Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions

  • Wenxiong Chen
  • Congming Li
  • Guanfeng Li


In this paper, we consider equations involving fully nonlinear non-local operators
$$\begin{aligned} F_{\alpha }(u(x)) \equiv C_{n,\alpha } PV \int _{{\mathbb {R}}^n} \frac{G(u(x)-u(z))}{|x-z|^{n+\alpha }} dz= f(x,u). \end{aligned}$$
We prove a maximum principle and obtain key ingredients for carrying on the method of moving planes, such as narrow region principle and decay at infinity. Then we establish radial symmetry and monotonicity for positive solutions to Dirichlet problems associated to such fully nonlinear fractional order equations in a unit ball and in the whole space, as well as non-existence of solutions on a half space. We believe that the methods developed here can be applied to a variety of problems involving fully nonlinear nonlocal operators. We also investigate the limit of this operator as \(\alpha {\rightarrow }2\) and show that
$$\begin{aligned} F_{\alpha }(u(x)) {\rightarrow }a(-\bigtriangleup u(x)) + b |{\bigtriangledown }u(x)|^2. \end{aligned}$$

Mathematics Subject Classification



  1. 1.
    Brandle, C., Colorado, E., de Pablo, A., Sanchez, U.: A concave–convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb. 143, 39–71 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, W., Fang, Y., Yang, R.: Liouville theorems involving the fractional Laplacian on a half space. Adv. Math. (2015) 274(2015), 167–198Google Scholar
  3. 3.
    Chen, W., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63, 615–622 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, W., Li, C.: Methods on Nonlinear Elliptic Equations. AIMS Book Series 4. American Institute of Mathematical Sciences, Springfield, MO (2010)Google Scholar
  5. 5.
    Chen, W., Li, C., Li, Y.: A direct method of moving planes for the fractional Laplacian. Adv. Math. 308, 404–437 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, W., Li, C., Ou, B.: Classification of solutions for an integral equation. CPAM 59, 330–343 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chen, W., Li, C., Ou, B.: Qualitative properties of solutions for an integral equation. Discrete Contin. Dyn. Syst. 12, 347–354 (2005)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chen, H., Lv, Z.: The properties of positive solutions to an integral system involving Wolff potential. Discrete Contin. Dyn. Syst. 34, 1879–1904 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. PDE 32, 1245–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62, 597–638 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, W., Zhu, J.: Indefinite fractional elliptic problem and Liouville theorems. J. Differ. Equ. 260, 4758–4785 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fang, Y., Chen, W.: A Liouville type theorem for poly-harmonic Dirichlet problem in a half space. Adv. Math. 229, 2835–2867 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Frank, R.L., Lieb, E.: Inversion positivityand the sharp Hardy–Littlewood–Sobolev inequality. Calc. Var. PDEs 39, 85–99 (2010)CrossRefzbMATHGoogle Scholar
  14. 14.
    Hang, F.: On the integral systems related to Hardy–Littlewood–Sobolev inequality. Math. Res. Lett. 14, 373–383 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Han, X., Lu, G., Zhu, J.: Characterization of balls in terms of Bessel-potential integral equation. J. Differ. Equ. 252, 1589–1602 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hang, F., Wang, X., Yan, X.: An integral equation in conformal geometry. Ann. H. Poincare Nonlinear Anal. 26, 1–21 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jarohs, S., Weth, T.: Symmetry via antisymmetric maximum principles in nonlocal problems of variable order. Ann. Mat. Pura Appl. 195, 273–291 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lei, Y.: Asymptotic properties of positive solutions of the Hardy–Sobolev type equations. J. Differ. Equ. 254, 1774–1799 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lei, Y., Lv, Z.: Axisymmetry of locally bounded solutions to an Euler–Lagrange system of the weighted Hardy–Littlewood–Sobolev inequality. Discrete Contin. Dyn. Syst. 33, 1987–2005 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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. PDEs 45, 43–61 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lu, G., Zhu, J.: An overdetermined problem in Riesz-potential and fractional Laplacian. Nonlinear Anal. 75, 3036–3048 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lu, G., Zhu, J.: The axial symmetry and regularity of solutions to an integral equation in a half space. Pac. J. Math. 253, 455–473 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lu, G., Zhu, J.: Symmetry and regularity of extremals of an integral equation related to the Hardy–Sobolev inequality. Calc. Var. PDEs 42, 563–577 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ma, L., Chen, D.: A Liouville type theorem for an integral system. Commun. Pure Appl. Anal. 5, 855–859 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhuo, R., Chen, W., Cui, X., Yuan, Z.: Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete Contin. Dyn. Syst. 36, 1125–1141 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anChina
  2. 2.Department of Mathematical SciencesYeshiva UniversityNew YorkUSA
  3. 3.School of Mathematical SciencesShanghai Jiao Tong UniversityShanghaiChina
  4. 4.Department of Applied MathematicsUniversity of ColoradoBoulderUSA
  5. 5.Department of MathematicsHarbin Institute of TechnologyHarbinChina

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