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2014, 13(6): 2493-2508. doi: 10.3934/cpaa.2014.13.2493

A fourth order elliptic equation with a singular nonlinearity

Zongming Guo 1, and  Long Wei 2,

1. 

Department of Mathematics, Henan Normal University, Xinxiang, 453007, China
2. 

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

Received  January 2014 Revised  May 2014 Published  July 2014

In this paper, we study the structure of solutions of a fourth order elliptic equation with a singular nonlinearity. For different boundary values $\kappa$, we establish the global bifurcation branches of solutions to the equation. More precisely, we show that $\kappa=1$ is a critical boundary value to change the structure of solutions to this problem.
Keywords: singular nonlinearities, Fourth order elliptic equations, branches of solutions., blow-up arguments.
Mathematics Subject Classification: Primary: 35B45, 35J4.
Citation: Zongming Guo, Long Wei. A fourth order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2493-2508. doi: 10.3934/cpaa.2014.13.2493
References:
[1]

R. Batra, M. Porfiri and D. Spinello, Effects of Casimir force on pull-in instability in micromembranes,, \emph{Europhys. Lett.}, 77 (2007).

[2]

D. S. Cohen and J. D. Murry, A generalized diffusion model for growth and dispersal in a population,, \emph{J. Math. Biology}, 12 (1981), 237. doi: 10.1007/BF00276132.

[3]

A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation,, \emph{Physica D}, 10 (1984), 277. doi: 10.1016/0167-2789(84)90180-5.

[4]

C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1033. doi: 10.3934/dcds.2010.28.1033.

[5]

C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity,, \emph{Arch. Ration. Mech. Anal.}, 198 (2010), 763. doi: 10.1007/s00205-010-0367-x.

[6]

E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large,, \emph{Proc. Lodon Math. Soc.}, 53 (1986), 429. doi: 10.1112/plms/s3-53.3.429.

[7]

E. N. Dancer, Moving plane methods for systems on half spaces,, \emph{Math. Ann.}, 342 (2008), 245. doi: 10.1007/s00208-008-0226-3.

[8]

E. N. Dancer, Infinitely many turning points for some supercritical problems,, \emph{Ann. Math. Pura Appl.}, 178 (2000), 225. doi: 10.1007/BF02505896.

[9]

Z. M. Guo and X. F. Bai, On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity,, \emph{Commun. Pure Appl. Anal.}, 7 (2008), 1091. doi: 10.3934/cpaa.2008.7.1091.

[10]

Z. M. Guo and Z. Y. Liu, Further study of a fourth-order elliptic equation with negative exponent,, \emph{Proc. R. Soc. Edinb. A}, 141 (2011), 537. doi: 10.1017/S0308210509001061.

[11]

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

[12]

Z. M. Guo and J. C. Wei, On a fourth order nonlinear elliptic equation with negative exponent,, \emph{SIAM J. Math. Anal.}, 40 (2009), 2034. doi: 10.1137/070703375.

[13]

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

[14]

R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations,, \emph{Arch. Ration. Mech. Anal.}, 154 (2000), 3. doi: 10.1007/PL00004234.

[15]

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

[16]

M. Moghimi Zand and M. T. Ahmadian, Dynamic pull-in instability of electrostatically actuated beams incorporating Casimir and van der Waals forces,, \emph{J. Mechanical Engineering Science}, 224 (2010), 2037.

[17]

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

[18]

P. Polácik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8.

[19]

G. I. Sivashinsky, On cellular instability in the solidification of a dilute binary alloy,, \emph{Phys. D}, 8 (1983), 243. doi: 10.1016/0167-2789(83)90321-4.

[20]

C. P. Vyasarayani, E. M. Abdel-Rahman and J. McPhee, Modeling of Contact and Stiction in Electrostatic Microcantilever Actuators,, \emph{J. Nanotechnol. Eng. Med.}, 3 (2012).

show all references

References:
[1]

R. Batra, M. Porfiri and D. Spinello, Effects of Casimir force on pull-in instability in micromembranes,, \emph{Europhys. Lett.}, 77 (2007).

[2]

D. S. Cohen and J. D. Murry, A generalized diffusion model for growth and dispersal in a population,, \emph{J. Math. Biology}, 12 (1981), 237. doi: 10.1007/BF00276132.

[3]

A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation,, \emph{Physica D}, 10 (1984), 277. doi: 10.1016/0167-2789(84)90180-5.

[4]

C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1033. doi: 10.3934/dcds.2010.28.1033.

[5]

C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity,, \emph{Arch. Ration. Mech. Anal.}, 198 (2010), 763. doi: 10.1007/s00205-010-0367-x.

[6]

E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large,, \emph{Proc. Lodon Math. Soc.}, 53 (1986), 429. doi: 10.1112/plms/s3-53.3.429.

[7]

E. N. Dancer, Moving plane methods for systems on half spaces,, \emph{Math. Ann.}, 342 (2008), 245. doi: 10.1007/s00208-008-0226-3.

[8]

E. N. Dancer, Infinitely many turning points for some supercritical problems,, \emph{Ann. Math. Pura Appl.}, 178 (2000), 225. doi: 10.1007/BF02505896.

[9]

Z. M. Guo and X. F. Bai, On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity,, \emph{Commun. Pure Appl. Anal.}, 7 (2008), 1091. doi: 10.3934/cpaa.2008.7.1091.

[10]

Z. M. Guo and Z. Y. Liu, Further study of a fourth-order elliptic equation with negative exponent,, \emph{Proc. R. Soc. Edinb. A}, 141 (2011), 537. doi: 10.1017/S0308210509001061.

[11]

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

[12]

Z. M. Guo and J. C. Wei, On a fourth order nonlinear elliptic equation with negative exponent,, \emph{SIAM J. Math. Anal.}, 40 (2009), 2034. doi: 10.1137/070703375.

[13]

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

[14]

R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations,, \emph{Arch. Ration. Mech. Anal.}, 154 (2000), 3. doi: 10.1007/PL00004234.

[15]

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

[16]

M. Moghimi Zand and M. T. Ahmadian, Dynamic pull-in instability of electrostatically actuated beams incorporating Casimir and van der Waals forces,, \emph{J. Mechanical Engineering Science}, 224 (2010), 2037.

[17]

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

[18]

P. Polácik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8.

[19]

G. I. Sivashinsky, On cellular instability in the solidification of a dilute binary alloy,, \emph{Phys. D}, 8 (1983), 243. doi: 10.1016/0167-2789(83)90321-4.

[20]

C. P. Vyasarayani, E. M. Abdel-Rahman and J. McPhee, Modeling of Contact and Stiction in Electrostatic Microcantilever Actuators,, \emph{J. Nanotechnol. Eng. Med.}, 3 (2012).

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