Abstract and Applied Analysis

Nonlinear Periodic Systems with the p -Laplacian: Existence and Multiplicity Results

Francesca Papalini

Full-text: Open access

Abstract

We study second-order nonlinear periodic systems driven by the vector p -Laplacian with a nonsmooth, locally Lipschitz potential function. Under minimal and natural hypotheses on the potential and using variational methods based on the nonsmooth critical point theory, we prove existence theorems and a multiplicity result. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (unbounded from both above and below).

Article information

Source
Abstr. Appl. Anal., Volume 2007 (2007), Article ID 80394, 23 pages.

Dates
First available in Project Euclid: 5 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1183666877

Digital Object Identifier
doi:10.1155/2007/80394

Mathematical Reviews number (MathSciNet)
MR2320799

Zentralblatt MATH identifier
1156.34011

Citation

Papalini, Francesca. Nonlinear Periodic Systems with the $p$ -Laplacian: Existence and Multiplicity Results. Abstr. Appl. Anal. 2007 (2007), Article ID 80394, 23 pages. doi:10.1155/2007/80394. https://projecteuclid.org/euclid.aaa/1183666877


Export citation

References

  • Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, vol. 188 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1995.
  • M S. Berger and M. Schechter, ``On the solvability of semilinear gradient operator equations,'' Advances in Mathematics, vol. 25, no. 2, pp. 97--132, 1977.
  • J. Mawhin, ``Semicoercive monotone variational problems,'' Académie Royale de Belgique. Bulletin de la Classe des Sciences, vol. 73, no. 3-4, pp. 118--130, 1987.
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
  • C.-L. Tang, ``Periodic solutions for nonautonomous second order systems with sublinear nonlinearity,'' Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3263--3270, 1998.
  • C.-L. Tang, ``Existence and multiplicity of periodic solutions for nonautonomous second order systems,'' Nonlinear Analysis. Theory, Methods & Applications, vol. 32, no. 3, pp. 299--304, 1998.
  • C.-L. Tang and X.-P. Wu, ``Periodic solutions for second order systems with not uniformly coercive potential,'' Journal of Mathematical Analysis and Applications, vol. 259, no. 2, pp. 386--397, 2001.
  • R. Manásevich and J. Mawhin, ``Periodic solutions for nonlinear systems with $p$-Laplacian-like operators,'' Journal of Differential Equations, vol. 145, no. 2, pp. 367--393, 1998.
  • čommentComment on ref. [13?]: We changed the year according to the MathSciNet database. Please check. J. Mawhin, ``Periodic solutions of systems with $p$-Laplacian-like operators,'' in Nonlinear Analysis and Its Applications to Differential Equations (Lisbon, 1998), vol. 43 of Progress in Nonlinear Differential Equations and Applications, pp. 37--63, Birkhäuser, Boston, Mass, USA, 2001.
  • J. Mawhin, ``Some boundary value problems for Hartman-type perturbations of the ordinary vector $p$-Laplacian,'' Nonlinear Analysis. Theory, Methods & Applications, vol. 40, no. 1--8, pp. 497--503, 2000.
  • čommentComment on ref. [10?]: We changed the year according to the MathSciNet database. Please check. S. Kyritsi, N. Matzakos, and N. S. Papageorgiou, ``Periodic problems for strongly nonlinear second-order differential inclusions,'' Journal of Differential Equations, vol. 183, no. 2, pp. 279--302, 2002.
  • E. H. Papageorgiou and N. S. Papageorgiou, ``Strongly nonlinear multivalued, periodic problems with maximal monotone terms,'' Differential and Integral Equations, vol. 17, no. 3-4, pp. 443--480, 2004.
  • L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, vol. 8 of Series in Mathematical Analysis and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005.
  • E. H. Papageorgiou and N. S. Papageorgiou, ``Existence of solutions and of multiple solutions for nonlinear nonsmooth periodic systems,'' Czechoslovak Mathematical Journal, vol. 54(129), no. 2, pp. 347--371, 2004.
  • E. H. Papageorgiou and N. S. Papageorgiou, ``Non-linear second-order periodic systems with non-smooth potential,'' Proceedings of the Indian Academy of Sciences. Mathematical Sciences, vol. 114, no. 3, pp. 269--298, 2004.
  • čommentComment on ref. [5?]: Please update the information of this reference, if possible. Z. Denkowski, L. Gasiński, and N. S. Papageorgiou, ``Positive solutions for nonlinear periodic problems with the scalar $p$-Laplacian,'' submitted.
  • S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, vol. 419 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
  • S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume II: Applications, vol. 500 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
  • L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, vol. 9 of Series in Mathematical Analysis and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2006.
  • Z. Denkowski, S. Migórski, and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Publishers, Boston, Mass, USA, 2003.
  • K. C. Chang, ``Variational methods for nondifferentiable functionals and their applications to partial differential equations,'' Journal of Mathematical Analysis and Applications, vol. 80, no. 1, pp. 102--129, 1981.
  • F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1983.