September  2011, 10(5): 1401-1414. doi: 10.3934/cpaa.2011.10.1401

Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential

1. 

Département de Mathématiques, Université de Perpignan, Avenue de Villeneuve 52, 66860 Perpignan Cedex

2. 

Ben Gurion University of the Negev, Department of Mathematics, Be'er Sheva 84105, Israel

3. 

Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received  March 2009 Revised  August 2010 Published  April 2011

A nonautonomous second order system with a nonsmooth potential is studied. It is assumed that the system is asymptotically linear at infinity and resonant (both at infinity and at the origin), with respect to the zero eigenvalue. Also, it is assumed that the linearization of the system is indefinite. Using a nonsmooth variant of the reduction method and the local linking theorem, we show that the system has at least two nontrivial solutions.
Citation: D. Motreanu, V. V. Motreanu, Nikolaos S. Papageorgiou. Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1401-1414. doi: 10.3934/cpaa.2011.10.1401
References:
[1]

H. Amann, Saddle points and multiple solutions of differential equations,, Math. Z., 169 (1979), 127.  doi: 10.1007/BF01215273.  Google Scholar

[2]

G. Barletta and R. Livrea, Existence of three periodic solutions for a nonautonomous second order system,, Le Mathematiche, 57 (2002), 205.   Google Scholar

[3]

G. Barletta and N. S. Papageorgiou, Nonautonomous second order periodic systems: existence and multiplicity of solutions,, J. Nonlinear Convex Anal., 8 (2007), 373.   Google Scholar

[4]

G. Bonanno and R. Livrea, Periodic solutions for a class of second order Hamiltonian systems,, Electronic J. Differential Equations, 115 (2005).   Google Scholar

[5]

A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem,, Annali Mat. Pura Appl., 120 (1979), 113.  doi: 10.1007/BF02411940.  Google Scholar

[6]

F. Clarke, "Optimization and Nonsmooth Analysis,", Wiley, (1983).   Google Scholar

[7]

G. Cordaro, Three periodic solutions to an eigenvalue problem for a class of second-order Hamiltonian systems,, Abstr. Appl. Anal., 115 (2003), 1037.  doi: 10.1155/S1085337503305044.  Google Scholar

[8]

F. Faraci, Three periodic solutions for a second order nonautonomous system,, J. Nonlinear Convex Anal., 3 (2002), 393.   Google Scholar

[9]

L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,", Chapman & Hall/CRC, (2005).   Google Scholar

[10]

L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis,", Chapman & Hall/CRC, (2006).   Google Scholar

[11]

S. Hu and N. S. Papageorgiou, Nontrivial solutions for superquadratic nonautonomous periodic systems,, Topol. Methods Nonlinear Anal., 34 (2009), 327.   Google Scholar

[12]

J. Mawhin, Forced second order conservative systems with periodic nonlinearity,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 6 (1989), 415.   Google Scholar

[13]

J. Mawhin and M. Willem, "Critical Point Theory And Hamiltonian Systems,", Springer-Verlag, (1989).   Google Scholar

[14]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Periodic solutions for nonautonomous systems with nonsmooth quadratic or superquadratic potential,, Topol. Methods Nonlinear Anal., 24 (2004), 269.   Google Scholar

[15]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Two nontrivial solutions for periodic systems with indefinite linear part,, Discrete Contin. Dyn. Syst., 19 (2007), 197.  doi: 10.3934/dcds.2007.19.197.  Google Scholar

[16]

D. Motreanu and V. Radulescu, "Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems,", Kluwer Academic Publishers, (2003).   Google Scholar

[17]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 31 (1978), 157.  doi: 10.1002/cpa.3160310203.  Google Scholar

[18]

R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations,", Pitman, (1977).   Google Scholar

[19]

C.-L. Tang and X.-P. Wu, Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems,, J. Math. Anal. Appl., 275 (2002), 870.  doi: 10.1016/S0022-247X(02)00442-0.  Google Scholar

[20]

K. Thews, Nontrivial solutions of elliptic equations at resonance,, Proc. Roy. Soc. Edinburgh Sect. A, 85 (1980), 119.   Google Scholar

show all references

References:
[1]

H. Amann, Saddle points and multiple solutions of differential equations,, Math. Z., 169 (1979), 127.  doi: 10.1007/BF01215273.  Google Scholar

[2]

G. Barletta and R. Livrea, Existence of three periodic solutions for a nonautonomous second order system,, Le Mathematiche, 57 (2002), 205.   Google Scholar

[3]

G. Barletta and N. S. Papageorgiou, Nonautonomous second order periodic systems: existence and multiplicity of solutions,, J. Nonlinear Convex Anal., 8 (2007), 373.   Google Scholar

[4]

G. Bonanno and R. Livrea, Periodic solutions for a class of second order Hamiltonian systems,, Electronic J. Differential Equations, 115 (2005).   Google Scholar

[5]

A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem,, Annali Mat. Pura Appl., 120 (1979), 113.  doi: 10.1007/BF02411940.  Google Scholar

[6]

F. Clarke, "Optimization and Nonsmooth Analysis,", Wiley, (1983).   Google Scholar

[7]

