Multiple critical points for a class of periodic lower semicontinuous functionals

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

Multiple critical points for a class of periodic lower semicontinuous functionals

Cristian Bereanu ^1, and Petru Jebelean ^2,

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

Abstract
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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.

Keywords: Lusternik-Schnirelman category., Ekeland's variational principle, Periodic problem, Szulkin's critical point theory.

Mathematics Subject Classification: 35J20, 35J60, 35J9.

Citation: Cristian Bereanu, Petru Jebelean. Multiple critical points for a class of periodic lower semicontinuous functionals. Discrete and Continuous Dynamical Systems, 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), 536-557. 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. Maró, Periodic 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. Mawhin, Resonance 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.

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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), 536-557. 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. Maró, Periodic 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. Mawhin, Resonance 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.

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