Skip to main content

Fractional p-Laplacian problems with negative powers in a ball or an exterior domain


This paper is concerned with the qualitative properties of positive solutions to fractional p-Laplacian problems

$$\begin{aligned}(-\Delta )^s_p u + |x|^{-\alpha }u^{-q} = 0\end{aligned}$$

in \(B_1 \setminus \{0\}\) and \(\mathbb {R}^n \setminus \overline{B_1}\), where \(0<s<1\), \(p\ge 2\), \(q>0\), \(\alpha \ge 0\) and \(B_1\) is the unit ball centered at the origin. By deriving a decay at infinity principle and exploiting the direct method of moving planes for the fractional p-Laplacian, we prove the symmetry or monotonicity of positive solutions to the above problems.

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


  1. Bertozzi, A.L., Pugh, M.C.: Long-wave instabilities and saturation in thin film equations. Commun. Pure Appl. Math. 51, 625–661 (1998)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  3. Cai, M., Ma, L.: Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete Contin. Dyn. Syst. 38, 4603–4615 (2018)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  5. Chen, W., Li, C., Ou, B.: Qualitative properties of solutions for an integral equation. Discrete Contin. Dyn. Syst. 12, 347–354 (2005)

    MathSciNet  Article  Google Scholar 

  6. Chen, W., Li, C.: Maximum principles for the fractional \(p\)-Laplacian and symmetry of solutions. Adv. Math. 335, 735–758 (2018)

    MathSciNet  Article  Google Scholar 

  7. Dal Passo, R., Giacomelli, L., Shishkov, A.: The thin film equation with nonlinear diffusion. Commun. Partial Differ. Equ. 26, 1509–1557 (2001)

    MathSciNet  Article  Google Scholar 

  8. Du, Y.H., Guo, Z.M.: Positive solutions of an elliptic equation with negative exponent: stability and critical power. J. Differ. Equ. 246, 2387–2414 (2009)

    MathSciNet  Article  Google Scholar 

  9. Duong, A.T., Le, P.: Symmetry and nonexistence results for a fractional Hénon-Hardy system on a half-space. Rocky Mountain J. Math. (2019). (to appear).

  10. Guo, Y.: Nonexistence and symmetry of solutions to some fractional Laplacian equations in the upper half space. Acta Math. Sci. 37, 836–851 (2017)

    MathSciNet  Article  Google Scholar 

  11. Guo, Z.M., Wei, J.C.: Symmetry of non-negative solutions of a semilinear elliptic equation with singular nonlinearity. Proc. R. Soc. Edinb. Sect. A 137, 963–994 (2007)

    MathSciNet  Article  Google Scholar 

  12. Laugesen, R.S., Pugh, M.C.: Linear stability of steady states for thin film and Cahn-Hilliard type equations. Arch. Ration. Mech. Anal. 154, 3–51 (2000)

    MathSciNet  Article  Google Scholar 

  13. Le, P.: Liouville theorem and classification of positive solutions for a fractional Choquard type equation. Nonlinear Anal. 185, 123–141 (2019)

    MathSciNet  Article  Google Scholar 

  14. Le, P., Nguyen, H.T., Nguyen, T.Y.: On positive stable solutions to weighted quasilinear problems with negative exponent. Complex Var. Elliptic Equ. 63, 1739–1751 (2018)

    MathSciNet  Article  Google Scholar 

  15. Meadows, A.: Stable and singular solutions of the equation \(\Delta u=\frac{1}{u}\). Indiana Univ. Math. J. 53, 1681–1703 (2004)

    MathSciNet  Article  Google Scholar 

  16. Ma, L., Wei, J.C.: Properties of positive solutions to an elliptic equation with negative exponent. J. Funct. Anal. 254, 1058–1087 (2008)

    MathSciNet  Article  Google Scholar 

  17. Ma, L., Wei, J.C.: Stability and multiple solutions to Einstein-scalar field Lichnerowicz equation on manifolds. J. Math. Pures Appl. 99, 174–186 (2013)

    MathSciNet  Article  Google Scholar 

  18. Song, X.F., Zhao, L.: Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds. Z. Angew. Math. Phys. 61, 655–662 (2010)

    MathSciNet  Article  Google Scholar 

  19. Wu, L., Niu, P.: Symmetry and nonexistence of positive solutions to fractional \(p\)-Laplace equations. Discrete Contin. Dyn. Syst. 39, 1573–1583 (2018)

    Article  Google Scholar 

  20. Xu, X.: Uniqueness theorem for integral equations and its application. J. Funct. Anal. 247, 95–109 (2007)

    MathSciNet  Article  Google Scholar 

  21. Zhang, L., Yu, M., He, J.: A Liouville theorem for a class of fractional systems in \(\mathbb{R}_+^n\). J. Differ. Equ. 263, 6025–6065 (2017)

    MathSciNet  Article  Google Scholar 

  22. Zhuo, R., Li, Y.: A Liouville theorem for the higher-order fractional Laplacian. Commun. Contemp. Math. 21, 1850005 (2019)

    MathSciNet  Article  Google Scholar 

Download references


The authors are grateful to the anonymous referees for their valuable suggestions and comments which lead to the improvement of the paper.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Vu Ho.

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

Verify currency and authenticity via CrossMark

Cite this article

Le, P., Ho, V. Fractional p-Laplacian problems with negative powers in a ball or an exterior domain. J. Pseudo-Differ. Oper. Appl. 11, 789–803 (2020).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Fractional p-Laplacian
  • Negative powers
  • Symmetry of solutions
  • Method of moving planes

Mathematics Subject Classification

  • 35R11
  • 35J92
  • 35B06
  • 35J75