Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

MULTIPLE PERIODIC SOLUTIONS FOR EIGENVALUE PROBLEMS WITH A p-LAPLACIAN AND NON-SMOOTH POTENTIAL

  • Zhang, Guoqing (COLLEGE OF SCIENCES UNIVERSITY OF SHANGHAI FOR SCIENCE AND TECHNOLOGY) ;
  • Liu, Sanyang (COLLEGE OF SCIENCES XIDIAN UNIVERSITY)
  • Received : 2009.06.28
  • Published : 2011.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we establish a multiple critical points theorem for a one-parameter family of non-smooth functionals. The obtained result is then exploited to prove a multiplicity result for a class of periodic eigenvalue problems driven by the p-Laplacian and with a non-smooth potential. Under suitable assumptions, we locate an open subinterval of the eigenvalue.

Keywords

References

  1. G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal. 54 (2003), no. 4, 651-665. https://doi.org/10.1016/S0362-546X(03)00092-0
  2. G. Bonanno and N. Giovannelli, An eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities, J. Math. Anal. Appl. 308 (2005), no. 2, 596-604. https://doi.org/10.1016/j.jmaa.2004.11.053
  3. K.-C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), no. 1, 102-129. https://doi.org/10.1016/0022-247X(81)90095-0
  4. H. Dang and S. F. Oppenheimer, Existence and uniqueness results for some nonlinear boundary value problems, J. Math. Anal. Appl. 198 (1996), no. 1, 35-48. https://doi.org/10.1006/jmaa.1996.0066
  5. M. A. del Pino, R. Manasevich, and A. Murua, Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE, Nonlinear Anal. 18 (1992), no. 1, 79-92. https://doi.org/10.1016/0362-546X(92)90048-J
  6. C. Fabry and D. Fayyad, Periodic solutions of second order differential equations with a p-Laplacian and asymmetric nonlinearities, Rend. Istit. Mat. Univ. Trieste 24 (1992), no. 1-2, 207-227.
  7. L. Gasinski, Multiplicity theorems for periodic systems with a p-Laplacian-like operator, Nonlinear Anal. 67 (2007), no. 9, 2632-2641. https://doi.org/10.1016/j.na.2006.09.028
  8. L. Gasinski and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman and Hall, CRC, Boca Raton, 2005.
  9. L. Gasinski and N. S. Papageorgiou, On the existence of multiple periodic solutions for equations driven by the p-Laplacian and with a non-smooth potential, Proc. Edinb. Math. Soc. (2) 46 (2003), no. 1, 229-249. https://doi.org/10.1017/S0013091502000159
  10. Z. M. Guo, Boundary value problems of a class of quasilinear ordinary differential equations, Differential Integral Equations 6 (1993), no. 3, 705-719.
  11. N. Kourogenis and N. S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance, Kodai Math. J. 23 (2000), no. 1, 108-135. https://doi.org/10.2996/kmj/1138044160
  12. J. Mawhin, Periodic solutions of systems with p-Laplacian-like operators, Nonlinear analysis and its applications to differential equations (Lisbon, 1998), 37-63, Progr. Nonlinear Differential Equations Appl., 43, Birkhauser Boston, Boston, MA, 2001.
  13. S. A. Marano and D. Motreanu, On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems, Nonlinear Anal. 48 (2002), no. 1, Ser. A: Theory Methods, 37-52. https://doi.org/10.1016/S0362-546X(00)00171-1
  14. E. H. Papageorgiou and N. S. Papageorgiou, Two nontrivial solutions for quasilinear periodic equations, Proc. Amer. Math. Soc. 132 (2004), no. 2, 429-434 https://doi.org/10.1090/S0002-9939-03-07076-X
  15. B. Ricceri, On a three critical points theorem, Arch. Math. (Basel) 75 (2000), no. 3, 220-226. https://doi.org/10.1007/s000130050496
  16. M. Struwe, Varaitional Methods, Springer-Verlag, Berlin, 1996.

Cited by

  1. Nonlinear, Nonhomogeneous Periodic Problems with no Growth Control on the Reaction vol.21, pp.3, 2015, https://doi.org/10.1007/s10883-014-9245-4