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A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach
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,, \emph{Europhys. Lett.}, 77 (2007). |
[2] |
D. S. Cohen and J. D. Murry, A generalized diffusion model for growth and dispersal in a population,, \emph{J. Math. Biology}, 12 (1981), 237.
doi: 10.1007/BF00276132. |
[3] |
A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation,, \emph{Physica D}, 10 (1984), 277.
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,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1033.
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,, \emph{Arch. Ration. Mech. Anal.}, 198 (2010), 763.
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,, \emph{Proc. Lodon Math. Soc.}, 53 (1986), 429.
doi: 10.1112/plms/s3-53.3.429. |
[7] |
E. N. Dancer, Moving plane methods for systems on half spaces,, \emph{Math. Ann.}, 342 (2008), 245.
doi: 10.1007/s00208-008-0226-3. |
[8] |
E. N. Dancer, Infinitely many turning points for some supercritical problems,, \emph{Ann. Math. Pura Appl.}, 178 (2000), 225.
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,, \emph{Commun. Pure Appl. Anal.}, 7 (2008), 1091.
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,, \emph{Proc. R. Soc. Edinb. A}, 141 (2011), 537.
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,, \emph{J. Math. Anal. Appl.}, 354 (2009), 134.
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,, \emph{SIAM J. Math. Anal.}, 40 (2009), 2034.
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 $\mathbbR^3$,, \emph{Adv. Differential Equations}, 13 (2008), 753.
|
[14] |
R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations,, \emph{Arch. Ration. Mech. Anal.}, 154 (2000), 3.
doi: 10.1007/PL00004234. |
[15] |
F. H. Lin and Y. S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation,, \emph{Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, 463 (2007), 1323.
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,, \emph{J. Mechanical Engineering Science}, 224 (2010), 2037. |
[17] |
A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent,, \emph{J. Differential Equations}, 248 (2010), 594.
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,, \emph{Duke Math. J.}, 139 (2007), 555.
doi: 10.1215/S0012-7094-07-13935-8. |
[19] |
G. I. Sivashinsky, On cellular instability in the solidification of a dilute binary alloy,, \emph{Phys. D}, 8 (1983), 243.
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,, \emph{J. Nanotechnol. Eng. Med.}, 3 (2012). |
show all references
References:
[1] |
R. Batra, M. Porfiri and D. Spinello, Effects of Casimir force on pull-in instability in micromembranes,, \emph{Europhys. Lett.}, 77 (2007). |
[2] |
D. S. Cohen and J. D. Murry, A generalized diffusion model for growth and dispersal in a population,, \emph{J. Math. Biology}, 12 (1981), 237.
doi: 10.1007/BF00276132. |
[3] |
A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation,, \emph{Physica D}, 10 (1984), 277.
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,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1033.
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,, \emph{Arch. Ration. Mech. Anal.}, 198 (2010), 763.
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,, \emph{Proc. Lodon Math. Soc.}, 53 (1986), 429.
doi: 10.1112/plms/s3-53.3.429. |
[7] |
E. N. Dancer, Moving plane methods for systems on half spaces,, \emph{Math. Ann.}, 342 (2008), 245.
doi: 10.1007/s00208-008-0226-3. |
[8] |
E. N. Dancer, Infinitely many turning points for some supercritical problems,, \emph{Ann. Math. Pura Appl.}, 178 (2000), 225.
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,, \emph{Commun. Pure Appl. Anal.}, 7 (2008), 1091.
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,, \emph{Proc. R. Soc. Edinb. A}, 141 (2011), 537.
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,, \emph{J. Math. Anal. Appl.}, 354 (2009), 134.
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,, \emph{SIAM J. Math. Anal.}, 40 (2009), 2034.
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 $\mathbbR^3$,, \emph{Adv. Differential Equations}, 13 (2008), 753.
|
[14] |
R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations,, \emph{Arch. Ration. Mech. Anal.}, 154 (2000), 3.
doi: 10.1007/PL00004234. |
[15] |
F. H. Lin and Y. S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation,, \emph{Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, 463 (2007), 1323.
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,, \emph{J. Mechanical Engineering Science}, 224 (2010), 2037. |
[17] |
A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent,, \emph{J. Differential Equations}, 248 (2010), 594.
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,, \emph{Duke Math. J.}, 139 (2007), 555.
doi: 10.1215/S0012-7094-07-13935-8. |
[19] |
G. I. Sivashinsky, On cellular instability in the solidification of a dilute binary alloy,, \emph{Phys. D}, 8 (1983), 243.
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,, \emph{J. Nanotechnol. Eng. Med.}, 3 (2012). |
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