Skip to main content
SpringerLink
Log in

Robin problems with indefinite, unbounded potential and reaction of arbitrary growth

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

Abstract

We study an elliptic Robin problem driven by the negative Laplacian plus an indefinite and unbounded potential and with a reaction of arbitrary growth which exhibits z-dependent zeros of constant sign. We prove multiplicity theorems producing three or four nontrivial solutions, all with precise sign information. As a particular case we consider a generalized equidiffusive logistic equation with potential.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aizicovici, S., Papageorgiou, N.S, Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Am. Math. Soc. 196, 915 (2008)

  2. Ambrosetti, A., Mancini, G.: Sharp nonuniqueness results for some nonlinear problems. Nonlinear Anal. 3, 635–645 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosetti, A., Lupo, D.: On a class of nonlinear Dirichlet problems with multiple solutions. Nonlinear Anal. 8, 1145–1150 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brezis, H., Nirenberg, L.: \(H^1\) versus \(C^1\) local minimizers. C. R. Acad. Sci. Paris Sér. I Math. 317, 465–472 (1993)

    MathSciNet  MATH  Google Scholar 

  5. Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston (1993)

    Book  MATH  Google Scholar 

  6. Dunford, N., Schwartz, J.: Linear Operators I. Wiley-Interscience, New York (1958)

    MATH  Google Scholar 

  7. de Figueiredo, D., Gossez, J.P.: Strong monotonicity of eigenvalues and unique continuation. Commun. Partial Diff. Equ. 17, 339–346 (1992)

    Article  MATH  Google Scholar 

  8. Filippakis, M., Kristaly, A., Papageorgiou, N.S.: Existence of five nonzero solutions with exact sign for a \(p\)-Laplacian operator. Discret. Contin. Dyn. Syst. 24, 405–440 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Filippakis, M., Papageorgiou, N.S.: Multiple constant sign and nodal solutions for nonlinear elliptic equations with the \(p\)-Laplacian. J. Differ. Equ. 245, 1883–1922 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton (2006)

    MATH  Google Scholar 

  11. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)

    Book  MATH  Google Scholar 

  12. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Volume I: Theory. Kluwer Academic Publishers, Dordrecht (1997)

  13. Kyritsi, S., Papageorgiou, N.S.: Multiple solutions for superlinear Dirichlet problems with an indefinite potential. Ann. Mat. Pura Appl. 192, 297–315 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, C., Li, S., Liu, J.: Splitting theorem, Poincaré–Hopf theorem and jumping nonlinear problems. J. Funct. Anal. 221, 439–455 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Michael, E.: Continuous selections I. Ann. Math. 63, 361–382 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  16. Moroz, V.: Solutions of superlinear at zero elliptic equations via Morse theory. Topol. Methods Nonlinear Anal. 10, 1–11 (1997)

    MathSciNet  Google Scholar 

  17. Motreanu, D., Motreanu, V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)

    Book  MATH  Google Scholar 

  18. Mugnai, D., Papageorgiou, N.S.: Resonant nonlinear Neumann problems with indefinite weight. Ann. Sc. Norm. Super. Pisa 11, 729–788 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Palais, R.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  20. Papageorgiou, N.S., Kyritsi, S.: Handbook of Applied Analysis. Springer, New York (2009)

    MATH  Google Scholar 

  21. Papageorgiou, N.S., Papalini, F.: Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries. Isr. J. Math. 201, 761–796 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Papageorgiou, N.S., Rădulescu, V.D.: Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity. Contemp. Math. 595, 293–315 (2013)

    Article  Google Scholar 

  23. Papageorgiou, N.S., Rădulescu, V.D.: Multiple solutions with precise sign information for parametric Robin problems. J. Differ. Equ. 256, 2449–2479 (2014)

    Article  MATH  Google Scholar 

  24. Papageorgiou, N.S., Rădulescu, V.D.: Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential. Trans. Am. Math. Soc. doi:10.1090/S0002-9947-2014-06518-5

  25. Papageorgiou, N.S., Smyrlis, G.: On a class of parametric Neumann problems with indefinite and unbounded potetial. Forum Math. doi:10.1515/forum-2012-0042

  26. Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)

    MATH  Google Scholar 

  27. Struwe, M.: A note on a result of Ambrosetti and Mancini. Ann. Mat. Pura Appl. 81, 107–115 (1982)

    Article  MathSciNet  Google Scholar 

  28. Struwe, M.: Variational Methods. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  29. Wang, X.: Neumann problems of semilinear elliptic equations involving critical Sobolev exponents. J. Differ. Equ. 93, 283–310 (1991)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors wish to thank two very knowledgeable referees for their corrections and helpful remarks which improved the paper considerably. V. Rǎdulescu acknowledges the support through Grant of the Executive Council for Funding Higher Education, Research and Innovation, Romania-UEFISCDI, Project Type: Advanced Collaborative Research Projects - PCCA, No 23/2014.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Technical University, Zografou Campus, 15780, Athens, Greece

    Nikolaos S. Papageorgiou

  2. Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Vicenţiu D. Rădulescu

  3. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700, Bucharest, Romania

    Vicenţiu D. Rădulescu

Authors
  1. Nikolaos S. Papageorgiou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vicenţiu D. Rădulescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vicenţiu D. Rădulescu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Papageorgiou, N.S., Rădulescu, V.D. Robin problems with indefinite, unbounded potential and reaction of arbitrary growth. Rev Mat Complut 29, 91–126 (2016). https://doi.org/10.1007/s13163-015-0181-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-015-0181-y

Keywords

Mathematical Subject Classification

Search

Navigation