# American Institute of Mathematical Sciences

• 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

 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.
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] Jinxing Liu, Xiongrui Wang, Jun Zhou, Huan Zhang. Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4321-4335. doi: 10.3934/dcdss.2021108 [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] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] 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 [20] 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

2020 Impact Factor: 1.916