American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach
  • CPAA Home
  • This Issue
  • Next Article
    Global dynamics of a non-local delayed differential equation in the half plane
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

[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

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

[1]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[2]

Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677
[3]

Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027
[4]

Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521
[5]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[6]

Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828
[7]

Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771
[8]

Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733
[9]

Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure & Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465
[10]

Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25.

[11]

Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661
[12]

Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170
[13]

Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089
[14]

Tokushi Sato, Tatsuya Watanabe. Singular positive solutions for a fourth order elliptic problem in $R$. Communications on Pure & Applied Analysis, 2011, 10 (1) : 245-268. doi: 10.3934/cpaa.2011.10.245
[15]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085
[16]

Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126
[17]

Jens Lorenz, Wilberclay G. Melo, Natã Firmino Rocha. The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3819-3841. doi: 10.3934/dcdsb.2018332
[18]

M. Grossi, P. Magrone, M. Matzeu. Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 703-718. doi: 10.3934/dcds.2001.7.703
[19]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[20]

Alan E. Lindsay. An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 189-215. doi: 10.3934/dcdsb.2014.19.189

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences