Symmetric Properties for Choquard Equations Involving Fully Nonlinear Nonlocal Operators

Abstract

In this paper we consider the following nonlinear nonlocal Choquard equation

$$\begin{aligned} {{\mathcal {F}}} _{\alpha }\left( u(x)\right) +\omega u(x) = C_{n,2s} \left( |x|^{2s-n}*u^q(x)\right) u^r(x), ~ x\in {\mathbb {R}}^n, \end{aligned}$$

where \(0<s<1, ~0< \alpha < 2, {\mathcal {F}_\alpha }\) is the fully nonlinear nonlocal operator:

$$\begin{aligned} {\mathcal {F}_\alpha }(u(x)) = {C_{n,\alpha }}P.V.\int _{{\mathbb {R}^n}} {\frac{{F(u(x) - u(y))}}{{{{\left| {x - y} \right| }^{n + \alpha }}}}dy}. \end{aligned}$$

The positive solution to nonlinear nonlocal Choquard equation is shown to be symmetric and monotone by using the moving plane method which has been introduced by Chen, Li and Li in 2015. We first turn single equation into equivalent system of equations. Then the key ingredients are to obtain the “narrow region principle” and “decay at infinity” for the corresponding problems. We also get radial symmetry results of positive solution for the Schrödinger-Maxwell nonlocal equation. Similar ideas can be easily applied to various nonlocal problems with more general nonlinearities.

This is a preview of subscription content, access via your institution.

References

  1. Alexandrov, A.D.: A characteristic property of spheres. Ann. Mat. Pura Appl. 58, 303–315 (1962)

    MathSciNet  Article  Google Scholar 

  2. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Part. Differ. Equ. 32, 1245–1260 (2007)

    MathSciNet  Article  Google Scholar 

  3. Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62, 597–638 (2009)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  5. Cao, L.F., Wang, X.S., Dai, Z.H.: Radial symmetry and monotonicity of solutions to a system involving fractional p-Laplacian in a ball. Adv. Math. Phys. 1565731, 6 (2018)

    MathSciNet  Google Scholar 

  6. Chang, S., del Mar-González, M.: Fractional Laplacian in conformal geometry. Adv. Math. 226, 1410–1432 (2011)

    MathSciNet  Article  Google Scholar 

  7. Chen, L., Lee, J.O.: Rate of convergence in nonlinear Hartree dynamics with factorized initial data. J. Math. Phys. 52(052108), 25 (2011)

    MathSciNet  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  9. Chen, W.X., Li, C.M., Ou, B.: Classification of solutions for a system of integral equations. Comm. Part. Differ. Equ. 30, 59–65 (2005)

    MathSciNet  Article  Google Scholar 

  10. Chen, L., Lee, J.O., Schlein, B.: Rate of convergence towards Hartree dynamics. J. Stat. Phys. 144, 872–903 (2011)

    MathSciNet  Article  Google Scholar 

  11. Chen, W.X., Li, C.M., Li, Y.: A direct method of moving planes for the fractional Laplacian. Adv. Math. 308, 404–437 (2017)

    MathSciNet  Article  Google Scholar 

  12. Chen, W.X., Li, C.M., Li, G.F.: Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions. Calc. Var. Part. Differ. Equ. 56, 29 (2017)

    MathSciNet  Article  Google Scholar 

  13. Chen, W.X., Li, Y., Ma, P.: The Fractional Laplacian. World Scientific Publishing Company, Singapore (2019)

    Google Scholar 

  14. Chen, W.X., Li, C.M., Zhu, J.Y.: Fractional equations with indefinite nonlinearities. Discrete Contin. Dyn. Syst. 39, 1257–1268 (2019)

    MathSciNet  Article  Google Scholar 

  15. Dai, W., Liu, Z.: Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities. Calc. Var. Part. Differ. Equ. 58, 156 (2019)

    Article  Google Scholar 

  16. Dai, W., Qin, G.L.: Classification of positive smooth solutions to third-order PDEs involving fractional Laplacians. Pacific J. Math. 295, 367–383 (2018)

    MathSciNet  Article  Google Scholar 

  17. Dai, W., Fang, Y.Q., Qin, G.L.: Classification of positive solutions to fractional order Hartree equations via a direct method of moving planes. J. Differ. Equ. 265, 2044–2063 (2018)

    MathSciNet  Article  Google Scholar 

  18. Dipierro, S., Palatucci, G., Valdinoci, E.: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Le Mate. 68, 201–216 (2012)

