Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
$p$-Laplacian; nonsmooth critical point theory; Clarke subdifferential; saddle point theorem; periodic solution; Poincare-Wirtinger inequality; Sobolev inequality; nonsmooth Palais-Smale condition
$p$-Laplacian; nonsmooth critical point theory; Clarke subdifferential; saddle point theorem; periodic solution; Poincare-Wirtinger inequality; Sobolev inequality; nonsmooth Palais-Smale condition
Summary:
In this paper we examine nonlinear periodic systems driven by the vectorial $p$-Laplacian and with a nondifferentiable, locally Lipschitz nonlinearity. Our approach is based on the nonsmooth critical point theory and uses the subdifferential theory for locally Lipschitz functions. We prove existence and multiplicity results for the “sublinear” problem. For the semilinear problem (i.e. $p = 2$) using a nonsmooth multidimensional version of the Ambrosetti-Rabinowitz condition, we prove an existence theorem for the “superlinear” problem. Our work generalizes some recent results of Tang (PAMS 126(1998)).
References:
[1] H. Brezis and L. Nirenberg: Remarks on finding critical points. Comm. Pure. Appl. Math. 44 (1991), 939–963. DOI 10.1002/cpa.3160440808 | MR 1127041
[2] K. C. Chang: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80 (1981), 102–129. DOI 10.1016/0022-247X(81)90095-0 | MR 0614246
[3] F. H. Clarke: Optimization and Nonsmooth Analysis. Wiley, New York, 1983. MR 0709590 | Zbl 0582.49001
[4] S. Hu and N. S. Papageorgiou: Handbook of Multivalued Analysis. Volume I: Theory. Kluwer, Dordrecht, 1997. MR 1485775
[5] S. Hu and N. S. Papageorgiou: Handbook of Multivalued Analysis. Volume II: Applications. Kluwer, Dordrecht, 1997. MR 1741926
[6] N. Kourogenis and N. S. Papageorgiou: Nonsmooth critical point theory and nonlinear elliptic equations at resonance. J. Austral. Math. Soc. (Series A) 69 (2000), 245–271. DOI 10.1017/S1446788700002202 | MR 1775181
[7] N. Kourogenis and N. S. Papageorgiou: Periodic solutions for quasilinear differential equations with discontinuous nonlinearities. Acta. Sci. Math. (Szeged) 65 (1999), 529–542. MR 1737269
[8] G. Lebourg: Valeur moyenne pour gradient généralisé. CRAS Paris 281 (1975), 795–797. MR 0388097 | Zbl 0317.46034
[9] J. Mawhin and M. Willem: Critical Point Theory and Hamiltonian Systems. Springer-Verlag, Berlin, 1989. MR 0982267
[10] Z. Naniewicz and P. Panagiotopoulos: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York, 1994. MR 1304257
[11] P. Rabinowitz: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics, No.45. AMS, Providence, 1986. MR 0845785
[12] C. L. Tang: Periodic solutions for nonautonomous second order systems with sublinear nonlinearity. Proc. AMS 126 (1998), 3263–3270. MR 1476396 | Zbl 0902.34036
[13] C. L. Tang: Existence and multiplicity of periodic solutions for nonautonomous second order systems. Nonlin. Anal. 32 (1998), 299–304. DOI 10.1016/S0362-546X(97)00493-8 | MR 1610641 | Zbl 0949.34032
[14] J. P. Aubin and H. Frankowska: Set-Valued Analysis. Birkhäuser-Verlag, Boston, 1990. MR 1048347
Search
Partner of
