American Institute of Mathematical Sciences

Advanced
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

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 & 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

show all references

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

[1]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
[2]

Roman Srzednicki. On periodic solutions in the Whitney's inverted pendulum problem. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2127-2141. doi: 10.3934/dcdss.2019137
[3]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161
[4]

Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571
[5]

Fabio Cipriani, Gabriele Grillo. On the $l^p$ -agmon's theory. Conference Publications, 1998, 1998 (Special) : 167-176. doi: 10.3934/proc.1998.1998.167
[6]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483
[7]

Antonio Ambrosetti, Massimiliano Berti. Applications of critical point theory to homoclinics and complex dynamics. Conference Publications, 1998, 1998 (Special) : 72-78. doi: 10.3934/proc.1998.1998.72
[8]

Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074
[9]

In-Soo Baek, Lars Olsen. Baire category and extremely non-normal points of invariant sets of IFS's. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 935-943. doi: 10.3934/dcds.2010.27.935
[10]

Olga Kharlampovich and Alexei Myasnikov. Tarski's problem about the elementary theory of free groups has a positive solution. Electronic Research Announcements, 1998, 4: 101-108.

[11]

Andrea Venturelli. A Variational proof of the existence of Von Schubart's orbit. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 699-717. doi: 10.3934/dcdsb.2008.10.699
[12]

Artur M. C. Brito da Cruz, Natália Martins, Delfim F. M. Torres. Hahn's symmetric quantum variational calculus. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 77-94. doi: 10.3934/naco.2013.3.77
[13]

Massimiliano Berti. Some remarks on a variational approach to Arnold's diffusion. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 307-314. doi: 10.3934/dcds.1996.2.307
[14]

Danilo Coelho, David Pérez-Castrillo. On Marilda Sotomayor's extraordinary contribution to matching theory. Journal of Dynamics & Games, 2015, 2 (3&4) : 201-206. doi: 10.3934/jdg.2015001
[15]

Takayoshi Ogawa, Kento Seraku. Logarithmic Sobolev and Shannon's inequalities and an application to the uncertainty principle. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1651-1669. doi: 10.3934/cpaa.2018079
[16]

Alexander Veretennikov. On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 523-549. doi: 10.3934/dcdsb.2013.18.523
[17]

Marc Deschamps, Olivier Poncelet. Complex ray in anisotropic solids: Extended Fermat's principle. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1623-1633. doi: 10.3934/dcdss.2019110
[18]

Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187
[19]

Szandra Beretka, Gabriella Vas. Stable periodic solutions for Nazarenko's equation. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3257-3281. doi: 10.3934/cpaa.2020144
[20]

E. N. Dancer, Zhitao Zhang. Critical point, anti-maximum principle and semipositone p-laplacian problems. Conference Publications, 2005, 2005 (Special) : 209-215. doi: 10.3934/proc.2005.2005.209

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences