Skip to main content
SpringerLink
Log in

Positive solutions of semilinear schrödinger equations in two-dimensional exterior domains

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We consider the existence of a positive (generally unbounded) solution to the semilinear Schrödinger equation in a two-dimensional exterior domain. Our method relies on the comparison method, that is, the technique of super/sub-solutions. An application to Emden-Fowler equation is also presented.

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. Agarwal, R.P., Djebali, S., Moussaoui, T., Mustafa, O.G.: On the asymptotic integration of nonlinear differential equations. J. Comput. Appl. Math. 202, 352–376 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aslanov, A.: A note on the positive solutions of second order differential equations. Monatsh. Math. 183, 229–239 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bihari, I.: Researches of the boundedness and stability of the solutions of non-linear differential equations. Acta Math. Acad. Sci. Hung. 8, 261–278 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boas, M.L., Boas, R.P., Jr., Levinson, N.: The growth of solutions of a differential equation. Duke Math. J. 9, 847–853 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caligo, O.: Comportamento asintotico degli integrali dell’equazione \( y^{\prime \prime }+a(t)y=0\), nell’ipotesi \(\lim _{t\rightarrow \infty }a(t)=0\). Boll. Un. Mat. Ital. 3, 286–295 (1941)

    MathSciNet  MATH  Google Scholar 

  6. Constantin, A.: Positive solutions of Schrödinger equations in two-dimensional exterior domains. Monatsh. Math. 123, 121–126 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Constantin, A.: On the existence of positive solutions of second-order differential equations. Ann. Mat. Pura Appl. 184, 131–138 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Constantin, A., Johnson, R.S.: Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates. Proc. Roy. Soc. London A 473, 20170063 (2017). https://doi.org/10.1098/rspa.2017.0063

    Article  MathSciNet  MATH  Google Scholar 

  9. Constantin, O., Kalaj, D.: Some symmetry properties of gyre flows. Nonlinear Anal. Real World Appl. 67, 103589 (2022). https://doi.org/10.1016/j.nonrwa.2022.103589

    Article  MathSciNet  MATH  Google Scholar 

  10. Constantin, A., Germain, P.: Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere. Arch. Ration. Mech. Anal. 245, 587–644 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dube, S., Mingarelli, A.B.: Note on a non-oscillation theorem of Atkinson. Electron. J. Differ. Equ. 2004(22), 1–6 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Eastham, M.S.P.: The Asymptotic Solution of Linear Differential Systems, Applications of the Levinson Theorem. Clarendon Press, Oxford (1989)

    MATH  Google Scholar 

  13. Ertem, T., Zafer, A.: Asymptotic integration of second-order nonlinear differential equations via principal and nonprincipal solutions. Appl. Math. Comput. 219, 5876–5886 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Ertem, T., Zafer, A.: Existence of solutions for a class of nonlinear boundary value problems on half-line. Bound. Value Probl. 43 (2012)

  15. Ertem, T., Zafer, A.: Monotone positive solutions for a class of second-order nonlinear differential equations. J. Comput. Appl. Math. 259, 672–681 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hale, J.K., Onuchic, N.: On the asymptotic behavior of solutions of a class of differential equations. Contrib. Diff. Equ. 2, 61–75 (1963)

    MathSciNet  MATH  Google Scholar 

  17. Haupt, O.: Über das asymptotische verhalten der lösungen gewisser linearer gewöhnlicher differentialgleichungen. Math. Z. 48, 289–292 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kiguradze, I.T., Chanturia, T.A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic Publishers, Dordrecht (1993)

    Book  Google Scholar 

  19. Lipovan, O.: On the asymptotic behavior of the solutions to a class of second order nonlinear differential equations. Glasgow Math. J. 45, 179–187 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, T., Rogovchenko, Y.V.: Oscillation criteria for second-order superlinear Emden-Fowler neutral differential equations. Monatsh. Math. 184, 489–500 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mustafa, O.G., Rogovchenko, Y.V.: Asymptotic integration of a class of nonlinear differential equations. Appl. Math. Lett. 19, 849–853 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Noussair, E.S., Swanson, C.A.: Positive solutions of quasilinear elliptic equations in exterior domains. J. Math. Anal. Appl. 75, 121–133 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  23. Taylor, M.: Euler equation on a rotating surface. J. Funct. Anal. 270, 3884–3945 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wahlen, E.: Positive solutions of second-order differential equations. Nonlinear Anal. 58, 359–366 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yin, Z.: Bounded Positive Solutions of Schrödinger Equations in Two-Dimensional Exterior Domains. Monatsh. Math. 141, 337–344 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yin, Z.: Monotone positive solutions of second-order nonlinear differential equations. Nonlinear Anal. 54, 391–403 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhao, Z.: Positive Solutions of Nonlinear Second Order Ordinary Differential Equations. Proc. Am. Math. Soc. 121, 465–469 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Engineering, Istanbul Esenyurt University, Istanbul, Turkey

    Afgan Aslanov

  2. Department of Mathematics, Khazar University, Baku, Azerbaijan

    Afgan Aslanov

Authors
  1. Afgan Aslanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Afgan Aslanov.

Ethics declarations

Conflicts of interest

The author declares that there is no conflict of interests regarding the publication of this paper.

Additional information

Communicated by Adrian Constantin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aslanov, A. Positive solutions of semilinear schrödinger equations in two-dimensional exterior domains. Monatsh Math 201, 963–973 (2023). https://doi.org/10.1007/s00605-022-01761-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-022-01761-7

Keywords

Mathematics Subject Classification

Search

Navigation