Skip to main content
SpringerLink
Log in

Existence of a positive solution for a logarithmic Schrödinger equation with saddle-like potential

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

In this article we use the variational method developed by Szulkin (Ann Inst H Poincaré Anal Non Linéire 3:77–109, 1986) to prove the existence of a positive solution for the following logarithmic Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} -\,{\epsilon }^2\Delta u+ V(x)u=u \log u^2, &{}\quad \text{ in } \,\, {\mathbb {R}}^{N}, \\ u \in H^1({\mathbb {R}}^{N}), &{} \\ \end{array} \right. \end{aligned}$$

where \(\epsilon >0, N \ge 1\) and V is a saddle-like 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. Alves, C.O.: Existence of a positive solution for a nonlinear elliptic equation with saddle-like potential and nonlinearity with exponential critical growth in \({\mathbb{R}}^{2}\). Milan J. Math. 84, 1–22 (2016)

    Article  MathSciNet  Google Scholar 

  2. Alves, C.O., de Morais Filho, D.C.: Existence of concentration of positive solutions for a Schrödinger logarithmic equation. Z. Angew. Math. Phys. 69, 144 (2018). https://doi.org/10.1007/s00033-018-1038-2

    Article  MATH  Google Scholar 

  3. Alves, C.O., de Morais Filho, D.C., Figueiredo, G.M.: On concentration of solution to a Schrödinger logarithmic equation with deepening potential well. Math. Methods Appl. Sci. 42, 4862–4875 (2019)

    Article  MathSciNet  Google Scholar 

  4. Alves, C.O., Ji, C.: Multiple positive solutions for a Schrödinger logarithmic equation. Discrete Contin. Dyn. Syst. 40, 2671–2685 (2020)

    Article  MathSciNet  Google Scholar 

  5. Alves, C.O., Miyagaki, O.H.: A critical nonlinear fractional elliptic equation with saddle-like potential in \({\mathbb{R}}^{N}\). J. Math. Phys. 57, 081501 (2016)

    Article  MathSciNet  Google Scholar 

  6. Bartsch, T., Pankov, A., Wang, Z.-Q.: Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 3, 1–21 (2001)

    Article  MathSciNet  Google Scholar 

  7. Bartsch, T., Wang, Z.-Q.: Existence and multiplicity results for some superlinear elliptic problems on \({\mathbb{R}}^{N}\). Commun. Partial Differ. Equ. 20, 1725–1741 (1995)

    Article  Google Scholar 

  8. d’Avenia, P., Montefusco, E., Squassina, M.: On the logarithmic Schrödinger equation. Commun. Contemp. Math. 16, 1350032 (2014)

    Article  MathSciNet  Google Scholar 

  9. d’Avenia, P., Squassina, M., Zenari, M.: Fractional logarithmic Schrödinger equations. Math. Methods Appl. Sci. 38, 5207–5216 (2015)

    Article  MathSciNet  Google Scholar 

  10. del Pino, M., Felmer, P.L.: Local Mountain Pass for semilinear elliptic problems in unbounded domains. Cal. Var. Partial Differ. Equ. 4, 121–137 (1996)

    Article  Google Scholar 

  11. del Pino, M., Felmer, P.L., Miyagaki, O.H.: Existence of positive bound states of nonlinear Schrödinger equations wwith saddle-like potential. Nonlinear Anal. 34, 979–989 (1998)

    Article  MathSciNet  Google Scholar 

  12. Degiovanni, M., Zani, S.: Multiple solutions of semilinear elliptic equations with one-sided growth conditions, nonlinear operator theory. Math. Comput. Model. 32, 1377–1393 (2000)

    Article  Google Scholar 

  13. Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equations with bounded potential. J. Funct. Anal. 69, 397–408 (1986)

    Article  MathSciNet  Google Scholar 

  14. Ji, C., Szulkin, A.: A logarithmic Schrödinger equation with asymptotic conditions on the potential. J. Math. Anal. Appl. 437, 241–254 (2016)

    Article  MathSciNet  Google Scholar 

  15. Lions, P.L.: The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 2. Anal. Inst. H. Poincaré Sect. C 1, 223–253 (1984)

    Article  Google Scholar 

  16. Lieb, E.H., Loss, M.L.: Analysis, Second Edition, Graduate Studies in Mathematics 14. AMS, Providence (2001)

  17. Oh, Y.J.: Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials on the class \((V)_{a}\). Commun. Partial Differ. Equ. 13, 1499–1519 (1988)

    Article  Google Scholar 

  18. Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)

    Article  MathSciNet  Google Scholar 

  19. Squassina, M., Szulkin, A.: Multiple solution to logarithmic Schrödinger equations with periodic potential. Cal. Var. Partial Differ. Equ. 54, 585–597 (2015)

    Article  Google Scholar 

  20. Squassina, M., Szulkin, A.: Erratum to: Multiple solutions to logarithmic Schrödinger equations with periodic potential. Cal. Var. Partial Differ. Equ. (2017). https://doi.org/10.1007/s00526-017-1127-7

    Article  MATH  Google Scholar 

  21. Szulkin, A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré Anal. Non Linéire 3, 77–109 (1986)

    Article  MathSciNet  Google Scholar 

  22. Tanaka, K., Zhang, C.: Multi-bump solutions for logarithmic Schrödinger equations. Cal. Var. Partial Differ. Equ. 56, 35 (2017). (Art. 33)

    Article  Google Scholar 

  23. Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 53, 229–244 (1993)

    Article  Google Scholar 

  24. Zloshchastiev, K.G.: Logarithmic nonlinearity in the theories of quantum gravity: origin of time and observational consequences. Grav. Cosmol. 16, 288–297 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions and comments.

Author information

Authors and Affiliations

  1. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, CEP: 58429-900, Brazil

    Claudianor O. Alves

  2. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, People’s Republic of China

    Chao Ji

Authors
  1. Claudianor O. Alves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chao Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chao Ji.

Additional information

Publisher's Note

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

C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7. C. Ji was partially supported by Shanghai Natural Science Foundation (18ZR1409100).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alves, C.O., Ji, C. Existence of a positive solution for a logarithmic Schrödinger equation with saddle-like potential. manuscripta math. 164, 555–575 (2021). https://doi.org/10.1007/s00229-020-01197-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-020-01197-z

Mathematics Subject Classification

Search

Navigation