    Article  Google Scholar 

  19. Erdős, L., Yau, H.T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5, 1169–1205 (2001)

    MathSciNet  Article  Google Scholar 

  20. Fall, M.M., Mahmoudi, F., Valdinoci, E.: Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity 28, 1937–1961 (2015)

    MathSciNet  Article  Google Scholar 

  21. Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209–243 (1979)

    MathSciNet  Article  Google Scholar 

  22. Lewin, M., Nam, P.T., Rougerie, N.: Derivation of Hartree’s theory for generic mean-field Bose systems. Adv. Math. 254, 570–621 (2014)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  24. Li, Y.Y., Zhu, M.J.: Uniqueness theorems through the method of moving spheres. Duke Math. J. 80, 383–417 (1995)

    MathSciNet  Article  Google Scholar 

  25. Lieb, E.H., Yau, H.T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Comm. Math. Phys. 112, 147–174 (1987)

    MathSciNet  Article  Google Scholar 

  26. Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose Gas and its Condensation. Birkhäuser, Basel (2005). (Oberwolfach Seminars)

    Google Scholar 

  27. Lin, C.S.: A classification of solutions of a conformally invariant fourth order equation in \({\mathbb{R}}^n\). Comment. Math. Helv. 73, 206–231 (1998)

    MathSciNet  Article  Google Scholar 

  28. Ma, L.W., Zhang, Z.Q.: Symmetry of positive solutions for Choquard equations with fractional \(p\)-Laplacian. Nonlinear Anal. 182, 248–262 (2019)

    MathSciNet  Article  Google Scholar 

  29. Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Rational Mech. Anal. 195, 455–467 (2010)

    MathSciNet  Article  Google Scholar 

  30. Ma, P., Shang, X., Zhang, J.H.: Symmetry and nonexistence of positive solutions for fractional Choquard equations. Pac. J. Math. 304, 143–167 (2020)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  32. Moroz, V., Van Schaftingen, J.: A guide to the Choquard equation. J. Fixed Point Theory Appl. 19, 773–813 (2017)

    MathSciNet  Article  Google Scholar 

  33. Niu, P.C., Wu, L.Y., Ji, X.X.: Positive solutions to nonlinear systems involving fully nonlinear fractional operators. Fract. Calc. Appl. Anal. 21, 552–574 (2018)

    MathSciNet  Article  Google Scholar 

  34. Pickl, P.: A simple derivation of mean field limits for quantum systems. Lett. Math. Phys. 97, 151–164 (2011)

    MathSciNet  Article  Google Scholar 

  35. Rodnianski, I., Schlein, B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Comm. Math. Phys. 291, 31–61 (2009)

    MathSciNet  Article  Google Scholar 

  36. Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318 (1971)

    MathSciNet  Article  Google Scholar 

  37. Wang, P.Y., Niu, P.C.: A direct method of moving planes for a fully nonlinear nonlocal system. Commun. Pure Appl. Anal. 16, 1707–1718 (2017)

    MathSciNet  Article  Google Scholar 

  38. Wang, P.Y., Niu, P.C.: Symmetric properties of positive solutions for fully nonlinear nonlocal system. Nonlinear Anal. 187, 134–146 (2019)

    MathSciNet  Article  Google Scholar 

  39. Wang, P.Y., Yu, M.: Solutions of fully nonlinear nonlocal systems. J. Math. Anal. Appl. 450, 982–995 (2017)

    MathSciNet  Article  Google Scholar 

  40. Wei, J., Wu, Y.: Ground states of Nonlinear Schrö dinger System with Mixed Couplings. arXiv:1903.05340 (2019)

Download references

Acknowledgements

The work was carried out when the first author visits University of Mannheim in Germany. Pengyan Wang is supported by the scholarship of NPU’s exchange funding program. Li Chen is partially supported by DFG Project CH 955/4-1. Pengcheng Niu is supported by National Natural Science Foundation of China (Grant no.11771354).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Pengcheng Niu.

Additional information

Publisher's Note

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

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, P., Chen, L. & Niu, P. Symmetric Properties for Choquard Equations Involving Fully Nonlinear Nonlocal Operators. Bull Braz Math Soc, New Series (2021). https://doi.org/10.1007/s00574-020-00234-5

Download citation

Keywords

  • Nonlinear nonlocal Choquard equation
  • Fully nonlinear nonlocal operator
  • Decay at infinity
  • Narrow region principle
  • Method of moving planes

Mathematics Subject Classification

  • 35R11
  • 35A09
  • 35B06
  • 35B09