# American Institute of Mathematical Sciences

January  2013, 33(1): 47-66. doi: 10.3934/dcds.2013.33.47

## Multiple critical points for a class of periodic lower semicontinuous functionals

 1 Institute of Mathematics "Simion Stoilow", Romanian Academy, 21, Calea Griviţei, RO-010702-Bucharest, Sector 1, Romania 2 Department of Mathematics, West University of Timişoara, 4, Blvd. V. Pârvan RO-300223-Timişoara, Romania

Received  October 2011 Revised  January 2012 Published  September 2012

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.
Citation: Cristian Bereanu, Petru Jebelean. Multiple critical points for a class of periodic lower semicontinuous functionals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 47-66. doi: 10.3934/dcds.2013.33.47
##### References:
 [1] C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian,, J. Differential Equations, 243 (2007), 536.  doi: 10.1016/j.jde.2007.05.014.  Google Scholar [2] C. Bereanu, P. Jebelean and J. Mawhin, Variational methods for nonlinear perturbation of singular $\phi$-Laplacians,, Rend. Lincei Mat. Appl., 22 (2011), 89.   Google Scholar [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.   Google Scholar [4] H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations, 23 (2010), 801.   Google Scholar [5] H. Brezis and J. Mawhin, Periodic solutions of Lagrangian systems of relativistic oscillators,, Comm. Appl. Anal., 15 (2011), 235.   Google Scholar [6] K. C. Chang, On the periodic nonlinearity and the multiplicity of solutions,, Nonlinear Anal., 13 (1989), 527.  doi: 10.1016/0362-546X(89)90062-X.  Google Scholar [7] E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems,, Ann. Mat. Pura Appl., 131 (1982), 167.  doi: 10.1007/BF01765151.  Google Scholar [8] W.-Y. Ding, A generalization of the Poincaré-Birkhoff theorem,, Proc. Amer. Math. Soc., 88 (1983), 341.  doi: 10.1090/S0002-9939-1983-0695272-2.  Google Scholar [9] P. Felmer, Periodic solutions of spatially periodic Hamiltonian systems,, J. Differential Equations, 98 (1992), 143.  doi: 10.1016/0022-0396(92)90109-Z.  Google Scholar [10] A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane,, Advanced Nonlinear Studies, 12 (2012), 395.   Google Scholar [11] G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 259.   Google Scholar [12] J. Franks, Generalizations of the Poincaré-Birkhoff theorem,, Ann. Math., 128 (1988), 139.  doi: 10.2307/1971464.  Google Scholar [13] G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden,, Math. Ann., 86 (1922), 1.  doi: 10.1007/BF01458566.  Google Scholar [14] J. Q. Liu, A generalized saddle point theorem,, J. Differential Equations, 82 (1989), 372.  doi: 10.1016/0022-0396(89)90139-3.  Google Scholar [15] R. Manásevich and J. R. Ward, On a result of Brezis and Mawhin,, Proc. Amer. Math. Soc., 140 (2012), 531.  doi: 10.1090/S0002-9939-2011-11311-X.  Google Scholar [16] S. Maró, Periodic solutions of a forced relativistic pendulum via twist dynamics,, Topol. Meth. Nonlin. Anal., ().   Google Scholar [17] J. Mawhin, Forced second order conservative systems with periodic nonlinearity,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 415.   Google Scholar [18] J. Mawhin, The forced pendulum: a paradigm for nonlinear analysis and dynamical systems,, Exposition Math., 6 (1988), 271.   Google Scholar [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.   Google Scholar [20] J. Mawhin, Resonance problems for some non-autonomous ordinary differential equations,, preprint., ().   Google Scholar [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.  doi: 10.1016/0022-0396(84)90180-3.  Google Scholar [22] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer, (1989).   Google Scholar [23] P. H. Rabinowitz, On a class of functionals invariant under a $Z_n$ action,, Trans. Amer. Math. Soc., 310 (1988), 303.  doi: 10.1090/S0002-9947-1988-0965755-5.  Google Scholar [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.   Google Scholar [25] A. Szulkin, A relative category and applications to critical point theory for strongly indefinite functionals,, Nonlinear Anal., 15 (1990), 725.  doi: 10.1016/0362-546X(90)90089-Y.  Google Scholar [26] M. Willem, Oscillations forcées de l'équation du pendule,, Pub. IRMA Lille, 3 (1981).   Google Scholar

