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Liouville theorems for nonlinear elliptic equations in half-spaces

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Abstract

In this paper we study the existence of nonnegative supersolutions of the nonlinear elliptic problem −Δu + |∇u|q = λup in the half-space ℝN+, where N = 2, q > 1, p > 0 and λ > 0. We obtain Liouville theorems for positive, bounded supersolutions, depending on the exponents q and p, the dimension N, and, in some critical cases, also on the parameter λ > 0.

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References

  1. S. Alarcón, M. Á. Burgos-Pérez, J. García-Melián and A. Quaas, Nonexistence results for elliptic equations with gradient terms, J. Differential Equations 260 (2016), 758–780.

    MathSciNet  MATH  Google Scholar 

  2. S. Alarcón, J. García-Melián and A. Quaas, Existence and uniqueness of solutions of nonlinear elliptic equations without growth conditions at infinity, J. Anal.Math. 118 (2012), 83–104.

    MathSciNet  MATH  Google Scholar 

  3. S. Alarcón, J. García-Melián and A. Quaas, Nonexistence of positive supersolutions to some nonlinear elliptic problems, J. Math. Pures Appl. 99 (2013), 618–634.

    MathSciNet  MATH  Google Scholar 

  4. S. Alarcón, J. García-Melián and A. Quaas, Liouville type theorems for elliptic equations with gradient terms, Milan J. Math. 81 (2013), 171–185.

    MathSciNet  MATH  Google Scholar 

  5. S. Alarcón, J. García-Melián and A. Quaas, Optimal Liouville theorems for supersolutions of elliptic equationswith the Laplacian, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), 129–158.

    MathSciNet  MATH  Google Scholar 

  6. S. N. Armstrong and B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equation 36 (2011), 2011–2047.

    MathSciNet  MATH  Google Scholar 

  7. S. N. Armstrong, B. Sirakov and C. K. Smart, Singular solutions of fully nonlinear elliptic equations and applications, Arch. Ration. Mech. Anal. 205 (2012), 345–394.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. M. F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannianmanifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), 489–539.

    MathSciNet  MATH  Google Scholar 

  14. L. Boccardo, F. Murat and J. P. Puel, Résultats d’existence pour certains problèmes elliptiques quasilinéaires, Ann. Sc. Norm. Super. Pisa 11 (1984), 213–235.

    MATH  Google Scholar 

  15. L. Boccardo, F. Murat and J. P. Puel, L ∞stimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal. 23 (1992), 326–333.

    MathSciNet  MATH  Google Scholar 

  16. L. Boccardo, L. Orsina and A. Porretta, Strongly coupled elliptic equations related to mean-field games systems, J. Differential Equations 261 (2016), 1796–1834.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. M. Chipot and F. B. Weissler, Some blow-up results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal. 20 (1989), 886–907.

    MathSciNet  MATH  Google Scholar 

  19. M. Cirant, Stationary focusing mean-field games, Comm. Partial Differential Equations 41 (2016), 1324–1346.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. D. G. de Figueiredo and J. Yang, A priori bounds for positive solutions of a non-variational elliptic system, Comm. Partial Differential Equations 26 (2001), 2305–2321.

    MathSciNet  MATH  Google Scholar 

  23. E. I. Galakhov, Solvability of an elliptic equation with a gradient nonlinearity, Differ. Eq. 41 (2005), 693–702.

    MathSciNet  MATH  Google Scholar 

  24. E. I. Galakhov, Positive solutions of quasilinear elliptic equations, Math. Notes 78, (2005), 185–193.

    MathSciNet  MATH  Google Scholar 

  25. B. Gidas, Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations, in Nonlinear Partial Differential Equations in Engineering and Applied Science, Marcel Dekker, New York, 1980, pp. 255–273.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  27. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin–Heidelberg, 1983.

    MATH  Google Scholar 

  28. J. L. Kazdan and R. J. Kramer, Invariant criteria for existence of solutions to second-order quasilinear elliptic equations. Comm. Pure Appl. Math. 31 (1978), 619–645.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  30. V. Kondratiev, V. Liskevich and Z. Sobol, Positive solutions to semi-linear and quasi-linear elliptic equations on unbounded domains, in Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 6, Elsevier, Amsterdam, 2008, pp. 255–273.

    Google Scholar 

  31. O. A. Ladyzhenskaya and N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.

    MATH  Google Scholar 

  32. Y. Li, L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math. 90 (2003), 27–87.

    MathSciNet  MATH  Google Scholar 

  33. P. L. Lions, Résolution des problèmes elliptiques quasilinéaires, Arch. Ration. Mech. Anal. 74 (1980), 335–353.

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  36. P. Poláčik, P. Quittner, P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J. 139 (2007), 555–579.

    MathSciNet  MATH  Google Scholar 

  37. P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.

    MATH  Google Scholar 

  38. P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser, Basel, 2007.

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  40. P. Souplet, Recent results and open problems on parabolic equationswith gradient nonlinearities, Electron. J. Differential Equations 2001 (2001), 1–19.

    Google Scholar 

  41. F. X. Voirol, Coexistence of singular and regular solutions for the equation of Chipot and Weissler, Acta Math. Univ. Comenianae 65 (1996), 53–64.

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

J. G.-M. and A. Q. were partially supported by Ministerio de Economía y Competitividad under grants MTM2011-27998 and MTM2014-52822-P (Spain). A. Q. was also partially supported by Fondecyt Grant No. 1151180 Programa Basal, CMM. U. de Chile and Millennium Nucleus Center for Analysis of PDE NC130017.

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Authors and Affiliations

  1. Departamento de Análisis Matemático and Instituto Universitario de Estudios Avanzados (IUdEA) en Física Atómica, Molecular y Fotónica, Universidad de La Laguna, Astrofísico Francisco Sánchez s/n, 38200, La Laguna, Spain

    Jorge García-Melián

  2. Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla V-110 Avda. España, 1680, Valparaíso, Chile

    Alexander Quaas

  3. Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marques De São Vicente, 225, Rio De Janeiro, RJ, CEP 22451-900, Brasil

    Boyan Sirakov

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  1. Jorge García-Melián
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  2. Alexander Quaas
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  3. Boyan Sirakov
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Correspondence to Jorge García-Melián.

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García-Melián, J., Quaas, A. & Sirakov, B. Liouville theorems for nonlinear elliptic equations in half-spaces. JAMA 139, 559–583 (2019). https://doi.org/10.1007/s11854-019-0066-y

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  • DOI: https://doi.org/10.1007/s11854-019-0066-y

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