Skip to main content
SpringerLink
Log in

Symmetry of Positive Solutions to Choquard Type Equations Involving the Fractional \(p\)-Laplacian

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We study symmetric properties of positive solutions to the Choquard type equation

$$ (-\Delta )^{s}_{p} u + |x|^{a} u = \left (\frac{1}{|x|^{n-\alpha }}*u^{q} \right ) u^{r} \quad \text{in}\ \mathbb{R}^{n}, $$

where \(0< s<1\), \(0<\alpha <n\), \(p\ge 2\), \(q>1\), \(r>0\), \(a\ge 0\) and \((-\Delta )^{s}_{p}\) is the fractional \(p\)-Laplacian. Via a direct method of moving planes, we prove that every positive solution \(u\) which has an appropriate decay property must be radially symmetric and monotone decreasing about some point, which is the origin if \(a>0\).

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. Belchior, P., Bueno, H., Miyagaki, O.H., Pereira, G.A.: Remarks about a fractional Choquard equation: ground state, regularity and polynomial decay. Nonlinear Anal. 164, 38–53 (2017). https://doi.org/10.1016/j.na.2017.08.005

    Article  MathSciNet  MATH  Google Scholar 

  2. Bjorland, C., Caffarelli, L., Figalli, A.: Nonlocal tug-of-war and the infinity fractional Laplacian. Commun. Pure Appl. Math. 65(3), 337–380 (2012). https://doi.org/10.1002/cpa.21379

    Article  MathSciNet  MATH  Google Scholar 

  3. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007). https://doi.org/10.1080/03605300600987306

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, W., Li, C.: Maximum principles for the fractional \(p\)-Laplacian and symmetry of solutions. Adv. Math. 335, 735–758 (2018). https://doi.org/10.1016/j.aim.2018.07.016

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Y.-H., Liu, C.: Ground state solutions for non-autonomous fractional Choquard equations. Nonlinearity 29(6), 1827–1842 (2016). https://doi.org/10.1088/0951-7715/29/6/1827

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, W., Li, C., Ou, B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59(3), 330–343 (2006). https://doi.org/10.1002/cpa.20116

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, W., Li, C., Li, Y.: A direct method of moving planes for the fractional Laplacian. Adv. Math. 308, 404–437 (2017). https://doi.org/10.1016/j.aim.2016.11.038

    Article  MathSciNet  MATH  Google Scholar 

  8. Dai, W., Fang, Y., Qin, G.: Classification of positive solutions to fractional order Hartree equations via a direct method of moving planes. J. Differ. Equ. 265(5), 2044–2063 (2018). https://doi.org/10.1016/j.jde.2018.04.026

    Article  MathSciNet  MATH  Google Scholar 

  9. d’Avenia, P., Siciliano, G., Squassina, M.: On fractional Choquard equations. Math. Models Methods Appl. Sci. 25(8), 1447–1476 (2015). https://doi.org/10.1142/S0218202515500384

    Article  MathSciNet  MATH  Google Scholar 

  10. Ghimenti, M., Van Schaftingen, J.: Nodal solutions for the Choquard equation. J. Funct. Anal. 271(1), 107–135 (2016). https://doi.org/10.1016/j.jfa.2016.04.019

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang, X., Zhang, Z.: Symmetry of positive solutions for the fractional Schrödinger equations with Choquard-type nonlinearities, arXiv e-prints (Jun 2019). arXiv:1906.02388

  12. Jin, L., Li, Y.: A Hopf’s lemma and the boundary regularity for the fractional \(p\)-Laplacian. Discrete Contin. Dyn. Syst. 39(3), 1477–1495 (2019). https://doi.org/10.3934/dcds.2019063

    Article  MathSciNet  MATH  Google Scholar 

  13. Le, P.: Liouville theorem and classification of positive solutions for a fractional Choquard type equation. Nonlinear Anal. 185, 123–141 (2019). https://doi.org/10.1016/j.na.2019.03.006

    Article  MathSciNet  MATH  Google Scholar 

  14. Le, P.: Symmetry of singular solutions for a weighted Choquard equation involving the fractional \(p\)-Laplacian. Commun. Pure Appl. Anal. 19(1), 527–539 (2020). https://doi.org/10.3934/cpaa.2020026

