Skip to main content
SpringerLink
Log in

Classification of Ground-states of a coupled Schrödinger system

  • Published:
Nonlinear Differential Equations and Applications NoDEA Aims and scope Submit manuscript

Abstract

The paper is concerned with the existence of nontrivial ground-state solutions for a coupled nonlinear Schrödinger system

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u_j+ \lambda u_j=\mu |u_j|^{2p}u_j+\sum _{i\ne j}^m \beta |u_i|^{p+1}|u_j|^{p-1}u_j, &{} \text {in}\ \mathbb {R}^n, \\ u_j(x)\rightarrow 0\ \text {as}\ |x|\ \rightarrow \infty , \quad j=1,2,\ldots , m, \end{array}\right. \end{aligned}$$

where \( m\ge 2\), \(0<p<\frac{2}{(n-2)^+}\), \(\lambda >0,\) \(\mu >0\) and \(\beta >0\). We establish a sufficient and necessary condition for the existence of nontrivial ground-state solutions which have the least energy among all the non-zero solutions of the system and whose components have the same modulus. This gives an affirmative answer to a conjecture raised in Correia (Nonlinear Anal. 140:112–129, 2016).

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Ambrosetti, A., Colorado, E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. 75, 67–82 (2007)

    Article  MathSciNet  Google Scholar 

  2. Bartsch, T., Wang, Z.-Q.: Note on ground states of nonlinear Schrödinger systems. J. Partial Differ. Equ. 19, 200–207 (2006)

    MATH  Google Scholar 

  3. Bartsch, T., Wang, Z.-Q., Wei, J.: Bound states for a coupled Schrödinger system. J. Fixed Point Theory Appl. 2, 353–367 (2007)

    Article  MathSciNet  Google Scholar 

  4. Benney, D.J., Newell, A.C.: The propagation of nonlinear wave envelopes. J. Math. Phys. 46, 133–139 (1967)

    Article  MathSciNet  Google Scholar 

  5. Chen, Z., Zou, W.: An optimal constant for the existence of least energy solutions of a coupled Schröinger system. Calc. Var. Partial Differ. Equ. 48, 695–711 (2013)

    Article  Google Scholar 

  6. Correia, S.: Ground-states for systems of m-coupled semilinear Schrödinger equations with attraction-repulsion effects: characterization and perturbation results. Nonlinear Anal. 140, 112–129 (2016)

    Article  MathSciNet  Google Scholar 

  7. Correia, S.: Characterization of ground-states for a system of m-coupled semilinear Schrödinger equations and applications. J. Differ. Equ. 260, 3302–3326 (2016)

    Article  Google Scholar 

  8. Correia, S., Oliveira, F., Tavares, H.: Semitrivial vs fully nontrivial ground states in cooperative cubic Schrödinger systems with \(d\ge 3\) equations. J. Funct. Anal. 271, 2247–2273 (2016)

    Article  MathSciNet  Google Scholar 

  9. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers I. Anormalous dispersion. Appl. Phys. Lett. 23, 142–144 (1973)

    Article  Google Scholar 

  10. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers II. Normal dispersion. Appl. Phys. Lett. 23, 171–172 (1973)

    Article  Google Scholar 

  11. Lin, T.C., Wei, J.: Ground state of N coupled nonlinear Schrödinger equations in \({\mathbb{R}}^n, n\le 3\). Commun. Math. Phys. 255, 629–653 (2005)

    Article  Google Scholar 

  12. Lin, T.C., Wei, J.: Ground State of N Coupled Nonlinear Schrödinger Equations in \({\mathbb{R}}^n, n\le 3\). Commun. Math. Phys. 277, 573–576 (2008)

    Article  Google Scholar 

  13. Liu, C., Wang, Z.-Q.: Synchronization of positive solutions for coupled Schrödinger equations. Discrete Contin. Dyn. Syst. 38(6), 2811–2824 (2018)

    Article  MathSciNet  Google Scholar 

  14. Liu, H., Liu, Z., Chang, J.: Existence and uniquiness of positive solutions of nonlinear Schrödinger systems. Proc. R. Soc. Edinburgh Sect. A Math. 145, 365–390 (2015)

    Article  Google Scholar 

  15. Liu, Z., Wang, Z.-Q.: Ground States and Bound States of a Nonlinear Schrödinger system. Adv. Nonlinear Stud. 10, 175–193 (2010)

    Article  MathSciNet  Google Scholar 

  16. Maia, L.A., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229, 743–767 (2006)

    Article  Google Scholar 

  17. Mandel, R.: Minimal energy solutions for cooperative nonlinear Schrodinger systems. NoDEA 22, 239–262 (2015)

    Article  MathSciNet  Google Scholar 

  18. Mitchell, M., Segev, M.: Self-trapping of inconherent white light. Nature 387, 880–882 (1997)

    Article  Google Scholar 

  19. Oliveira, F., Tavares, H.: Ground States for a nonlinear Schrödinger system with sublinear coupling terms. Adv. Nonlinear Stud. 16, 381–387 (2015)

    MATH  Google Scholar 

  20. Roskes, G.J.: Some nonlinear multiphase interactions. Stud. Appl. Math. 55, 231–238 (1976)

    Article  Google Scholar 

  21. Rüegg, C., et al.: Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl\(_3\). Nature 423, 62–65 (2003)

    Article  Google Scholar 

  22. Sirakov, B.: Least Energy Solitary Waves for a System of Nonlinear Schrödinger Equations in \({\mathbb{R}}^n\). Commun. Math. Phys. 271, 199–221 (2007)

    Article  Google Scholar 

  23. Yang, J.: Multiple permanent-wave trains in nonlinear systems. Stud. Appl. Math. 100, 127–152 (1998)

    Article  MathSciNet  Google Scholar 

  24. Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. Sov. Phys. J. Appl. Mech. Tech. Phys. 4, 190–194 (1968)

    Google Scholar 

  25. Zakharov, V.E.: Collapse of Langmuir waves. Sov. Phys. Jetp. 35, 908–914 (1972)

    Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referee’s thoughtful and valuable comments and suggestions which improved the presentation of the paper. The work is supported by NSFC 11971191, 11771324, 11831009. The first author was also supported by the Fundamental Research Funds for the Central Universities (Nos. KJ02072020-0319 and CCNU19QN079) and he gratefully acknowledges the financial support from China Scholarship Council and would like to thank Utah State University for hosting his visit during which this work was carried out.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Luo-Yu Road 152, Wuhan, 430079, People’s Republic of China

    Chuangye Liu

  2. College of Mathematics and Informatics, Fujian Normal University, Fuzhou, 350117, China

    Zhi-Qiang Wang

  3. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    Zhi-Qiang Wang

Authors
  1. Chuangye Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi-Qiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi-Qiang Wang.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, C., Wang, ZQ. Classification of Ground-states of a coupled Schrödinger system. Nonlinear Differ. Equ. Appl. 28, 21 (2021). https://doi.org/10.1007/s00030-021-00685-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00030-021-00685-9

Keywords

Mathematics Subject Classification

Search

Navigation