Skip to main content

Advertisement

SpringerLink
Log in

Uniqueness of Nonnegative Solutions to Elliptic Differential Inequalities on Finsler Manifolds

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We consider a class of elliptic differential inequalities involving Finsler p-Laplacian and a positive potential function on forward geodesically complete noncompact Finsler measure spaces with finite reversibility. Under various volume growth conditions concerning geodesic balls with a given center and the potential function, we prove that the only nonnegative weak solution of the differential inequalities is identically zero.

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. Bao, D., Chern, S.-S., Shen, Z.: An introduction to Riemann-Finsler geometry. Springer-Verlag, New York (2000)

    Book  Google Scholar 

  2. Caristi, G., D’Ambrosio, L., Mitidieri, E.: Liouville theorems for some nonlinear inequalities. Proc. Steklov Inst. Math. 260, 90–111 (2008). In Russian; translated in Tr. Mat. Inst. Steklova 260 (2008), 97–118

    Article  MathSciNet  Google Scholar 

  3. Caristi, G., Mitidieri, E.: Nonexistence of positive solutions of quasilinear equations. Adv. Differential Equ. 2(3), 319–359 (1997)

    MathSciNet  MATH  Google Scholar 

  4. Caristi, G., Mitidieri, E., Pokhozhaev, S.I.: Liouville theorems for quasilinear elliptic inequalities. Dokl. Akad. Nauk 424(6), 741–747 (2009). translation in Dokl. Math. 79 (2009), no. 1, 118–124

    MathSciNet  MATH  Google Scholar 

  5. Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28(3), 333–354 (1975)

    Article  MathSciNet  Google Scholar 

  6. D’Ambrosio, L.: Liouville theorems for anisotropic quasilinear inequalities. Nonlinear Anal. 70(8), 2855–2869 (2009)

    Article  MathSciNet  Google Scholar 

  7. D’Ambrosio, L., Lucente, S.: Nonlinear Liouville theorems for Grushin and Tricomi operators. J. Differential Equ. 193(2), 511–541 (2003)

    Article  MathSciNet  Google Scholar 

  8. D’Ambrosio, L., Mitidieri, E.: A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities. Adv. Math. 224 (3), 967–1020 (2010)

    Article  MathSciNet  Google Scholar 

  9. Farina, A., Valdinoci, E.: Gradient bounds for anisotropic partial differential equations, Calc. Var. Partial Differential Equ. 49(3-4), 923–936 (2014)

    Article  Google Scholar 

  10. Gidas, B.: Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations, Nonlinear partial differential equations in engineering and applied science (Proc. Conf., Univ. Rhode Island, Kingston, R.I., 1979), pp. 255–273, Lecture Notes in Pure and Appl Math. 54, Dekker, New York (1980)

  11. Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34(4), 525–598 (1981)

    Article  MathSciNet  Google Scholar 

  12. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, 2nd edn. Springer-Verlag, Berlin (1983)

    MATH  Google Scholar 

  13. Grigor’yan, A., Kondratiev, V.A.: On the existence of positive solutions of semilinear elliptic inequalities on Riemannian manifolds, Around the research of Vladimir Maz’ya. II, 203–218, Int. Math. Ser. (N Y.), vol. 12. Springer, New York (2010)

    Google Scholar 

  14. Grigor’yan, A., Sun, Y.: On nonnegative solutions of the inequality Δu + uσ ≤ 0 on Riemannian manifolds. Comm. Pure Appl. Math. 67(8), 1336–1352 (2014)

    Article  MathSciNet  Google Scholar 

  15. Huang, J.-C.: Liouville type theorem for quasi-linear elliptic inequality Δpu + uσ ≤ 0 on Riemannian manifolds. J. Math. Anal. Appl. 428(1), 12–31 (2015)

    Article  MathSciNet  Google Scholar 

  16. Kurta, V.V.: On the absence of positive solutions to semilinear elliptic equations. Tr. Mat. Inst. Steklova 227, 162–169 (1999)

    MathSciNet  MATH  Google Scholar 

  17. Mastrolia, P., Monticelli, D.D., Punzo, F.: Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds. Calc. Var. Partial Differential Equ. 54(2), 1345–1372 (2015)