G. Cordaro, Three periodic solutions to an eigenvalue problem for a class of second-order Hamiltonian systems,, Abstr. Appl. Anal., 115 (2003), 1037.  doi: 10.1155/S1085337503305044.  Google Scholar

[8]

F. Faraci, Three periodic solutions for a second order nonautonomous system,, J. Nonlinear Convex Anal., 3 (2002), 393.   Google Scholar

[9]

L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,", Chapman & Hall/CRC, (2005).   Google Scholar

[10]

L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis,", Chapman & Hall/CRC, (2006).   Google Scholar

[11]

S. Hu and N. S. Papageorgiou, Nontrivial solutions for superquadratic nonautonomous periodic systems,, Topol. Methods Nonlinear Anal., 34 (2009), 327.   Google Scholar

[12]

J. Mawhin, Forced second order conservative systems with periodic nonlinearity,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 6 (1989), 415.   Google Scholar

[13]

J. Mawhin and M. Willem, "Critical Point Theory And Hamiltonian Systems,", Springer-Verlag, (1989).   Google Scholar

[14]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Periodic solutions for nonautonomous systems with nonsmooth quadratic or superquadratic potential,, Topol. Methods Nonlinear Anal., 24 (2004), 269.   Google Scholar

[15]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Two nontrivial solutions for periodic systems with indefinite linear part,, Discrete Contin. Dyn. Syst., 19 (2007), 197.  doi: 10.3934/dcds.2007.19.197.  Google Scholar

[16]

D. Motreanu and V. Radulescu, "Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems,", Kluwer Academic Publishers, (2003).   Google Scholar

[17]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 31 (1978), 157.  doi: 10.1002/cpa.3160310203.  Google Scholar

[18]

R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations,", Pitman, (1977).   Google Scholar

[19]

C.-L. Tang and X.-P. Wu, Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems,, J. Math. Anal. Appl., 275 (2002), 870.  doi: 10.1016/S0022-247X(02)00442-0.  Google Scholar

[20]

K. Thews, Nontrivial solutions of elliptic equations at resonance,, Proc. Roy. Soc. Edinburgh Sect. A, 85 (1980), 119.   Google Scholar

[1]

Nikolaos S. Papageorgiou, Vicenšiu D. Rădulescu, Dušan D. Repovš. Robin problems with indefinite linear part and competition phenomena. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1293-1314. doi: 10.3934/cpaa.2017063

[2]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929

[3]

D. Motreanu, V. V. Motreanu, Nikolaos S. Papageorgiou. Two nontrivial solutions for periodic systems with indefinite linear part. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 197-210. doi: 10.3934/dcds.2007.19.197

[4]

Diego Averna, Nikolaos S. Papageorgiou, Elisabetta Tornatore. Multiple solutions for nonlinear nonhomogeneous resonant coercive problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 155-178. doi: 10.3934/dcdss.2018010

[5]

Nobu Kishimoto. Resonant decomposition and the $I$-method for the two-dimensional Zakharov system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4095-4122. doi: 10.3934/dcds.2013.33.4095

[6]

M. Grossi, P. Magrone, M. Matzeu. Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 703-718. doi: 10.3934/dcds.2001.7.703

[7]

Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial & Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259

[8]

Fabiana Maria Ferreira, Francisco Odair de Paiva. On a resonant and superlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5775-5784. doi: 10.3934/dcds.2019253

[9]

Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure & Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571

[10]

Hongxiu Zhong, Guoliang Chen, Xueping Guo. Semi-local convergence of the Newton-HSS method under the center Lipschitz condition. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 85-99. doi: 10.3934/naco.2019007

[11]

Zhirong He, Weinian Zhang. Critical periods of a periodic annulus linking to equilibria at infinity in a cubic system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 841-854. doi: 10.3934/dcds.2009.24.841

[12]

Thierry Champion, Luigi De Pascale. On the twist condition and $c$-monotone transport plans. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1339-1353. doi: 10.3934/dcds.2014.34.1339

[13]

Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829

[14]

Kunio Hidano, Dongbing Zha. Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1735-1767. doi: 10.3934/cpaa.2019082

[15]

Tahereh Salimi Siahkolaei, Davod Khojasteh Salkuyeh. A preconditioned SSOR iteration method for solving complex symmetric system of linear equations. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 483-492. doi: 10.3934/naco.2019033

[16]

D. Ruiz, J. R. Ward. Some notes on periodic systems with linear part at resonance. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 337-350. doi: 10.3934/dcds.2004.11.337

[17]

Henrik Garde, Stratos Staboulis. The regularized monotonicity method: Detecting irregular indefinite inclusions. Inverse Problems & Imaging, 2019, 13 (1) : 93-116. doi: 10.3934/ipi.2019006

[18]

Yunhua Zhou. The local $C^1$-density of stable ergodicity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2621-2629. doi: 10.3934/dcds.2013.33.2621

[19]

Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049

[20]

A. Azzollini. Erratum to: "On a functional satisfying a weak Palais-Smale condition". Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4987-4987. doi: 10.3934/dcds.2014.34.4987

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

[Back to Top]