Skip to main content
SpringerLink
Log in

Study of the Existence of Supersolutions for Nonlocal Equations with Gradient Terms

  • Published:
Milan Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study the existence of positive supersolutions of nonlocal equations of type \((-\Delta)^{s} u + |\Delta u|^{q} = \lambda f(u)\) posed in exterior domains where the datum f can be comparable with \(u^p\) near the origin. We prove that the existence of bounded supersolutions depends on the values of p, q and s.

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. Alarcón, S., García-Melián, J., Quaas, A.: Liouville type theorems for elliptic equations with gradient terms. Milan J. Math. 81(1), 171–185 (2013)

    Article  MathSciNet  Google Scholar 

  2. S. Alarcón, J. García-Melián, and A. Quaas. Nonexistence of positive supersolutions to some nonlinear elliptic problems. J. Math. Pures Appl. (9), 99(5):618–634, 2013

  3. G. Alberti and G. Bellettini. A nonlocal anisotropic model for phase transitions. I. The optimal profile problem. Math. Ann., 310(3):527–560, 1998

  4. D. Applebaum. Lévy processes and stochastic calculus, volume 116 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2009

  5. S. N. Armstrong and B. Sirakov. Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, 3(711–728), 2011

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

  7. Barrios, B., Figalli, A., Ros-Oton, X.: Free boundary regularity in the parabolic fractional obstacle problem. Comm. Pure Appl. Math. 71(10), 2129–2159 (2018)

    Article  MathSciNet  Google Scholar 

  8. Barrios, B., Figalli, A., Ros-Oton, X.: Global regularity for the free boundary in the obstacle problem for the fractional Laplacian. Amer. J. Math. 140(2), 415–447 (2018)

    Article  MathSciNet  Google Scholar 

  9. Barrios, B., Peral, I., Vita, S.: Some remarks about the summability of nonlocal nonlinear problems. Adv. Nonlinear Anal. 4(2), 91–107 (2015)

    Article  MathSciNet  Google Scholar 

  10. Barrios, B., Quaas, A.: The sharp exponent in the study of the nonlocal Hénon equation in RN: a Liouville theorem and an existence result. Calc. Var. 59, 114 (2020). https://doi.org/10.1007/s00526-020-01763-z

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Bjorland, C., Caffarelli, L., Figalli, A.: Nonlocal tug-of-war and the infinity fractional Laplacian. Comm. Pure Appl. Math. 65(3), 337–380 (2012)

    Article  MathSciNet  Google Scholar 

  14. Bonforte, M., Vázquez, J.L.: Quantitative local and global a priori estimates for fractional nonlinear diffusion equations. Adv. Math. 250, 242–284 (2014)

    Article  MathSciNet  Google Scholar 

  15. J. P. Bouchaud and A. Georges. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Physics Reports. A Review Section of Physics Letters, 195(4-5):127–293, 1990

  16. Burgos-Pérez, M.A., García-Melián, J., Quaas, A.: Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems. Discrete Contin. Dyn. Syst. 36(9), 4703–4721 (2016)

    Article  MathSciNet  Google Scholar 

  17. Chen, H., Véron, L.: Semilinear fractional elliptic equations with gradient nonlinearity involving measures. J. Funct. Anal. 266(8), 5467–5492 (2014)

    Article  MathSciNet  Google Scholar 

  18. P. Constantin. Euler equations, Navier-Stokes equations and turbulence. In Mathematical foundation of turbulent viscous flows, volume 1871 of Lecture Notes in Math., pages 1–43. Springer, Berlin, 2006

  19. Cont, R., Tankov, P.: Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL (2004)

    MATH  Google Scholar 

  20. Cui, X., Yu, M.: Non-existence of positive solutions for a higher order fractional equation. Discrete Contin. Dyn. Syst. 39(3), 1379–1387 (2019)

    Article  MathSciNet  Google Scholar 

  21. A. Cutríand F. Leoni. On the Liouville property for fully nonlinear equations. Ann. Inst. H. Poincar'e Anal. Non Lin'eaire, 17(2):219–245, 2000

  22. Del Pezzo, L.M., Quaas, A.: A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian. J. Differential Equations 263(1), 765–778 (2017)

    Article  MathSciNet  Google Scholar 

  23. L. Dupaigne and Y. Sire. A Liouville theorem for non local elliptic equations. In Symmetry for elliptic PDEs, volume 528 of Contemp. Math., pages 105–114. Amer. Math. Soc., Providence, RI, 2010

  24. M. M. Fall. Semilinear elliptic equations for the fractional laplacian with hardy potential. Nonlinear Analysis, 2018

