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April  2011, 30(1): 227-241. doi: 10.3934/dcds.2011.30.227

Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity

Baishun Lai 1, and  Qing Luo 2,

1. 

Institute of Contemporary Mathematics, Henan University, School of Mathematics and Information Science, Henan University, Kaifeng 475004, China
2. 

School of Mathematics and Information Science, Henan University, Kaifeng 475004, China

Received  January 2010 Revised  May 2010 Published  February 2011

In this paper, we consider the relation between $p > 1$ and critical dimension of the extremal solution of the semilinear equation

$\beta \Delta^{2}u-\tau \Delta u=\frac{\lambda}{(1-u)^{p}} \mbox{in} B$,
$0 < u \leq 1 \mbox{in} B$,
$u=\Delta u=0 \mbox{on} \partial B$,

where $B$ is the unit ball in $R^{n}$, $\lambda>0$ is a parameter, $\tau>0, \beta>0,p>1$ are fixed constants. By Hardy-Rellich inequality, we find that when $p$ is large enough, the critical dimension is 13.

Keywords: regularity, critical dimension, fourth order., stability, Minimal solutions.
Mathematics Subject Classification: Primary 35B45; Secondary 35J4.
Citation: Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227
References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a forth order elliptic problem with singular nonlineartiy, Arch. Ration. Mech. Anal., 198 (2010), 763-787. doi: 10.1007/s00205-010-0367-x.  Google Scholar

[3]

M. G. Crandall and P. H. Rabinawitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal., 58 (1975), 207-218. doi: 10.1007/BF00280741.  Google Scholar

[4]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Research Monograph, under review (2007). Google Scholar

[5]

A. Ferrero and G. Warnault, On solutions of second and fourth order elliptic equations with power-type nonlinearities, Nonlinear Anal., 70 (2009), 2889-2902. doi: 10.1016/j.na.2008.12.041.  Google Scholar

[6]

Z. M. Guo and J. C. Wei, Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscript Math., 120 (2006), 193-209. doi: 10.1007/s00229-006-0001-2.  Google Scholar

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Z. Gui and J. Wei, On a fourth order nonlinear elliptic equation with negative exponent, SLAM J. Math. Anal., 40 (2009), 2034-2054. doi: 10.1137/070703375.  Google Scholar

[8]

H. Jian and F. Lin, Zero set of Sobolev functions with negative power of integrability, Chin. Ann. Math B, 25 (2004), 65-72. doi: 10.1142/S0252959904000068.  Google Scholar

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F. Lin and Y. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. A, 463 (2007), 1323-1337. doi: 10.1098/rspa.2007.1816.  Google Scholar

[10]

A. M. Meadows, Stable and singular solutions of the equation $\Delta u=\frac{1}{u}$, Indiana Univ Math. J., 53 (2004), 1681-1703. doi: 10.1512/iumj.2004.53.2560.  Google Scholar

[11]

A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. of Differential Equations, 248 (2010), 594-616. doi: 10.1016/j.jde.2009.09.011.  Google Scholar

[12]

F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung, in "Proceedings of the International Congress of Mathematicians Amsterdam (1954)" (J. C. H. Gerretsen et al. eds.), 3, Nordhoff, Groningen, (1956), 243-250.  Google Scholar

[13]

D. Ye and F. Zhou, On a general family of nonautonomous elliptic and parabolic equations, Calc. Var. and PDE, 37 (2010), 259-274. doi: 10.1007/s00526-009-0262-1.  Google Scholar

show all references

References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a forth order elliptic problem with singular nonlineartiy, Arch. Ration. Mech. Anal., 198 (2010), 763-787. doi: 10.1007/s00205-010-0367-x.  Google Scholar

[3]

M. G. Crandall and P. H. Rabinawitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal., 58 (1975), 207-218. doi: 10.1007/BF00280741.  Google Scholar

[4]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Research Monograph, under review (2007). Google Scholar

[5]

A. Ferrero and G. Warnault, On solutions of second and fourth order elliptic equations with power-type nonlinearities, Nonlinear Anal., 70 (2009), 2889-2902. doi: 10.1016/j.na.2008.12.041.  Google Scholar

[6]

Z. M. Guo and J. C. Wei, Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscript Math., 120 (2006), 193-209. doi: 10.1007/s00229-006-0001-2.  Google Scholar

[7]

Z. Gui and J. Wei, On a fourth order nonlinear elliptic equation with negative exponent, SLAM J. Math. Anal., 40 (2009), 2034-2054. doi: 10.1137/070703375.  Google Scholar

[8]

H. Jian and F. Lin, Zero set of Sobolev functions with negative power of integrability, Chin. Ann. Math B, 25 (2004), 65-72. doi: 10.1142/S0252959904000068.  Google Scholar

[9]

F. Lin and Y. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. A, 463 (2007), 1323-1337. doi: 10.1098/rspa.2007.1816.  Google Scholar

[10]

A. M. Meadows, Stable and singular solutions of the equation $\Delta u=\frac{1}{u}$, Indiana Univ Math. J., 53 (2004), 1681-1703. doi: 10.1512/iumj.2004.53.2560.  Google Scholar

[11]

A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. of Differential Equations, 248 (2010), 594-616. doi: 10.1016/j.jde.2009.09.011.  Google Scholar

[12]

F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung, in "Proceedings of the International Congress of Mathematicians Amsterdam (1954)" (J. C. H. Gerretsen et al. eds.), 3, Nordhoff, Groningen, (1956), 243-250.  Google Scholar

[13]

D. Ye and F. Zhou, On a general family of nonautonomous elliptic and parabolic equations, Calc. Var. and PDE, 37 (2010), 259-274. doi: 10.1007/s00526-009-0262-1.  Google Scholar

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