A fourth order elliptic equation with a singular nonlinearity

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	E. N. Dancer, Infinitely many turning points for some supercritical problems, Ann. Math. Pura Appl., 178 (2000), 225-233. 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, Commun. Pure Appl. Anal., 7 (2008), 1091-1107. 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, Proc. R. Soc. Edinb. A, 141 (2011), 537-549. 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, J. Math. Anal. Appl., 354 (2009), 134-146. 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, SIAM J. Math. Anal., 40 (2009), 2034-2054. 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 $\mathbb{R}^3$, Adv. Differential Equations, 13 (2008), 753-780.
[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.
[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.
[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.
[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.
[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.
[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.
[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).

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References:

[1]	R. Batra, M. Porfiri and D. Spinello, Effects of Casimir force on pull-in instability in micromembranes, Europhys. Lett., 77 (2007), 20010.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	E. N. Dancer, Infinitely many turning points for some supercritical problems, Ann. Math. Pura Appl., 178 (2000), 225-233. 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, Commun. Pure Appl. Anal., 7 (2008), 1091-1107. 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, Proc. R. Soc. Edinb. A, 141 (2011), 537-549. 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, J. Math. Anal. Appl., 354 (2009), 134-146. 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, SIAM J. Math. Anal., 40 (2009), 2034-2054. 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 $\mathbb{R}^3$, Adv. Differential Equations, 13 (2008), 753-780.
[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.
[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.
[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.
[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.
[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.
[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.
[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).

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