×

Nonlinear periodic systems with the \(p\)-Laplacian: existence and multiplicity results. (English) Zbl 1156.34011

The author studies second-order nonlinear periodic systems driven by the vector \(p\)-Laplacian with a nonsmooth, locally Lipschitz potential function. He proves the existence of nontrivial solutions requiring, among others, that the potential function \(j(t,x)\) satisfies a locally, nonuniform anticoercivity. Moreover, she does not assume any polynomial growth of the subdifferential \(\partial j(t,x)\).

MSC:

34A60 Ordinary differential inclusions
34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
47J05 Equations involving nonlinear operators (general)
47N20 Applications of operator theory to differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, vol. 188 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1995. · Zbl 0968.49008
[2] M S. Berger and M. Schechter, “On the solvability of semilinear gradient operator equations,” Advances in Mathematics, vol. 25, no. 2, pp. 97-132, 1977. · Zbl 0354.47025 · doi:10.1016/0001-8708(77)90001-9
[3] J. Mawhin, “Semicoercive monotone variational problems,” Académie Royale de Belgique. Bulletin de la Classe des Sciences, vol. 73, no. 3-4, pp. 118-130, 1987. · Zbl 0647.49007
[4] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. · Zbl 0676.58017
[5] C.-L. Tang, “Periodic solutions for nonautonomous second order systems with sublinear nonlinearity,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3263-3270, 1998. · Zbl 0902.34036 · doi:10.1090/S0002-9939-98-04706-6
[6] C.-L. Tang, “Existence and multiplicity of periodic solutions for nonautonomous second order systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 32, no. 3, pp. 299-304, 1998. · Zbl 0949.34032 · doi:10.1016/S0362-546X(97)00493-8
[7] C.-L. Tang and X.-P. Wu, “Periodic solutions for second order systems with not uniformly coercive potential,” Journal of Mathematical Analysis and Applications, vol. 259, no. 2, pp. 386-397, 2001. · Zbl 0999.34039 · doi:10.1006/jmaa.2000.7401
[8] R. Manásevich and J. Mawhin, “Periodic solutions for nonlinear systems with p-Laplacian-like operators,” Journal of Differential Equations, vol. 145, no. 2, pp. 367-393, 1998. · Zbl 0910.34051 · doi:10.1006/jdeq.1998.3425
[9] J. Mawhin, “Periodic solutions of systems with p-Laplacian-like operators,” in Nonlinear Analysis and Its Applications to Differential Equations (Lisbon, 1998), vol. 43 of Progress in Nonlinear Differential Equations and Applications, pp. 37-63, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 1016.34042
[10] J. Mawhin, “Some boundary value problems for Hartman-type perturbations of the ordinary vector p-Laplacian,” Nonlinear Analysis. Theory, Methods & Applications, vol. 40, no. 1-8, pp. 497-503, 2000. · Zbl 0959.34014 · doi:10.1016/S0362-546X(00)85028-2
[11] S. Kyritsi, N. Matzakos, and N. S. Papageorgiou, “Periodic problems for strongly nonlinear second-order differential inclusions,” Journal of Differential Equations, vol. 183, no. 2, pp. 279-302, 2002. · Zbl 1022.34008 · doi:10.1006/jdeq.2001.4110
[12] E. H. Papageorgiou and N. S. Papageorgiou, “Strongly nonlinear multivalued, periodic problems with maximal monotone terms,” Differential and Integral Equations, vol. 17, no. 3-4, pp. 443-480, 2004. · Zbl 1224.34024
[13] L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, vol. 8 of Series in Mathematical Analysis and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005. · Zbl 1058.58005
[14] E. H. Papageorgiou and N. S. Papageorgiou, “Existence of solutions and of multiple solutions for nonlinear nonsmooth periodic systems,” Czechoslovak Mathematical Journal, vol. 54(129), no. 2, pp. 347-371, 2004. · Zbl 1080.34532 · doi:10.1023/B:CMAJ.0000042374.53530.7e
[15] E. H. Papageorgiou and N. S. Papageorgiou, “Non-linear second-order periodic systems with non-smooth potential,” Proceedings of the Indian Academy of Sciences. Mathematical Sciences, vol. 114, no. 3, pp. 269-298, 2004. · Zbl 1059.34009 · doi:10.1007/BF02830004
[16] Z. Denkowski, L. Gasiński, and N. S. Papageorgiou, “Positive solutions for nonlinear periodic problems with the scalar p-Laplacian,” submitted. · Zbl 1172.34005 · doi:10.1007/s11228-007-0059-3
[17] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, vol. 419 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. · Zbl 0887.47001
[18] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume II: Applications, vol. 500 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. · Zbl 0943.47037
[19] L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, vol. 9 of Series in Mathematical Analysis and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2006. · Zbl 1086.47001
[20] Z. Denkowski, S. Migórski, and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Publishers, Boston, Mass, USA, 2003. · Zbl 1040.46001
[21] K. C. Chang, “Variational methods for nondifferentiable functionals and their applications to partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 80, no. 1, pp. 102-129, 1981. · Zbl 0487.49027 · doi:10.1016/0022-247X(81)90095-0
[22] F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1983. · Zbl 0582.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.