-
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
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 |
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). |
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. |
[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). |
[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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]