Skip to main content
SpringerLink
Log in

On a Coupled Schrödinger System with Stein–Weiss Type Convolution Part

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In this paper, we are interested in a coupled Schrödinger system with Stein–Weiss type convolution part. Firstly we study the existence and nonexistence of the solutions by variational methods. Second, by changing the system into an equivalent integral form, we study the symmetry, regularity and asymptotic behaviors of the solutions by moving plane arguments.

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. Akhmediev, N., Ankiewicz, A.: Partially coherent solitons on a finite background. Phys. Rev. Lett. 82, 26–61 (1999)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Alves, C.O., Figueiredo, G.M., Yang, M.: Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity. Adv. Nonlinear Anal. 5, 331–345 (2016)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Beckner, W.: Pitt’s inequality with sharp convolution estimates. Proc. Am. Math. Soc. 136, 1871–1885 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beckner, W.: Weighted inequalities and Stein-Weiss potentials. Forum. Math. 20, 587–606 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, L., Liu, Z., Lu, G.: Symmetry and regularity of solutions to the weighted Hardy-Sobolev type system. Adv. Nonlinear Stud. 16, 1–13 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, W., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63, 615–622 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, W., Li, C., Ou, B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59, 330–343 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, W., Li, C.: Methods on nonlinear elliptic equations, AIMS Book Series on Diff. Equa. Dyn. Sys. 4, (2010)

  12. Chen, W., Jin, C., Li, C., Lim, J.: Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Discret. Contin. Dyn. Sys. 2005, 164–172 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Chen, W., Li, C.: The best constant in a weighted Hardy-Littlewood-Sobolev inequality. Calc. Var. Partial Differ. Equ. 136, 955–962 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Chen, Z., Zou, W.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case. Calc. Var. Partial Differ. Equ. 52, 423–467 (2015)

    Article  MATH  Google Scholar 

  15. Chen, Z., Zou, W.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent. Arch. Ration. Mech. Anal. 205, 515–551 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, Z., Lin, C., Zou, W.: Sign-changing solutions and phase separation for an elliptic system with critical exponent. Commun. Partial Differ. Equ. 39, 1827–1859 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dancer, E., Wei, J.: Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction. Tran. Am. Math. Soc. 361, 1189–1208 (2009)

    Article  MATH  Google Scholar 

  18. Ding, Y., Gao, F., Yang, M.: Semiclassical states for Choquard type equations with critical growth: critical frequency case. Nonlinearity 33, 6695–6728 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Du, L., Gao, F., Yang, M.: Existence and qualitative analysis for nonlinear weighted Choquard equation, arXiv:1810.11759

  20. Du, L., Yang, M.: Uniqueness and nondegeneracy of solutions for a critical nonlocal equation. Discret. Contin. Dyn. Syst. 39, 5847–5866 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Elgart, A., Schlein, B.: Mean field dynamics of boson stars. Commun. Pure Appl. Math. 60, 500–545 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gidas, B., Ni, W., Nirenberg, : Symmetry of positive solutions of nonlinear elliptic equations in \({\mathbb{R}}^{N}\). Math. Anal. Appl. part A, 369–402 (1981)

    MATH  Google Scholar 

  23. Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34, 525–598 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gao, F., Silva, E., Yang, M., Zhou, J.: Existence of solutions for critical Choquard equations via the concentration compactness method. Proc. R. Soc. Edinb. Sect. A 150, 921–954 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gao, F., Liu, H., Moroz, V., Yang, M.: High energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents, arXiv:2009.03102

  26. Gao, F., Yang, M., Zhou, J.: Existence of multiple semiclassical solutions for a critical Choquard equation with indefinite potential. Nonlinear Anal. TMA 195, 111817 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Giacomoni, J., Wei, Y., Yang, M.: Nondegeneracy of solutions for a critical Hartree equation. Nonlinear Anal. TMA 199, 111969 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Giacomoni, J., Goel, D., Sreenadh, K.: Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities. Adv. Nonlinear Anal. https://doi.org/10.1007/s12220-020-00441-y (2020)

