# 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 & Applied Analysis, 2011, 10 (5) : 1401-1414. doi: 10.3934/cpaa.2011.10.1401
##### References:

show all references

##### References:
 [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] Ting Guo, Xianhua Tang, Qi Zhang, Zu Gao. Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1563-1579. doi: 10.3934/cpaa.2020078 [4] D. Motreanu, V. V. Motreanu, Nikolaos S. Papageorgiou. Two nontrivial solutions for periodic systems with indefinite linear part. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 197-210. doi: 10.3934/dcds.2007.19.197 [5] 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 [6] Nobu Kishimoto. Resonant decomposition and the $I$-method for the two-dimensional Zakharov system. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4095-4122. doi: 10.3934/dcds.2013.33.4095 [7] M. Grossi, P. Magrone, M. Matzeu. Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 703-718. doi: 10.3934/dcds.2001.7.703 [8] Hamid Maarouf. Local Kalman rank condition for linear time varying systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021029 [9] 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 [10] Fabiana Maria Ferreira, Francisco Odair de Paiva. On a resonant and superlinear elliptic system. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5775-5784. doi: 10.3934/dcds.2019253 [11] 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 [12] 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 [13] Jianguo Huang, Sen Lin. A $C^0P_2$ time-stepping virtual element method for linear wave equations on polygonal meshes. Electronic Research Archive, 2020, 28 (2) : 911-933. doi: 10.3934/era.2020048 [14] Zhirong He, Weinian Zhang. Critical periods of a periodic annulus linking to equilibria at infinity in a cubic system. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 841-854. doi: 10.3934/dcds.2009.24.841 [15] Thierry Champion, Luigi De Pascale. On the twist condition and $c$-monotone transport plans. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1339-1353. doi: 10.3934/dcds.2014.34.1339 [16] Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829 [17] 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 [18] D. Ruiz, J. R. Ward. Some notes on periodic systems with linear part at resonance. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 337-350. doi: 10.3934/dcds.2004.11.337 [19] Leyu Hu, Xingju Cai. Convergence of a randomized Douglas-Rachford method for linear system. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 463-474. doi: 10.3934/naco.2020045 [20] 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

2020 Impact Factor: 1.916