    Article  MathSciNet  MATH  Google Scholar 

  15. Le, P.: Symmetry of solutions for a fractional \(p\)-Laplacian equation of Choquard type. Int. J. Math. 31, 2050026 (2020). https://doi.org/10.1142/S0129167X20500263

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, C., Wu, Z., Xu, H.: Maximum principles and Bôcher type theorems. Proc. Natl. Acad. Sci. USA 115(27), 6976–6979 (2018). https://doi.org/10.1073/pnas.1804225115

    Article  MathSciNet  MATH  Google Scholar 

  17. Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2), 93–105 (1976/1977). https://doi.org/10.1002/sapm197757293

    Article  MathSciNet  MATH  Google Scholar 

  18. Lions, P.-L.: The Choquard equation and related questions. Nonlinear Anal. 4(6), 1063–1072 (1980). https://doi.org/10.1016/0362-546X(80)90016-4

    Article  MathSciNet  MATH  Google Scholar 

  19. Lions, P.-L.: Compactness and topological methods for some nonlinear variational problems of mathematical physics. In: Nonlinear Problems: Present and Future, Los Alamos, N.M., 1981. North-Holland Math. Stud., vol. 61, pp. 17–34. North-Holland, Amsterdam (1982)

    Chapter  Google Scholar 

  20. Liu, X.: Symmetry of positive solutions for the fractional Hartree equation. Acta Math. Sci. Ser. B 39(6), 1508–1516 (2019). https://doi.org/10.1007/s10473-019-0603-x (Engl. Ed.)

    Article  MathSciNet  Google Scholar 

  21. Ma, P., Shang, X., Zhang, J.: Symmetry and nonexistence of positive solutions for fractional Choquard equations. Pac. J. Math. 304, 143–167 (2020). https://doi.org/10.2140/pjm.2020.304.143

    Article  MathSciNet  MATH  Google Scholar 

  22. Ma, L., Zhang, Z.: Symmetry of positive solutions for Choquard equations with fractional \(p\)-Laplacian. Nonlinear Anal. 182, 248–262 (2019). https://doi.org/10.1016/j.na.2018.12.015

    Article  MathSciNet  MATH  Google Scholar 

  23. Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195(2), 455–467 (2010). https://doi.org/10.1007/s00205-008-0208-3

    Article  MathSciNet  MATH  Google Scholar 

  24. Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265(2), 153–184 (2013). https://doi.org/10.1016/j.jfa.2013.04.007

    Article  MathSciNet  MATH  Google Scholar 

  25. Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent. Commun. Contemp. Math. 17(5), 1550005 (2015). https://doi.org/10.1142/S0219199715500054

    Article  MathSciNet  MATH  Google Scholar 

  26. Pekar, S.: Untersuchung über die Elektronentheorie der Kristalle. Akademie Verlag, Berlin (1954)

    MATH  Google Scholar 

  27. Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relativ. Gravit. 28(5), 581–600 (1996). https://doi.org/10.1007/BF02105068

    Article  MathSciNet  MATH  Google Scholar 

  28. Saanouni, T.: A note on the fractional Schrödinger equation of Choquard type. J. Math. Anal. Appl. 470(2), 1004–1029 (2019). https://doi.org/10.1016/j.jmaa.2018.10.045

    Article  MathSciNet  MATH  Google Scholar 

  29. Wu, L., Niu, P.: Symmetry and nonexistence of positive solutions to fractional \(p\)-Laplacian equations. Discrete Contin. Dyn. Syst. 39(3), 1573–1583 (2019). https://doi.org/10.3934/dcds.2019069

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Phuong Le

  2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Phuong Le

Authors
  1. Phuong Le
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Phuong Le.

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

Le, P. Symmetry of Positive Solutions to Choquard Type Equations Involving the Fractional \(p\)-Laplacian. Acta Appl Math 170, 387–398 (2020). https://doi.org/10.1007/s10440-020-00338-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-020-00338-6

Mathematics Subject Classification (2010)

Keywords

Search

Navigation