Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-04T18:41:54.411Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE MULTIPLICITY OF SOLUTIONS FOR NON-LINEAR PERIODIC PROBLEMS WITH THE NON-LINEARITY CROSSING SEVERAL EIGENVALUES

Published online by Cambridge University Press:  25 November 2009

SOPHIA TH. KYRITSI  and
NIKOLAOS S. PAPAGEORGIOU
SOPHIA TH. KYRITSI
Affiliation:
Department of Mathematics, Hellenic Naval Academy, Pireaus 18539, Greece e-mail: [email protected]
NIKOLAOS S. PAPAGEORGIOU
Affiliation:
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider a non-linear periodic problem driven by the scalar p-Laplacian and with a non-smooth potential. We assume that the multi-valued right-hand-side non-linearity exhibits an asymmetric behaviour at ±∞ and crosses a finite number of eigenvalues as we move from −∞ to +∞. Using a variational approach based on the non-smooth critical-point theory, we show that the problem has at least two non-trivial solutions, one of which has constant sign. For the semi-linear (p = 2), smooth problem, using Morse theory, we show that the problem has at least three non-trivial solutions, again one with constant sign.

Keywords

34B1534C25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 2 , May 2010 , pp. 271 - 302
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Aizicovici, S., Papageorgiou, N. S. and Staicu, V., Periodic solutions for second order differential inclusions with the p-Laplacian, J. Math. Anal. Appl. 322 (2006), 913–929.CrossRefGoogle Scholar
2.Aizicovici, S., Papageorgiou, N. S. and Staicu, V., Multiple nontrival solutions for nonlinear periodic problems with the p-Laplacian, J. Diff. Eq. 243 (2007), 504–535.CrossRefGoogle Scholar
3.Allegretto, W. and Huang, Y., A Picone's identity for the p-Laplacian and applications, Nonlin. Anal. 32 (1998), 819–830.CrossRefGoogle Scholar
4.Bartsch, T. and Li, S., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlin. Anal. 28 (1997), 419–441.CrossRefGoogle Scholar
5.Chang, K. C., Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129.CrossRefGoogle Scholar
6.Chang, K. C., Infinite dimensional Morse theory and multiple solution problems (Birkhäuser, Boston, 1993).CrossRefGoogle Scholar
7.Clarke, F., A new approach to Lagrange multipliers, Math. Oper. Res. 1 (1976), 165–174.CrossRefGoogle Scholar
8.Clarke, F. H., Optimization and nonsmooth analysis (Wiley, New York, 1983).Google Scholar
9.Corvellec, J. N., On the second deformation lemma, Topol. Meth. Nonlin. Anal. 17 (2001), 55–66.CrossRefGoogle Scholar
10.del Pino, M. A., Manásevich, R. and Murúa, A., Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE, Nonlin. Anal. 18 (1992), 79–92.CrossRefGoogle Scholar
11.Denkowski, Z., Migórski, S. and Papageorgiou, N. S., An introduction to nonlinear analysis: applications (Kluwer Academic, Boston, 2003).Google Scholar
12.Drabek, P. and Manásevich, R., On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian, Diff. Integral Eq. 12 (1999), 773–788.Google Scholar
13.Gasiński, L. and Papageorgiou, N. S., On the existence of multiple periodic solutions for equations driven by the p-Laplacian and a nonsmooth potential, Proc. Edinburgh Math. Soc. 46 (2003), 229–249.CrossRefGoogle Scholar
14.Gasiński, L. and Papageorgiou, N. S., Nonsmooth critical point Ttheory and nonlinear boundary value problems (Chapman & Hall/CRC Press, Boca Raton, FL, 2005).Google Scholar
15.Gasiński, L. and Papageorgiou, N. S., Nonlinear analysis (Chapman & Hall/CRC Press, Boca Raton, Fl, 2006).Google Scholar
16.Kandilakis, D., Kourogenis, N. and Papageorgiou, N. S., Two nontrivial critical points for nonsmooth functionals via local linking and applications, J. Global Optim. 34 (2006), 219–244.CrossRefGoogle Scholar
17.Magnus, W. and Winkler, S., Hill's equation (Dover, New York, 1979).Google Scholar
18.Manásevich, P. and Mawhin, J., Boundary value problems for nonlinear perturbations of vector p-Lapalcian-like operators, J. Kor. Math. Soc. 37 (2000), 665–685.Google Scholar
19.Mawhin, J., Topological degree methods in nonlinear boundary value problems, Regional Conference Series in Mathematics, Vol. 40 (AMS, Providence, RI, 1979).CrossRefGoogle Scholar
20.Mawhin, J., Periodic solutions of systems with p-Lapalcian-like operators. In Nonlinear Analysis and Its Applications to Differential Equations (Lisbon 1998) (Ramos, M. and Sanchez, L., Editors) (Birkhäuser Verlag, Boston, MA, 2001), 37–63.CrossRefGoogle Scholar
21.Mawhin, J. and Willem, M., Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, vol. 74, (Springer, New York, 1989).CrossRefGoogle Scholar
22.Papageorgiou, E. H. and Papageorgiou, N. S., Two nontrivial solutions for quasilinear periodic problems, Proc. AMS 132 (2004), 429–434.CrossRefGoogle Scholar
23.Perera, K. and Schechter, M., Solution of nonlinear equations having asymptotic limits at zero and infinity, Calc. Var. Partial Diff. Eq. 12 (2001), 359–369.CrossRefGoogle Scholar
24.Vazquez, J., A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202.CrossRefGoogle Scholar
25.Yang, X., Multiple periodic solutions of a class of p-Laplacian, J. Math. Anal. Appl. 314 (2006), 17–29.CrossRefGoogle Scholar
26.Zhang, M., The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials, J. Lond. Math. Soc. 64 (2001), 125–143.CrossRefGoogle Scholar