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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 896902, 11 pages
http://dx.doi.org/10.1155/2014/896902
Research Article

Some Remarks on Biharmonic Elliptic Problems with a Singular Nonlinearity

1Institute of Contemporary Mathematics, Henan University, Kaifeng 475004, China
2School of Mathematics and Information Science, Henan University, Kaifeng 475004, China

Received 3 December 2013; Accepted 26 February 2014; Published 31 March 2014

Academic Editor: Alberto Fiorenza

Copyright © 2014 Baishun Lai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. A. Pelesko and A. A. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC Press, 2002.
  2. M. Ghergu, “A biharmonic equation with singular nonlinearity,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 55, no. 1, pp. 155–166, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. D. Cassani, J. M. do Ó, and N. Ghoussoub, “On a fourth order elliptic problem with a singular nonlinearity,” Advanced Nonlinear Studies, vol. 9, no. 1, pp. 177–197, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. C. Cowan, P. Esposito, N. Ghoussoub, and A. Moradifam, “The critical dimension for a fourth order elliptic problem with singular nonlinearity,” Archive for Rational Mechanics and Analysis, vol. 198, no. 3, pp. 763–787, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Dávila, I. Flores, and I. Guerra, “Multiplicity of solutions for a fourth order equation with power-type nonlinearity,” Mathematische Annalen, vol. 348, no. 1, pp. 143–193, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. P. Esposito, N. Ghoussoub, and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, vol. 20 of American Mathematical Society, Courant Institute of Mathematical Sciences, New York, NY, USA, 2010. View at Zentralblatt MATH · View at MathSciNet
  7. Z. Guo and J. Wei, “On a fourth order nonlinear elliptic equation with negative exponent,” SIAM Journal on Mathematical Analysis, vol. 40, no. 5, pp. 2034–2054, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  8. F. Lin and Y. Yang, “Nonlinear non-local elliptic equation modelling electrostatic actuation,” Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, vol. 463, no. 2081, pp. 1323–1337, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. Moradifam, “On the critical dimension of a fourth order elliptic problem with negative exponent,” Journal of Differential Equations, vol. 248, no. 3, pp. 594–616, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. Boggio, “Sulle funzioni di green d'ordine m,” Rendiconti del Circolo Matematico di Palermo, vol. 20, pp. 97–135, 1905. View at Google Scholar
  11. A. M. Meadows, “Stable and singular solutions of the equation Δu=1/u,” Indiana University Mathematics Journal, vol. 53, no. 6, pp. 1681–1703, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,” Communications on Pure and Applied Mathematics, vol. 12, pp. 623–727, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. H. Brezis, T. Cazenave, Y. Martel, and A. Ramiandrisoa, “Blow up for ut-Δu=g(u) revisited,” Advances in Differential Equations, vol. 1, no. 1, pp. 73–90, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y. Martel, “Uniqueness of weak extremal solutions of nonlinear elliptic problems,” Houston Journal of Mathematics, vol. 23, no. 1, pp. 161–168, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J. Dávila, L. Dupaigne, I. Guerra, and M. Montenegro, “Stable solutions for the bilaplacian with exponential nonlinearity,” SIAM Journal on Mathematical Analysis, vol. 39, no. 2, pp. 565–592, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. G. Arioli, F. Gazzola, H.-C. Grunau, and E. Mitidieri, “A semilinear fourth order elliptic problem with exponential nonlinearity,” SIAM Journal on Mathematical Analysis, vol. 36, no. 4, pp. 1226–1258, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. H.-C. Grunau and G. Sweers, “Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions,” Mathematische Annalen, vol. 307, no. 4, pp. 589–626, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet