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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[35]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[35]

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

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