    Article  MathSciNet  Google Scholar 

  18. Mitidieri, E., Pokhozhaev, S.I.: Absence of global positive solutions of quasilinear elliptic inequalities. Dokl. Akad. Nauk 359(4), 456–460 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Mitidieri, E., Pokhozhaev, S.I.: Absence of positive solutions for quasilinear elliptic problems in \(\mathbb {R}^{N}\). Tr. Mat. Inst. Steklova 227, 192–222 (1999). Issled. po Teor. Differ. Funkts. Mnogikh Perem. i ee Prilozh. 18

    MathSciNet  Google Scholar 

  20. Mitidieri, E., Pokhozhaev, S.I.: A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Tr. Mat. Inst. Steklova 234, 1–384 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Mitidieri, E., Pokhozhaev, S.I.: Towards a unified approach to nonexistence of solutions for a class of differential inequalities. Milan J. Math. 72, 129–162 (2004)

    Article  MathSciNet  Google Scholar 

  22. Monticelli, D.D.: Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators. J. Eur. Math Soc. 12(3), 611–654 (2010)

    Article  MathSciNet  Google Scholar 

  23. Ohta, S.: Finsler interpolation inequalities. Calc. Var. Partial Differential Equ. 36, 211–249 (2009)

    Article  MathSciNet  Google Scholar 

  24. Ohta, S.-I.: Nonlinear geometric analysis on Finsler manifolds. Eur. J. Math. 3 (4), 916–952 (2017)

    Article  MathSciNet  Google Scholar 

  25. Ohta, S., Sturm, K.: Heat flow on Finsler manifolds. Comm. Pure Appl. Math. 62, 1386–1433 (2009)

    Article  MathSciNet  Google Scholar 

  26. Shen, Z.: Lectures on Finsler geometry. World Scientific Publishing Co., Singapore (2001)

    Book  Google Scholar 

  27. Sun, Y.: Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds. J. Math. Anal. Appl. 419(1), 643–661 (2014)

    Article  MathSciNet  Google Scholar 

  28. Sun, Y.: On nonexistence of positive solutions of quasi-linear inequality on Riemannian manifolds. Proc. Amer. Math. Soc. 143(7), 2969–2984 (2015)

    Article  MathSciNet  Google Scholar 

  29. Sun, Y.: Uniqueness result on nonnegative solutions of a large class of differential inequalities on Riemannian manifolds. Pacific J. Math. 280(1), 241–254 (2016)

    Article  MathSciNet  Google Scholar 

  30. Xia, C.: Local gradient estimate for harmonic functions on Finsler manifolds. Calc. Var. Partial Differential Equ. 51(3-4), 849–865 (2014)

    Article  MathSciNet  Google Scholar 

  31. Xia, C.: Inverse anisotropic mean curvature flow and a Minkowski type inequality. Adv Math. 315, 102–129 (2017)

    Article  MathSciNet  Google Scholar 

  32. Zhang, H.-C.: A note on Liouville type theorem of elliptic inequality Δu + uσ ≤ 0 on Riemannian manifolds. Potential Anal. 43(2), 269–276 (2015)

    Article  MathSciNet  Google Scholar 

  33. Zhang, F.-E., Xia, Q.: Some Liouville-type theorems for harmonic functions on Finsler manifolds. J. Math. Anal. Appl. 417(2), 979–995 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Sciences Institute, Australian National University, Canberra, ACT 2601, Australia

    Changwei Xiong

Authors
  1. Changwei Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changwei Xiong.

Additional information

Publisher’s Note

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

This research was supported by Australian Laureate Fellowship FL150100126 of the Australian Research Council. We would like to thank the referee for careful reading of the paper and for valuable suggestions and comments which made this paper better and more readable. We are also grateful to Ben Andrews for his support.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiong, C. Uniqueness of Nonnegative Solutions to Elliptic Differential Inequalities on Finsler Manifolds. Potential Anal 53, 1145–1163 (2020). https://doi.org/10.1007/s11118-019-09801-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-019-09801-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation