In this paper we consider nonlinear periodic systems driven by the one-dimensional $p$-Laplacian and having a nonsmooth locally Lipschitz potential. Using a variational approach based on the nonsmooth Critical Point Theory, we establish the existence of a solution. We also prove a multiplicity result based on a nonsmooth extension of the result of Brezis-Nirenberg (Brezis, H., Nirenberg, L., Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), 939–963.) due to Kandilakis-Kourogenis-Papageorgiou (Kandilakis, D., Kourogenis, N., Papageorgiou, N., Two nontrivial critical point for nosmooth functional via local linking and applications, J. Global Optim., to appear.).
@article{108000,
author = {Michael E. Filippakis},
title = {Periodic solutions for systems with nonsmooth and partially coercive potential},
journal = {Archivum Mathematicum},
volume = {042},
year = {2006},
pages = {225-232},
zbl = {1164.34319},
mrnumber = {2260380},
language = {en},
url = {http://dml.mathdoc.fr/item/108000}
}
Filippakis, Michael E. Periodic solutions for systems with nonsmooth and partially coercive potential. Archivum Mathematicum, Tome 042 (2006) pp. 225-232. http://gdmltest.u-ga.fr/item/108000/
Sobolev Spaces, Pure and Applied Mathematics 65, Academic Press, New York/London 1975. (1975) | MR 0450957 | Zbl 0314.46030
On the solvability of semilinear gradient operator equations, Adv. Math. 25 (1977), 97–132. (1977) | MR 0500336
Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), 939–963. (1991) | MR 1127041 | Zbl 0751.58006
Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129. (1981) | MR 0614246
Optimization and Nonsmooth Analysis, Wiley, New York 1983. (1983) | MR 0709590 | Zbl 0582.49001
Existence and uniqueness results for some nonlinear boundary value problems, J. Math. Anal. Appl. 198, (1996) 35–48. (198,) | MR 1373525
Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE, Nonlinear Anal. 18, (1992) 79–92. (1992) | MR 1138643
Periodic solutions of second order differential equations with a $p$-Laplacian and asymetric nonlinearities, Rend. Istit. Mat. Univ. Trieste 24 (1992), 207–227. (1992) | MR 1310080
A multiplicity result for nonlinear second order periodic equations with nonsmooth potential, Bull. Belg. Math. Soc. Simon Stevin 9 (2002a), 245–258. | MR 2017079 | Zbl 1056.47056
Boundary value problems for a class of quasilinear ordinary differential equations, Differential Integral Equations 6 (1993), 705–719. (1993) | MR 1202567
Handbook of Multivalued Analysis. Volume I: Theory, Kluwer, Dordrecht, The Netherlands 1997. (1997) | MR 1485775 | Zbl 0887.47001
Handbook of Multivalued Analysis. Volume II: Applications, Kluwer, Dordrecht, The Netherlands 2000. | MR 1741926 | Zbl 0943.47037
Two nontrivial critical point for nosmooth functional via local linking and applications, J. Global Optim., to appear. | MR 2210278
Nonsmooth critical point theory and nonlinear elliptic equations at resonance, J. Austral. Math. Soc. Ser. A 69 (2000), 245–271. | MR 1775181 | Zbl 0999.58006
A weak nonsmooth Palais-Smale condition and coercivity, Rend. Circ. Mat. Palermo 49 (2000), 521–526. | MR 1809092 | Zbl 1225.49021
Periodic solutions for nonlinear systems with $p$-Laplacian-like operators, J. Differential Equations 145 (1998), 367–393. (1998) | MR 1621038
Critical Point Theory and Hamiltonian Systems, Vol. 74 of Applied Mathematics Sciences, Springer-Verlag, New York 1989. (1989) | MR 0982267 | Zbl 0676.58017
Periodic solutions for second order systems with not uniformly coercive potential, J. Math. Anal. Appl. 259 (2001), 386–397. | MR 1842066 | Zbl 0999.34039