# American Institute of Mathematical Sciences

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 and 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), 127-166. doi: 10.1007/BF01215273. [2] G. Barletta and R. Livrea, Existence of three periodic solutions for a nonautonomous second order system, Le Mathematiche, 57 (2002), 205-215. [3] G. Barletta and N. S. Papageorgiou, Nonautonomous second order periodic systems: existence and multiplicity of solutions, J. Nonlinear Convex Anal., 8 (2007), 373-390. [4] G. Bonanno and R. Livrea, Periodic solutions for a class of second order Hamiltonian systems, Electronic J. Differential Equations, 115 (2005), 13 pp. [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), 113-137. doi: 10.1007/BF02411940. [6] F. Clarke, "Optimization and Nonsmooth Analysis," Wiley, New York, 1983. [7] G. Cordaro, Three periodic solutions to an eigenvalue problem for a class of second-order Hamiltonian systems, Abstr. Appl. Anal., 115 (2003), 1037-1045. doi: 10.1155/S1085337503305044. [8] F. Faraci, Three periodic solutions for a second order nonautonomous system, J. Nonlinear Convex Anal., 3 (2002), 393-399. [9] L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems," Chapman & Hall/CRC, Boca Raton, FL, 2005. [10] L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, FL, 2006. [11] S. Hu and N. S. Papageorgiou, Nontrivial solutions for superquadratic nonautonomous periodic systems, Topol. Methods Nonlinear Anal., 34 (2009), 327-338. [12] J. Mawhin, Forced second order conservative systems with periodic nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 415-434. [13] J. Mawhin and M. Willem, "Critical Point Theory And Hamiltonian Systems," Springer-Verlag, New York, 1989. [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), 269-296. [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), 197-210. doi: 10.3934/dcds.2007.19.197. [16] D. Motreanu and V. Radulescu, "Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems," Kluwer Academic Publishers, Dordrecht, 2003. [17] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203. [18] R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations," Pitman, London-San Francisco, Calif.-Melbourne, 1977. [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), 870-882. doi: 10.1016/S0022-247X(02)00442-0. [20] K. Thews, Nontrivial solutions of elliptic equations at resonance, Proc. Roy. Soc. Edinburgh Sect. A, 85 (1980), 119-129.

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##### References:
 [1] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z., 169 (1979), 127-166. doi: 10.1007/BF01215273. [2] G. Barletta and R. Livrea, Existence of three periodic solutions for a nonautonomous second order system, Le Mathematiche, 57 (2002), 205-215. [3] G. Barletta and N. S. Papageorgiou, Nonautonomous second order periodic systems: existence and multiplicity of solutions, J. Nonlinear Convex Anal., 8 (2007), 373-390. [4] G. Bonanno and R. Livrea, Periodic solutions for a class of second order Hamiltonian systems, Electronic J. Differential Equations, 115 (2005), 13 pp. [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), 113-137. doi: 10.1007/BF02411940. [6] F. Clarke, "Optimization and Nonsmooth Analysis," Wiley, New York, 1983. [7] G. Cordaro, Three periodic solutions to an eigenvalue problem for a class of second-order Hamiltonian systems, Abstr. Appl. Anal., 115 (2003), 1037-1045. doi: 10.1155/S1085337503305044. [8] F. Faraci, Three periodic solutions for a second order nonautonomous system, J. Nonlinear Convex Anal., 3 (2002), 393-399. [9] L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems," Chapman & Hall/CRC, Boca Raton, FL, 2005. [10] L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, FL, 2006. [11] S. Hu and N. S. Papageorgiou, Nontrivial solutions for superquadratic nonautonomous periodic systems, Topol. Methods Nonlinear Anal., 34 (2009), 327-338. [12] J. Mawhin, Forced second order conservative systems with periodic nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 415-434. [13] J. Mawhin and M. Willem, "Critical Point Theory And Hamiltonian Systems," Springer-Verlag, New York, 1989. [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), 269-296. [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), 197-210. doi: 10.3934/dcds.2007.19.197. [16] D. Motreanu and V. Radulescu, "Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems," Kluwer Academic Publishers, Dordrecht, 2003. [17] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203. [18] R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations," Pitman, London-San Francisco, Calif.-Melbourne, 1977. [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), 870-882. doi: 10.1016/S0022-247X(02)00442-0. [20] K. Thews, Nontrivial solutions of elliptic equations at resonance, Proc. Roy. Soc. Edinburgh Sect. A, 85 (1980), 119-129.
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