  25. Fall, M.M., Weth, T.: Liouville theorems for a general class of nonlocal operators. Potential Anal. 45(1), 187–200 (2016)

    Article  MathSciNet  Google Scholar 

  26. Felmer, P., Quaas, A.: Fundamental solutions and Liouville type theorems for nonlinear integral operators. Adv. Math. 226(3), 2712–2738 (2011)

    Article  MathSciNet  Google Scholar 

  27. Ferrari, F., Verbitsky, I.E.: Radial fractional Laplace operators and Hessian inequalities. J. Differential Equations 253(1), 244–272 (2012)

    Article  MathSciNet  Google Scholar 

  28. Frank, R.L., Lieb, E.H., Seiringer, R.: Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. J. Amer. Math. Soc 21(4), 925–950 (2008)

    Article  MathSciNet  Google Scholar 

  29. 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 

  30. Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6(8), 883–901 (1981)

    Article  MathSciNet  Google Scholar 

  31. D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1983

  32. Graczyk, P., Jakubowski, T., Luks, T.: Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation. Positivity 17(4), 1043–1070 (2013)

    Article  MathSciNet  Google Scholar 

  33. V. Kondratiev, V. Liskevich, and Z. Sobol. Positive solutions to semi-linear and quasilinear elliptic equations on unbounded domains. Handbook of differential equations: stationary partial differential equations. Vol. VI, pages 177–267, 2008

  34. Korvenpää, J., Kuusi, T., Palatucci, G.: Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations. Math. Ann. 369(3–4), 1443–1489 (2017)

    Article  MathSciNet  Google Scholar 

  35. P. Poláčik, P. Quittner, and P. Souplet. Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems. Duke Math. J., 139(3):555–579, 2007

  36. Rossi, L.: Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure and Applied Analysis 7(1), 125–141 (2008)

    Article  MathSciNet  Google Scholar 

  37. Serrin, J., Zou, H.: Existence and nonexistence results for ground states of quasilinear elliptic equations. Arch. Rational Mech. Anal. 121(2), 101–130 (1992)

    Article  MathSciNet  Google Scholar 

  38. A. Signorini. Questioni di elasticitá. Statica non lineare; Vincoli unilaterali, statica semilinearizzata; Complementi. Confer. Sem. Mat. Univ. Bari, 48-49-50:42 pp. (1959), 1959

  39. Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60(1), 67–112 (2007)

    Article  MathSciNet  Google Scholar 

  40. Silvestre, L.: On the differentiability of the solution to an equation with drift and fractional diffusion. Indiana Univ. Math. J. 61(2), 557–584 (2012)

    Article  MathSciNet  Google Scholar 

  41. L. Silvestre, V. Vicol, and A. Zlatoš. On the loss of continuity for super-critical driftdiffusion equations. Arch. Ration. Mech. Anal., 207(3):845–877, 2013

  42. Toland, J.F.: The Peierls-Nabarro and Benjamin-Ono equations. J. Funct. Anal 145(1), 136–150 (1997)

    Article  MathSciNet  Google Scholar 

  43. F. X. Voirol. Coexistence of singular and regular solutions for the equation of Chipot and Weissler. Acta Math. Univ. Comenian. (N.S.), 65(1):53–64, 1996

  44. Wang, J.: Sub-Markovian C0-semigroups generated by fractional Laplacian with gradient perturbation. Integral Equations Operator Theory 76(2), 151–161 (2013)

    Article  MathSciNet  Google Scholar 

  45. R. Zhuo, W. Chen, X. Cui, and Z. Yuan. A Liouville Theorem for the Fractional Laplacian. arXiv e-prints, page arXiv:1401.7402, Jan 2014

Download references

Acknowledgements

B.B. was partially supported by AEI grant MTM2016-80474- P and Ramón y Cajal fellowship RYC2018-026098-I (Spain). L.DP. was partially supported by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 777822.

Author information

Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38200, La Laguna, Spain

    Begoña Barrios

  2. UBA – CONICET, Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I – Ciudad Universitaria (C1428BCW), Buenos Aires, Argentina

    Leandro M. Del Pezzo

Authors
  1. Begoña Barrios
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leandro M. Del Pezzo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Begoña Barrios.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barrios, B., Del Pezzo, L.M. Study of the Existence of Supersolutions for Nonlocal Equations with Gradient Terms. Milan J. Math. 88, 267–294 (2020). https://doi.org/10.1007/s00032-020-00314-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00032-020-00314-7

Keywords

Mathematics Subject Classification (2010)

Search

Navigation