Abstract and Applied Analysis

Multiplicity of Solutions for Perturbed Nonhomogeneous Neumann Problem through Orlicz-Sobolev Spaces

Liu Yang

Full-text: Open access

Abstract

We investigate the existence of multiple solutions for a class of nonhomogeneous Neumann problem with a perturbed term. By using variational methods and three critical point theorems of B. Ricceri, we establish some new sufficient conditions under which such a problem possesses three solutions in an appropriate Orlicz-Sobolev space.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 236712, 10 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365168327

Digital Object Identifier
doi:10.1155/2012/236712

Mathematical Reviews number (MathSciNet)
MR2984020

Zentralblatt MATH identifier
1256.49011

Citation

Yang, Liu. Multiplicity of Solutions for Perturbed Nonhomogeneous Neumann Problem through Orlicz-Sobolev Spaces. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 236712, 10 pages. doi:10.1155/2012/236712. https://projecteuclid.org/euclid.aaa/1365168327


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References

  • V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity,” Mathematics of the USSR-Izvestiya, vol. 29, pp. 33–66, 1987.
  • T. C. Halsey, “Electrorheological fluids,” Science, vol. 258, no. 5083, pp. 761–766, 1992.
  • M. Mihăilescu and V. Rădulescu, “A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids,” Proceedings of The Royal Society of London A, vol. 462, no. 2073, pp. 2625–2641, 2006.
  • Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406, 2006.
  • H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, Japan, 1950.
  • J. Musielak and W. Orlicz, “On modular spaces,” Studia Mathematica, vol. 18, pp. 49–65, 1959.
  • R. A. Adams, “On the Orlicz-Sobolev imbedding theorem,” vol. 24, no. 3, pp. 241–257, 1977.
  • A. Cianchi, “A sharp embedding theorem for Orlicz-Sobolev spaces,” Indiana University Mathematics Journal, vol. 45, no. 1, pp. 39–65, 1996.
  • T. K. Donaldson and N. S. Trudinger, “Orlicz-Sobolev spaces and imbedding theorems,” vol. 8, pp. 52–75, 1971.
  • M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, vol. 146, Marcel Dekker, New York, NY, USA, 1991.
  • P. Clément, B. de Pagter, G. Sweers, and F. de Thélin, “Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces,” Mediterranean Journal of Mathematics, vol. 1, no. 3, pp. 241–267, 2004.
  • M. Mihăilescu and D. Repovš, “Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces,” Applied Mathematics and Computation, vol. 217, no. 14, pp. 6624–6632, 2011.
  • G. Bonanno, G. M. Bisci, and V. Rădulescu, “Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 74, no. 14, pp. 4785–4795, 2011.
  • N. Halidias and V. K. Le, “Multiple solutions for quasilinear elliptic Neumann problems in Orlicz-Sobolev spaces,” Boundary Value Problems, vol. 3, pp. 299–306, 2005.
  • A. Kristály, M. Mihăilescu, and V. Rădulescu, “Two non-trivial solutions for a non-homogeneous Neumann problem: an Orlicz-Sobolev space setting,” Proceedings of the Royal Society of Edinburgh A, vol. 139, no. 2, pp. 367–379, 2009.
  • B. Ricceri, “A further three critical points theorem,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 71, no. 9, pp. 4151–4157, 2009.
  • B. Ricceri, “A three critical points theorem revisited,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 70, no. 9, pp. 3084–3089, 2009.
  • B. Ricceri, “Existence of three solutions for a class of elliptic eigenvalue problems,” Mathematical and Computer Modelling, vol. 32, no. 11–13, pp. 1485–1494, 2000.
  • J. Lamperti, “On the isometries of certain function-spaces,” Pacific Journal of Mathematics, vol. 8, pp. 459–466, 1958.