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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

 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.

Citation: Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete and 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. [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. [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. [4] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Research Monograph, under review (2007). [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. [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. [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. [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. [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. [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. [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. [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. [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.

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