Skip to main content
SpringerLink
Log in
Book cover

Elliptic and Parabolic Problems pp 111–118Cite as

Gaeta 2004. Elliptic Resonant Problems with a Periodic Nonlinearity

  • Chapter

  • 1340 Accesses

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 63))

Abstract

We study the existence of solution for a nonlinear PDE problem at resonance under Dirichlet boundary conditions. The nonlinear term considered comes from a periodic function: in particular, the problem is strongly resonant at infinity. In our proofs we shall use variational methods together with some asymptotic analysis.

The authors have been supported by the Ministry of Science and Technology of Spain (BFM2002-02649), and by J. Andalucía (FQM 116).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Arcoya, Periodic solutions of Hamiltonian systems with strong resonance at infinity, Diff. and Int. Eq., 3 (1990), 909–921.

    MathSciNet  MATH  Google Scholar 

  2. D. Arcoya and A. Cañada, Critical point theorems and applications to nonlinear boundary value problems, Nonl. Anal. TMA, 14 (1990), 393–411.

    MATH  Google Scholar 

  3. E.J. Banning, J.P. van der Weele, J.C. Ross and M.M. Kettenis, Mode competition in a system of two parametrically driven pendulums with nonlinear coupling, Physica A, 245 (1997), 49–98.

    MathSciNet  Google Scholar 

  4. P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonl. Anal. TMA, 7 (1983), 981–1012.

    MathSciNet  MATH  Google Scholar 

  5. D. Bonheure, C. Fabry and D. Ruiz, Problems at resonance for equations with periodic nonlinearities, Nonl. Anal. 55 (2003), 557–581.

    MathSciNet  MATH  Google Scholar 

  6. A. Cañada, A note on the existence of global minimum for a noncoercive functional, Nonl. Anal. 21 (1993) 161–166.

    MATH  Google Scholar 

  7. A. Cañada, Nonlinear ordinary boundary value problems under a combined effect of periodic and attractive nonlinearities, J. Math. Anal. Appl., 243 (2000), 174–189.

    MathSciNet  MATH  Google Scholar 

  8. A. Cañada and F. Roca, Existence and multiplicity of solutions of some conservative pendulum-type equations with homogeneous Dirichlet conditions Diff. Int. Eqns. 10 (1997) 1113–1122

    MATH  Google Scholar 

  9. A. Cañada and D. Ruiz, Resonant problems with multi-dimensional kernel and periodic nonlinearities, Diff. Int. Eq. vol 16, No. 4 (2003), 499–512.

    MATH  Google Scholar 

  10. A. Cañada and D. Ruiz, Asymptotic analysis of oscillating parametric integrals and ordinary boundary value problems at resonance, preprint.

    Google Scholar 

  11. A. Cañada and D. Ruiz, Periodic perturbations of resonant problems, to appear in Calc. Var PDE.

    Google Scholar 

  12. I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York 1984.

    MATH  Google Scholar 

  13. D. Costa, H. Jeggle, R. Schaaf and K. Schmitt, Oscillatory perturbations of linear problems at resonance, Results Math. 14 (1988), No. 3–4, 275–287.

    MathSciNet  MATH  Google Scholar 

  14. E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems Ann. Mat. Pura Appl. 131 (1982) 167–185.

    Article  MathSciNet  MATH  Google Scholar 

  15. P. Drábek and S. Invernizzi, Periodic solutions for systems of forced coupled pendulum-like equations, J. Diff. Eq., 70, (1987), 390–402.

    MATH  Google Scholar 

  16. D. Lupo and S. Solimini, A note on a resonance problem, Proc. Royal Soc. Edinburgh 102 A (1986), 1–7.

    MathSciNet  Google Scholar 

  17. J. Mawhin, Problèmes non linéaires, Sémin. Math. Sup. No. 104, Presses Univ. Montréal, 1987.

    Google Scholar 

  18. J. Mawhin, Forced second order conservative systems with periodic nonlinearity, Ann. Inst. H. Poincaré, special issue dedicated to J.J. Moreau, 6, (1989), Suppl., 415–434.

    MathSciNet  MATH  Google Scholar 

  19. R. Schaaf and K. Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. AMS 306 (1988), 853–859.

    MathSciNet  MATH  Google Scholar 

  20. R. Schaaf and K. Schmitt, Periodic perturbations of linear problems at resonance in convex domains, Rocky Mount. J. Math. 20 (1990), 541–559.

    Article  MathSciNet  Google Scholar 

  21. R. Schaaf and K. Schmitt, Asymptotic behavior of positive branches of elliptic problems with linear part at resonance, Z. angew. Math. Phys. 43 (1992), 645–676.

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Solimini, On the solvability of some elliptic partial differential equations with the linear part at resonance, J. Math. Anal. Appl. 117 (1986), 138–152.

    Article  MathSciNet  MATH  Google Scholar 

  23. J.R. Ward, A boundary value problem with a periodic nonlinearity, Nonlinear Analysis 10 (1986), 207–213.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Granada, E-18071, Granada, Spain

    A. Cañada & D. Ruiz

Authors
  1. A. Cañada
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Ruiz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051, Basel, Switzerland

    Catherine Bandle

  2. Ecole des hautes études en sciences sociales (EHESS), CAMS, 54, Boulevard Raspail, 75006, Paris, France

    Henri Berestycki

  3. Faculté des Sciences et Techniques, Université de Haute-Alsace, 4 rue des frères Lumières, 68093, Mulhouse Cedex, France

    Bernard Brighi

  4. Laboratoire de Gestion des Risques et Environnement, Université de Haute-Alsace, 25, rue de Chemnitz, 68200, Mulhouse, France

    Alain Brillard

  5. Angewandte Mathematik, Universität Zürich, Winterthurerstr. 190, 8057, Zürich, Switzerland

    Michel Chipot

  6. Département de mathématiques, Université de Paris-Sud, Bâtiment 425, 91405, Orsay, France

    Jean-Michel Coron

  7. Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Via Cintia, 80126, Napoli, Italy

    Carlo Sbordone

  8. Department of Mathematics, Technion — Israel Institute of Technology, 32000, Haifa, Israel

    Itai Shafrir

  9. CNR-IAC, Viale del Policlinico, 137, 00161, Roma, Italy

    Vanda Valente

  10. Dipartimento di metodi e modelli matematici per le Scienze Aplicate, Università di Roma “La Sapienza”, Via A. Scarpa 16, 00161, Roma, Italy

    Giorgio Vergara Caffarelli

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Cañada, A., Ruiz, D. (2005). Gaeta 2004. Elliptic Resonant Problems with a Periodic Nonlinearity. In: Bandle, C., et al. Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 63. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7384-9_12

Download citation

Publish with us

Policies and ethics

Search

Navigation