  29. Ikoma, N., Tanaka, K.: A local mountain pass type result for a system of nonlinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 40, 449–480 (2011)

    Article  MATH  Google Scholar 

  30. Jin, C., Li, C.: Quantitative analysis of some system of integral equations. Calc. Var. Partial Differ. Equ. 26, 447–457 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, C.: Local asymptotic symmetry of singular solutions to nonlinear elliptic equations. Invent. Math. 123, 221–231 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu, S.: Regularity, symmetry, and uniqueness of some integral type quasilinear equations. Nonlinear Anal. 71, 1796–1806 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lei, Y.: Qualitative analysis for the static Hartree-type equations. SIAM J. Math. Anal. 45, 388–406 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lei, Y.: Liouville theorems and classification results for a nonlocal Schrödinger equation. Discret. Contin. Dyn. Syst. 38, 5351–5377 (2018)

    Article  MATH  Google Scholar 

  35. Liu, X., Lei, Y.: Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discret. Contin. Dyn. Syst. 40, 467–489 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lei, Y., Li, C., Ma, C.: Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations. Calc. Var. Partial Differ. Equ. 16, 1–13 (2016)

    Google Scholar 

  37. Li, Y.: Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. 6, 153–180 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lin, T., Wei, J.: Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéare 221, 403–439 (2005)

    Article  MATH  Google Scholar 

  39. Lin, T., 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  MATH  Google Scholar 

  40. Lin, T., Wei, J.: Spikes in two-component systems of Nonlinear Schrödinger equations with trapping potentials. J. Differ. Equ. 299, 538–569 (2006)

    Article  MATH  Google Scholar 

  41. Lieb, E., Loss, M.: Analysis. Gradute Studies in Mathematics. AMS, Providence (2001)

    Google Scholar 

  42. Lieb, E.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math. 118, 349–374 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lewin, M., Lenzmann, E.: On singularity formation for the \(L^{2}\)-critical boson star equation. Nonlinearity 24, 3515–3540 (2011)

    MathSciNet  MATH  Google Scholar 

  44. Liu, S.: Regularity, symmetry, and uniqueness of some integral type quasilinear equations. Nonlinear Anal. TMA 71, 1796–1806 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Miao, C., Xu, G., Zhao, L.: Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case. Colloq. Math. 114, 213–236 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. Stein, E., Weiss, G.: Fractional integrals on n-dimensional Euclidean space. J. Math. Mech. 7, 503–514 (1958)

    MathSciNet  MATH  Google Scholar 

  48. Wei, J., Xu, X.: Classification of solutions of higher order conformally invariant equations. Math. Ann. 313, 207–228 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  49. You, S., Zhao, P., Wang, Q.: Positive ground states for coupled nonlinear Choquard equations involving Hardy-Littlewood-Sobolev critical exponent. Nonlinear Anal. Real World Appl. 48, 182–211 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  50. You, S., Wang, Q., Zhao, P.: Positive least energy solutions for coupled nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponent. Topol. Methods Nonlinear Anal. 53, 623–657 (2019)

    MathSciNet  MATH  Google Scholar 

  51. Yang, M., Wei, Y., Ding, Y.: Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities. Z. Angew. Math. Phys. 65, 41–68 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zheng, Y., Santos, C.A., Shen, Z., Yang, M.: Least energy solutions for coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents. Commun. Pure Appl. Anal. 19, 329–369 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  53. Zhen, Y., Gao, F., Shen, Z., Yang, M.: On a class of coupled critical Hartree system with deepening potential. Math. Methods Appl. Sci. 44, 772–798 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, People’s Republic of China

    Minbo Yang & Xianmei Zhou

Authors
  1. Minbo Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xianmei Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minbo Yang.

Additional information

Publisher's Note

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

Minbo Yang is the corresponding author who was partially supported by NSFC (11971436, 12011530199) and ZJNSF (LD19A010003).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, M., Zhou, X. On a Coupled Schrödinger System with Stein–Weiss Type Convolution Part. J Geom Anal 31, 10263–10303 (2021). https://doi.org/10.1007/s12220-021-00645-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-021-00645-w

Keywords

Mathematics Subject Classification

Search

Navigation