Positivity

, Volume 14, Issue 2, pp 335–351 | Cite as

On variational methods to a generalized Emden–Fowler equation

Article
First Online:
Received:
Accepted:

Abstract

In the present paper, the variational principle to the boundary value problems for a generalized Emden-Fowler equation is given and some existence results of solutions are obtained by using the critical point theory.

Keywords

Critical point Emden–Fowler equation Poincare-Wirtinger’s inequality 

Mathematics Subject Classification (2000)

34C25 58E30 47H04 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Antonacci F., Magrone P.: Second order nonautonomous systems with symmetric potential changing sign. Rendiconti di Matematica Seie VII 18, 367–379 (1998)MATHMathSciNetGoogle Scholar
  2. 2.
    Bhutani O.P., Vijayakumar K.: On certain new and exact solutions of the Emden-Fowler equation and Emden equation via invariant variational principles and group invariance. J. Aust. Math. Soc. Ser. B 32, 457–468 (1991)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brezis H., Nirenberg L.: Remarks on finding critical points. Comm. Pure Appl. Math. 44, 939–963 (1991)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Drabek P.: Strongly nonlinear degenerated and singular elliptic problems. Pitman Res. Notes Math. Ser. 343, 112–146 (1996)MathSciNetGoogle Scholar
  5. 5.
    Egnell H.: Existence and nonexistence results for m-laplace equations involving critical sobolev exponents. Arch. Rat. Mech. Anal. 104, 57–77 (1988)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Faraci F., Livrea R.: Infinitely many periodic solutions for a second-order nonautonomous system. Nonlinear Anal. TMA 54, 417–429 (2003)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gebeily M.E., Regan D.O.: A quasilinearization methods for a class of second order singular nonlinear differential equations with nonlinear boundary conditions. Nonlinear Anal. TMA 8, 174–186 (2007)MATHGoogle Scholar
  8. 8.
    Mawhin J., Willem M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)MATHGoogle Scholar
  9. 9.
    Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Conf. Ser. in Math. No., vol. 65. Amer. Math. Soc., Providence (1986)Google Scholar
  10. 10.
    Tang C.: Periodic solutions of non-autonomous second order systems with γ-quasisubadditive potential. J. Math. Anal. Appl. 189, 671–675 (1995)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Tang C.: Existence and multiplicity of periodic solutions for nonautonomous second order systems. Nonlinear Anal. TMA 32(3), 299–304 (1998)MATHCrossRefGoogle Scholar
  12. 12.
    Tang C.: Periodic solutions of non-autonomous second order systems. J. Math. Anal. Appl. 202, 465–469 (1996)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Tang C., Wu X.: Periodic solutions for second order systems with not uniformly coercive potential. J. Math. Anal. Appl. 259, 386–397 (2001)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Tang C.: Periodic solutions for non-autonomous second order systems with sublinear nonlinearity. Proc. Am. Math. Soc. 126(11), 3263–3270 (1998)MATHCrossRefGoogle Scholar
  15. 15.
    Tang C., Wu X.: Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems. J. Math. Anal. Appl. 275, 870–882 (2002)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Xu X.: Positive solutions of generalized Emden-Fowler equation, Nonlinear Analysis. TMA 53, 23–44 (2003)MATHGoogle Scholar
  17. 17.
    Yu J., Guo Z.: on bopundary value problems for a discrete generalized Emden-Fowler equation. J. Diff. Equ. 231, 18–31 (2006)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Wong J.: On the generalized Emden-Fowler equation. SIAM Rev. 17, 339–360 (1975)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Wu X., Tang C.: Periodic solutions of a class of non-autonomous second order systems. J. Math. Anal. Appl. 236, 227–235 (1999)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Wu X.: Saddle point characterization and multiplicity of periodic solutions of non-autonomous second order systems. Nonlinear Anal. TMA 58, 899–907 (2004)MATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.LAMA, The School of MathematicsPeking UniversityBeijingPeople’s Republic of China

Personalised recommendations