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November  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, Europhys. Lett., 77 (2007), 20010. Google Scholar

[2]

D. S. Cohen and J. D. Murry, A generalized diffusion model for growth and dispersal in a population, J. Math. Biology, 12 (1981), 237-249. doi: 10.1007/BF00276132.  Google Scholar

[3]

A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica D, 10 (1984), 277-298. doi: 10.1016/0167-2789(84)90180-5.  Google Scholar

[4]

C. Cowan, P. 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

[5]

C. Cowan, P. Esposito, N. 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

[6]

E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. Lodon Math. Soc., 53 (1986), 429-452. doi: 10.1112/plms/s3-53.3.429.  Google Scholar

[7]

E. N. Dancer, Moving plane methods for systems on half spaces, Math. Ann., 342 (2008), 245-254. doi: 10.1007/s00208-008-0226-3.  Google Scholar

[8]

E. N. Dancer, Infinitely many turning points for some supercritical problems, Ann. Math. Pura Appl., 178 (2000), 225-233. doi: 10.1007/BF02505896.  Google Scholar

[9]

Z. M. Guo and X. F. Bai, On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity, Commun. Pure Appl. Anal., 7 (2008), 1091-1107. doi: 10.3934/cpaa.2008.7.1091.  Google Scholar

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Z. M. Guo and Z. Y. Liu, Further study of a fourth-order elliptic equation with negative exponent, Proc. R. Soc. Edinb. A, 141 (2011), 537-549. doi: 10.1017/S0308210509001061.  Google Scholar

[11]

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

[12]

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. doi: 10.1137/070703375.  Google Scholar

[13]

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

[14]

R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations, Arch. Ration. Mech. Anal., 154 (2000), 3-51. doi: 10.1007/PL00004234.  Google Scholar

[15]

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

[16]

M. Moghimi Zand and M. T. Ahmadian, Dynamic pull-in instability of electrostatically actuated beams incorporating Casimir and van der Waals forces, J. Mechanical Engineering Science, 224 (2010), 2037-2047. Google Scholar

[17]

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

[18]

P. Polácik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

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G. I. Sivashinsky, On cellular instability in the solidification of a dilute binary alloy, Phys. D, 8 (1983), 243-248. doi: 10.1016/0167-2789(83)90321-4.  Google Scholar

[20]

C. P. Vyasarayani, E. M. Abdel-Rahman and J. McPhee, Modeling of Contact and Stiction in Electrostatic Microcantilever Actuators, J. Nanotechnol. Eng. Med., 3 (2012), 011003 (8 pages). Google Scholar

show all references

References:
[1]

R. Batra, M. Porfiri and D. Spinello, Effects of Casimir force on pull-in instability in micromembranes, Europhys. Lett., 77 (2007), 20010. Google Scholar

[2]

D. S. Cohen and J. D. Murry, A generalized diffusion model for growth and dispersal in a population, J. Math. Biology, 12 (1981), 237-249. doi: 10.1007/BF00276132.  Google Scholar

[3]

A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica D, 10 (1984), 277-298. doi: 10.1016/0167-2789(84)90180-5.  Google Scholar

[4]

C. Cowan, P. 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

[5]

C. Cowan, P. Esposito, N. 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

[6]

E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. Lodon Math. Soc., 53 (1986), 429-452. doi: 10.1112/plms/s3-53.3.429.  Google Scholar

[7]

E. N. Dancer, Moving plane methods for systems on half spaces, Math. Ann., 342 (2008), 245-254. doi: 10.1007/s00208-008-0226-3.  Google Scholar

[8]

E. N. Dancer, Infinitely many turning points for some supercritical problems, Ann. Math. Pura Appl., 178 (2000), 225-233. doi: 10.1007/BF02505896.  Google Scholar

[9]

Z. M. Guo and X. F. Bai, On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity, Commun. Pure Appl. Anal., 7 (2008), 1091-1107. doi: 10.3934/cpaa.2008.7.1091.  Google Scholar

[10]

Z. M. Guo and Z. Y. Liu, Further study of a fourth-order elliptic equation with negative exponent, Proc. R. Soc. Edinb. A, 141 (2011), 537-549. doi: 10.1017/S0308210509001061.  Google Scholar

[11]

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

[12]

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. doi: 10.1137/070703375.  Google Scholar

[13]

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

[14]

R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations, Arch. Ration. Mech. Anal., 154 (2000), 3-51. doi: 10.1007/PL00004234.  Google Scholar

[15]

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

[16]

M. Moghimi Zand and M. T. Ahmadian, Dynamic pull-in instability of electrostatically actuated beams incorporating Casimir and van der Waals forces, J. Mechanical Engineering Science, 224 (2010), 2037-2047. Google Scholar

[17]

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

[18]

P. Polácik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[19]

G. I. Sivashinsky, On cellular instability in the solidification of a dilute binary alloy, Phys. D, 8 (1983), 243-248. doi: 10.1016/0167-2789(83)90321-4.  Google Scholar

[20]

C. P. Vyasarayani, E. M. Abdel-Rahman and J. McPhee, Modeling of Contact and Stiction in Electrostatic Microcantilever Actuators, J. Nanotechnol. Eng. Med., 3 (2012), 011003 (8 pages). Google Scholar

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