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##### References:
 [1] C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian,, J. Differential Equations, 243 (2007), 536.  doi: 10.1016/j.jde.2007.05.014.  Google Scholar [2] C. Bereanu, P. Jebelean and J. Mawhin, Variational methods for nonlinear perturbation of singular $\phi$-Laplacians,, Rend. Lincei Mat. Appl., 22 (2011), 89.   Google Scholar [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.   Google Scholar [4] H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations, 23 (2010), 801.   Google Scholar [5] H. Brezis and J. Mawhin, Periodic solutions of Lagrangian systems of relativistic oscillators,, Comm. Appl. Anal., 15 (2011), 235.   Google Scholar [6] K. C. Chang, On the periodic nonlinearity and the multiplicity of solutions,, Nonlinear Anal., 13 (1989), 527.  doi: 10.1016/0362-546X(89)90062-X.  Google Scholar [7] E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems,, Ann. Mat. Pura Appl., 131 (1982), 167.  doi: 10.1007/BF01765151.  Google Scholar [8] W.-Y. Ding, A generalization of the Poincaré-Birkhoff theorem,, Proc. Amer. Math. Soc., 88 (1983), 341.  doi: 10.1090/S0002-9939-1983-0695272-2.  Google Scholar [9] P. Felmer, Periodic solutions of spatially periodic Hamiltonian systems,, J. Differential Equations, 98 (1992), 143.  doi: 10.1016/0022-0396(92)90109-Z.  Google Scholar [10] A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane,, Advanced Nonlinear Studies, 12 (2012), 395.   Google Scholar [11] G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 259.   Google Scholar [12] J. Franks, Generalizations of the Poincaré-Birkhoff theorem,, Ann. Math., 128 (1988), 139.  doi: 10.2307/1971464.  Google Scholar [13] G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden,, Math. Ann., 86 (1922), 1.  doi: 10.1007/BF01458566.  Google Scholar [14] J. Q. Liu, A generalized saddle point theorem,, J. Differential Equations, 82 (1989), 372.  doi: 10.1016/0022-0396(89)90139-3.  Google Scholar [15] R. Manásevich and J. R. Ward, On a result of Brezis and Mawhin,, Proc. Amer. Math. Soc., 140 (2012), 531.  doi: 10.1090/S0002-9939-2011-11311-X.  Google Scholar [16] S. Maró, Periodic solutions of a forced relativistic pendulum via twist dynamics,, Topol. Meth. Nonlin. Anal., ().   Google Scholar [17] J. Mawhin, Forced second order conservative systems with periodic nonlinearity,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 415.   Google Scholar [18] J. Mawhin, The forced pendulum: a paradigm for nonlinear analysis and dynamical systems,, Exposition Math., 6 (1988), 271.   Google Scholar [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.   Google Scholar [20] J. Mawhin, Resonance problems for some non-autonomous ordinary differential equations,, preprint., ().   Google Scholar [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.  doi: 10.1016/0022-0396(84)90180-3.  Google Scholar [22] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer, (1989).   Google Scholar [23] P. H. Rabinowitz, On a class of functionals invariant under a $Z_n$ action,, Trans. Amer. Math. Soc., 310 (1988), 303.  doi: 10.1090/S0002-9947-1988-0965755-5.  Google Scholar [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.   Google Scholar [25] A. Szulkin, A relative category and applications to critical point theory for strongly indefinite functionals,, Nonlinear Anal., 15 (1990), 725.  doi: 10.1016/0362-546X(90)90089-Y.  Google Scholar [26] M. Willem, Oscillations forcées de l'équation du pendule,, Pub. IRMA Lille, 3 (1981).   Google Scholar
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