Proceedings of ICMEMS, (2003), 182-187.

[10]

J. A. GaticaV. Oliker and P. Walyman, Singular nonlinear boundary value problems for second-order ordinary differential equations, J. Differential Equations, 79 (1989), 62-78. doi: 10.1016/0022-0396(89)90113-7.

[11]

J. A. GaticaG. E. Hernandez and P. Walyman, Radially symmetric solutions of a class of sigular elliptic equations, Proc. Edinburgh Math. Soc., 33 (1990), 169-180. doi: 10.1017/S0013091500018101.

[12]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math.Anal., 38 (2007), 1423-1449.

[13]

I. Guerra, A note on nonlinear biharmonic equations with negative exponents, J. Differential Equations, 253 (2012), 3147-3157. doi: 10.1016/j.jde.2012.08.037.

[14]

Z. M. Guo and L. Ma, Finite Morse index steady states of van der Waals force driven thin film equations, J. Math. Anal. Appl., 368 (2010), 559-572. doi: 10.1016/j.jmaa.2010.04.012.

[15]

Z. M. Guo and X. Z. Peng, On the structure of positive solutions to an elliptic problem with a singular nonlinearity, J. Math. Anal.Appl., 354 (2009), 134-146. doi: 10.1016/j.jmaa.2009.01.001.

[16]

Z. M. Guo and J. C. Wei, Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math., 120 (2006), 193-209. doi: 10.1007/s00229-006-0001-2.

[17]

Z. M. Guo and J. C. Wei, Symmetry of non-negative solutions of a semilinear elliptic equation with singular nonlinearity, Proc. R. Soc. Edinburgh, 137 (2007), 963-994. doi: 10.1017/S0308210505001083.

[18]

Z. M. Guo and J. C. Wei, The Cauchy problem for a reaction-diffusion equation with a singular nonlinearity, J. Differential Equations, 240 (2007), 279-323. doi: 10.1016/j.jde.2007.06.012.

[19]

Z. M. Guo and J. C. Wei, Infinitely many turning points for an elliptic problem with a singular nonlinearity, J. London Math. Soc., 78 (2008), 21-35. doi: 10.1112/jlms/jdm121.

[20]

Z. M. Guo and J. C. Wei, Rupture solutions of an elliptic equation with a singular nonlinearity, Proc. R. Soc. Edinb., 144 (2014), 905-924. doi: 10.1017/S0308210512001151.

[21]

Z. M. Guo and J. C. Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal., 40 (2009), 2034-2054.

[22]

Z. M. Guo and J. C. Wei, Entire solutions and global bifurcation for a biharmonic equation with singular nonlinearity in $\mathbb{R}^3$, Adv. Differential Equations, 13 (2008), 753-780.

[23]

Z. M. Guo and J. C. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equations with negative exponents, Disc. Conti. Dyn. Syst., 34 (2014), 2561-2580.

[24]

Z. M. Guo and L. Wei, A fourth order elliptic equation with a singular nonlinearity, Comm. Pure Appl. Anal., 13 (2014), 2493-2508. doi: 10.3934/cpaa.2014.13.2493.

[25]

Z. M. GuoB. S. Lai and D. Ye, Revisiting the biharmonic equation modeling electrostatic actuation in lower dimensions, Proc. Amer. Math. Soc., 142 (2014), 2027-2034. doi: 10.1090/S0002-9939-2014-11895-8.

[26]

Z. M. GuoD. Ye and F. Zhou, Existence of singular positive solutions for some semilinear elliptic equations, Pacific J. Math., 236 (2008), 57-71. doi: 10.2140/pjm.2008.236.57.

[27]

Z. M. Guo and Y. T. Yu, Boundary value problems for a semilinear elliptic equation with singular nonlinearity, Comm. Pure Appl. Anal., 15 (2016), 399-412. doi: 10.3934/cpaa.2016.15.399.

[28]

H. Jiang and W. M. Ni, On steady states of van der Waals force driven thin film equations, European J. Appl. Math., 18 (2007), 153-180. doi: 10.1017/S0956792507006936.

[29]

B. S. Lai and D. Ye, Remarks on entire solutions for fourth-order elliptic problems, Proc. Edinb. Math. Soc., 59 (2016), 777-786.

[30]

F. H. Lin and Y. S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. Lond., Ser. A Math. Phys. Eng. Sci., 463 (2007), 1323-1337. doi: 10.1098/rspa.2007.1816.

[31]

X. LuoD. Ye and F. Zhou, Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection, J. Differential Equations, 251 (2011), 2082-2099. doi: 10.1016/j.jde.2011.07.011.

[32]

A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. Differential Equations, 248 (2010), 594-616. doi: 10.1016/j.jde.2009.09.011.

[33]

A. Nachman and A. Callegari, A nonlinear boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275-281. doi: 10.1137/0138024.

[34]

J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908.

[35]

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

show all references

References:
[1]

C. CowanP. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, Discrete Contin. Dyn. Syst., 28 (2010), 1033-1050. doi: 10.3934/dcds.2010.28.1033.

[2]

C. CowanP. EspositoN. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., 198 (2010), 763-787. doi: 10.1007/s00205-010-0367-x.

[3]

J. DávilaI Flores and I. Guerra, Multiplicity of solutions for a fourth order problem with power-type nonlinearity, Math. Ann., 348 (2010), 143-193. doi: 10.1007/s00208-009-0476-8.

[4]

J. Dávila and D. Ye, On finite Morse index solutions of two equations with negative exponent, Proc. R. Soc. Edinb., 143 (2013), 121-128. doi: 10.1017/S0308210511001144.

