\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Multiple critical points for a class of periodic lower semicontinuous functionals

Abstract Related Papers Cited by
  • We deal with a class of functionals $I$ on a Banach space $X,$ having the structure $I=\Psi+\mathcal G,$ with $\Psi : X \to (- \infty , + \infty ]$ proper, convex, lower semicontinuous and $\mathcal G: X \to \mathbb{R} $ of class $C^1.$ Also, $I$ is $G$-invariant with respect to a discrete subgroup $G\subset X$ with $\mbox{dim (span}\ G)=N$. Under some appropriate additional assumptions we prove that $I$ has at least $N+1$ critical orbits. As a consequence, we obtain that the periodically perturbed $N$-dimensional relativistic pendulum equation has at least $N+1$ geometrically distinct periodic solutions.
    Mathematics Subject Classification: 35J20, 35J60, 35J93.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian, J. Differential Equations, 243 (2007), 536-557.doi: 10.1016/j.jde.2007.05.014.

    [2]

    C. Bereanu, P. Jebelean and J. Mawhin, Variational methods for nonlinear perturbation of singular $\phi$-Laplacians, Rend. Lincei Mat. Appl., 22 (2011), 89-111.

    [3]

    C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Amer. Math. Soc., 140 (2012), 2713-2719.

    [4]

    H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810.

    [5]

    H. Brezis and J. Mawhin, Periodic solutions of Lagrangian systems of relativistic oscillators, Comm. Appl. Anal., 15 (2011), 235-250.

    [6]

    K. C. Chang, On the periodic nonlinearity and the multiplicity of solutions, Nonlinear Anal., 13 (1989), 527-537.doi: 10.1016/0362-546X(89)90062-X.

    [7]

    E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl., 131 (1982), 167-185.doi: 10.1007/BF01765151.

    [8]

    W.-Y. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88 (1983), 341-346.doi: 10.1090/S0002-9939-1983-0695272-2.

    [9]

    P. Felmer, Periodic solutions of spatially periodic Hamiltonian systems, J. Differential Equations, 98 (1992), 143-168.doi: 10.1016/0022-0396(92)90109-Z.

    [10]

    A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane, Advanced Nonlinear Studies, 12 (2012), 395-408.

    [11]

    G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 259-281.

    [12]

    J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. Math., 128 (1988), 139-151.doi: 10.2307/1971464.

    [13]

    G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden, Math. Ann., 86 (1922), 1-13.doi: 10.1007/BF01458566.

    [14]

    J. Q. Liu, A generalized saddle point theorem, J. Differential Equations, 82 (1989), 372-385.doi: 10.1016/0022-0396(89)90139-3.

    [15]

    R. Manásevich and J. R. Ward, On a result of Brezis and Mawhin, Proc. Amer. Math. Soc., 140 (2012), 531-539.doi: 10.1090/S0002-9939-2011-11311-X.

    [16]

    S. MaróPeriodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Meth. Nonlin. Anal., to appear. arXiv:1110.0851

    [17]

    J. Mawhin, Forced second order conservative systems with periodic nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 415-434.

    [18]

    J. Mawhin, The forced pendulum: a paradigm for nonlinear analysis and dynamical systems, Exposition Math., 6 (1988), 271-287.

    [19]

    J. Mawhin, Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 4015-4026.

    [20]

    J. MawhinResonance problems for some non-autonomous ordinary differential equations, preprint.

    [21]

    J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations, 52 (1984), 264-287.doi: 10.1016/0022-0396(84)90180-3.

    [22]

    J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer, New York, 1989.

    [23]

    P. H. Rabinowitz, On a class of functionals invariant under a $Z_n$ action, Trans. Amer. Math. Soc., 310 (1988), 303-311.doi: 10.1090/S0002-9947-1988-0965755-5.

    [24]

    A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77-109.

    [25]

    A. Szulkin, A relative category and applications to critical point theory for strongly indefinite functionals, Nonlinear Anal., 15 (1990), 725-739.doi: 10.1016/0362-546X(90)90089-Y.

    [26]

    M. Willem, Oscillations forcées de l'équation du pendule, Pub. IRMA Lille, 3 (1981), V-1-V-3.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(120) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return