Liouville Type Theorems for Elliptic Equations with Gradient Terms

Abstract

In this paper we obtain Liouville type theorems for nonnegative supersolutions of the elliptic problem \({-\Delta u + b(x)|\nabla u| = c(x)u}\) in exterior domains of \({\mathbb{R}^N}\). We show that if lim \({{\rm inf}_{x \longrightarrow \infty} 4c(x) - b(x)^2 > 0}\) then no positive supersolutions can exist, provided the coefficients b and c verify a further restriction related to the fundamental solutions of the homogeneous problem. The weights b and c are allowed to be unbounded. As an application, we also consider supersolutions to the problems \({-\Delta u + b|x|^{\lambda}|{\nabla} u| = c|x|^{\mu} u^p}\) and \({-\Delta u + be^{\lambda |x|}|\nabla u| = ce^{\mu |x|}u^p}\), where p > 0 and λ, μ ≥ 0, and obtain nonexistence results which are shown to be optimal.

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References

  1. 1.

    S. Alarcón, J. García-Melián, A. Quaas, Nonexistence of positive supersolutions to some nonlinear elliptic equations, to appear in J. Math. Pures Appl.

  2. 2.

    S. N. Armstrong, B. Sirakov, Liouville results for fully nonlinear elliptic equations with power growth nonlinearities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) X (2011), 711–728.

  3. 3.

    S.N Armstrong, Sirakov B.: Nonexistence of positive supersolutions of elliptic equations via the maximum principle. Comm. Part. Diff. Eqns. 36, 2011–2047 (2011)

    Article  Google Scholar 

  4. 4.

    Bandle C., Essén M.: On positive solutions of Emden equations in cone-like domains. Arch. Rational Mech. Anal. 112(4), 319–338 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Bandle C., Levine H.: On the existence and nonexistence of global solutions of reactiondiffusion equations in sectorial domains. Trans. Amer. Math. Soc. 316(2), 595–622 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    H. Berestycki, Capuzzo-Dolcetta I., Nirenberg L.: Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Topol. Methods Nonlinear Anal. 4(1), 59–78 (1994)

    MathSciNet  Google Scholar 

  7. 7.

    Berestycki H., Hamel F., Nadirashvili N.: The speed of propagation for KPP type problems I. Periodic framework, J. Europ. Math. Soc. 7, 173–213 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    H. Berestycki, F. Hamel, L. Rossi, Liouville type results for semilinear elliptic equations in unbounded domains. Ann. Mat. Pura Appl. (4) 186 469–507 (2007).

    Google Scholar 

  9. 9.

    M. F. Bidaut-Véron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type. Arch. Rational Mech. Anal. 107 (4) (1989), 293–324.

  10. 10.

    M. F. Bidaut-Véron, S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems. J. Anal. Math. 84 (2001), 1-49.

    Google Scholar 

  11. 11.

    I. Capuzzo Dolcetta, A. Cutrì, Hadamard and Liouville type results for fully nonlinear partial differential inequalities, Commun. Contemp. Math. 5 (2003), no. 3, 435–448.

  12. 12.

    H. Chen, P. Felmer, On Liouville type theorems for fully nonlinear elliptic equations with gradient term, preprint.

  13. 13.

    A. Cutrì, F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré (C) An. Non Linéaire 17 (2000), 219–245.

    Google Scholar 

  14. 14.

    Felmer P. L., Quaas A: Fundamental solutions and Liouville type theorems for nonlinear integral operators. Adv. Math. 226, 2712–2738 (2011)

    MathSciNet  Article  Google Scholar 

  15. 15.

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

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Gidas B., Spruck J: A priori bounds for positive solutions of semilinear elliptic equations, Comm. Partial Differential Equations 6, 883–901 (1981)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer Verlag, 1983.

  18. 18.

    V. Kondratiev, V. Liskevich, V. Moroz, Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (1) (2005) 25–43.

    Google Scholar 

  19. 19.

    Kondratiev V., Liskevich V., Sobol Z: Positive supersolutions to semi-linear secondorder non-divergence type elliptic equations in exterior domains. Trans. Amer. Math. Soc. 361(2), 697–713 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    V. Kondratiev, V. Liskevich, Z. Sobol, Positive solutions to semi-linear and quasi-linear elliptic equations on unbounded domains. In “Handbook of differential equations: stationary partial differential equations”, volume 6, pages 255–273. Elsevier, 2008.

  21. 21.

    Liskevich V., Skrypnik I. I., Skrypnik I. V: Positive supersolutions to general nonlinear elliptic equations in exterior domains. Manuscripta Math. 115(4), 521–538 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Polacik P., Quittner P., Soupplet P: Singularity and decay estimates in super linear problems via Liouville-type theorems. Part I: elliptic equations and systems.. Duke Math. J. 139, 555–579 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    P. Pucci, J. Serrin, The maximum principle. Birkhäuser, 2007.

  24. 24.

    Rossi L: Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains, Commun. Pure Appl. Anal. 7, 125–141 (2008)

    MATH  Article  Google Scholar 

  25. 25.

    Serrin J., Zou H: Existence and non-existence results for ground states of quasilinear elliptic equations, Arch. Rat. Mech. Anal. 121, 101–130 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Serrin J., Zou H: Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189(1), 79–142 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    L. Véron, Singularities of solutions of second order quasilinear equations, vol. 353, Pitman Research Notes in Mathematics Series. Longman, Harlow, 1996.

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Correspondence to Alexander Quaas.

Additional information

S. A. was partially supported by USM Grant No. 121002 and Fondecyt grant 11110482. J. G-M and A. Q. were partially supported by Ministerio de Ciencia e Innovaci´on under grant MTM2011-27998 (Spain) and A. Q. was partially supported by Fondecyt Grant No. 1110210 and CAPDE, Anillo ACT-125. All three authors were partially supported by Programa Basal CMM, U. de Chile.

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Alarcón, S., García-Melián, J. & Quaas, A. Liouville Type Theorems for Elliptic Equations with Gradient Terms. Milan J. Math. 81, 171–185 (2013). https://doi.org/10.1007/s00032-013-0197-z

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Mathematics Subject Classification (2010)

  • Primary 35J60
  • Secondary 35B53

Keywords

  • Liouville theorems
  • Hadamard property
  • supersolutions
  • gradient term