[5]

J. D. DiazJ. M. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations, 12 (1987), 1333-1344. doi: 10.1080/03605308708820531.

[6]

P. Esposito, Compactness of a nonlinear eigenvalue problem with a singular nonlinearity, Comm. Contemp. Math., 10 (2008), 17-45. doi: 10.1142/S0219199708002697.

[7]

P. EspositoN. Ghoussoub and Y. Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2007), 1731-1768. doi: 10.1002/cpa.20189.

[8]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differetial Equations Modeling Electrostatic MEMS, Courant Lecture Notes in Mathematics, 20 (2010), Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010.

[9]

G. FloresG. A. Mercado and J. A. Pelesko, Dynamics and touchdown in electrostatic MEMS, Proceedings of ICMEMS, (2003), 182-187.

[10]

J. A. GaticaV. Oliker and P. Walyman, Singular nonlinear boundary value problems for second-order ordinary differential equations, J. Differential Equations, 79 (1989), 62-78. doi: 10.1016/0022-0396(89)90113-7.

[11]

J. A. GaticaG. E. Hernandez and P. Walyman, Radially symmetric solutions of a class of sigular elliptic equations, Proc. Edinburgh Math. Soc., 33 (1990), 169-180. doi: 10.1017/S0013091500018101.

[12]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math.Anal., 38 (2007), 1423-1449.

[13]

I. Guerra, A note on nonlinear biharmonic equations with negative exponents, J. Differential Equations, 253 (2012), 3147-3157. doi: 10.1016/j.jde.2012.08.037.

[14]

Z. M. Guo and L. Ma, Finite Morse index steady states of van der Waals force driven thin film equations, J. Math. Anal. Appl., 368 (2010), 559-572. doi: 10.1016/j.jmaa.2010.04.012.

[15]

Z. M. Guo and X. Z. Peng, On the structure of positive solutions to an elliptic problem with a singular nonlinearity, J. Math. Anal.Appl., 354 (2009), 134-146. doi: 10.1016/j.jmaa.2009.01.001.

[16]

Z. M. Guo and J. C. Wei, Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math., 120 (2006), 193-209. doi: 10.1007/s00229-006-0001-2.

[17]

Z. M. Guo and J. C. Wei, Symmetry of non-negative solutions of a semilinear elliptic equation with singular nonlinearity, Proc. R. Soc. Edinburgh, 137 (2007), 963-994. doi: 10.1017/S0308210505001083.

[18]

Z. M. Guo and J. C. Wei, The Cauchy problem for a reaction-diffusion equation with a singular nonlinearity, J. Differential Equations, 240 (2007), 279-323. doi: 10.1016/j.jde.2007.06.012.

[19]

Z. M. Guo and J. C. Wei, Infinitely many turning points for an elliptic problem with a singular nonlinearity, J. London Math. Soc., 78 (2008), 21-35. doi: 10.1112/jlms/jdm121.

[20]

Z. M. Guo and J. C. Wei, Rupture solutions of an elliptic equation with a singular nonlinearity, Proc. R. Soc. Edinb., 144 (2014), 905-924. doi: 10.1017/S0308210512001151.

[21]

Z. M. Guo and J. C. Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal., 40 (2009), 2034-2054.

[22]

Z. M. Guo and J. C. Wei, Entire solutions and global bifurcation for a biharmonic equation with singular nonlinearity in $\mathbb{R}^3$, Adv. Differential Equations, 13 (2008), 753-780.

[23]

Z. M. Guo and J. C. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equations with negative exponents, Disc. Conti. Dyn. Syst., 34 (2014), 2561-2580.

[24]

Z. M. Guo and L. Wei, A fourth order elliptic equation with a singular nonlinearity, Comm. Pure Appl. Anal., 13 (2014), 2493-2508. doi: 10.3934/cpaa.2014.13.2493.

[25]

Z. M. GuoB. S. Lai and D. Ye, Revisiting the biharmonic equation modeling electrostatic actuation in lower dimensions, Proc. Amer. Math. Soc., 142 (2014), 2027-2034. doi: 10.1090/S0002-9939-2014-11895-8.

[26]

Z. M. GuoD. Ye and F. Zhou, Existence of singular positive solutions for some semilinear elliptic equations, Pacific J. Math., 236 (2008), 57-71. doi: 10.2140/pjm.2008.236.57.

[27]

Z. M. Guo and Y. T. Yu, Boundary value problems for a semilinear elliptic equation with singular nonlinearity, Comm. Pure Appl. Anal., 15 (2016), 399-412. doi: 10.3934/cpaa.2016.15.399.

[28]

H. Jiang and W. M. Ni, On steady states of van der Waals force driven thin film equations, European J. Appl. Math., 18 (2007), 153-180. doi: 10.1017/S0956792507006936.

[29]

B. S. Lai and D. Ye, Remarks on entire solutions for fourth-order elliptic problems, Proc. Edinb. Math. Soc., 59 (2016), 777-786.

[30]

F. H. Lin and Y. S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. Lond., Ser. A Math. Phys. Eng. Sci., 463 (2007), 1323-1337. doi: 10.1098/rspa.2007.1816.

[31]

X. LuoD. Ye and F. Zhou, Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection, J. Differential Equations, 251 (2011), 2082-2099. doi: 10.1016/j.jde.2011.07.011.

[32]

A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. Differential Equations, 248 (2010), 594-616. doi: 10.1016/j.jde.2009.09.011.

[33]

A. Nachman and A. Callegari, A nonlinear boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275-281. doi: 10.1137/0138024.

[34]

J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908.

[35]

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

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