Skip to main content
SpringerLink
Log in

Liouville Type Theorems for Non-linear Differential Inequalities on Carnot Groups

  • Conference paper
Geometric Methods in PDE’s

Part of the book series: Springer INdAM Series ((SINDAMS,volume 13))

  • 1195 Accesses

Abstract

We overview some recent results on the existence and non-existence of positive solutions for differential inequalities of the kind

$$\displaystyle{\mathop{\mathrm{div}}\nolimits _{0}\left (\frac{\varphi (\vert \nabla _{0}u\vert )} {\vert \nabla _{0}u\vert } \nabla _{0}u\right )\geqslant f(u)\ell(\vert \nabla _{0}u\vert )}$$

in the setting of Carnot groups under the Keller-Osserman condition.

Dedicated to Ermanno Lanconelli on the occasion of his 70th birthday

Mathematical Subject Classification: Primary: 35R03, Secondary: 35R45, 35B53

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Balogh, Z.M., Tyson, J.T.: Polar coordinates in Carnot groups. Math. Z. 241(4), 697–730 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bandle, C., Greco, A., Porru, G.: Large solutions of quasilinear elliptic equations: existence and qualitative properties. Boll. Unione Mat. Ital. B (7) 11(1), 227–252 (1997)

    Google Scholar 

  3. Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin (2007)

    MATH  Google Scholar 

  4. Brandolini, L., Magliaro, M.: A note on Keller-Osserman conditions on Carnot groups. Nonlinear Anal. 75(4), 2326–2337 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brandolini, L., Magliaro, M.: Liouville type results and a maximum principle for non-linear differential operators on the heisenberg group. Preprint (2013)

    Google Scholar 

  6. Burago, Yu.D., Zalgaller, V.A.: Geometric Inequalities. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285. Springer, Berlin (1988) [Translated from the Russian by A.B. Sosinskiĭ, Springer Series in Soviet Mathematics]

    Google Scholar 

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

    Article  MathSciNet  MATH  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  MATH  Google Scholar 

  9. Farina, A., Serrin, J.: Entire solutions of completely coercive quasilinear elliptic equations, II. J. Differ. Equ. 250(12), 4409–4436 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Federer, H.: Geometric Measure Theory. Die Grundlehren der Mathematischen Wissenschaften, Band 153. Springer, New York (1969)

    Google Scholar 

  11. Filippucci, R., Pucci, P., Rigoli, M.: On entire solutions of degenerate elliptic differential inequalities with nonlinear gradient terms. J. Math. Anal. Appl. 356(2), 689–697 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Filippucci, R., Pucci, P., Rigoli, M.: On weak solutions of nonlinear weighted p-Laplacian elliptic inequalities. Nonlinear Anal. 70(8), 3008–3019 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Folland, G.B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13(2), 161–207 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Franchi, B., Serapioni, R., Serra Cassano, F.: Regular submanifolds, graphs and area formula in Heisenberg groups. Adv. Math. 211(1), 152–203 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kaplan, A.: Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans. Am. Math. Soc. 258(1), 147–153 (1980)

    Article  MATH  Google Scholar 

  17. Keller, J.B.: Electrohydrodynamics, I. The equilibrium of a charged gas in a container. J. Ration. Mech. Anal. 5, 715–724 (1956)

    MATH  Google Scholar 

  18. Keller, J.B.: On solutions of Δ u = f(u). Commun. Pure Appl. Math. 10, 503–510 (1957)

    Article  MATH  Google Scholar 

  19. Magliaro, M., Mari, L., Mastrolia, P., Rigoli, M.: Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group. J. Differ. Equ. 250(6), 2643–2670 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mari, L., Rigoli, M., Setti, A.G.: Keller-Osserman conditions for diffusion-type operators on Riemannian manifolds. J. Funct. Anal. 258(2), 665–712 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Naito, Y., Usami, H.: Entire solutions of the inequality \(\mathrm{div}(A(\vert Du\vert )Du) \geq f(u)\). Math. Z. 225(1), 167–175 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Osserman, R.: On the inequality Δ u ≥ f(u). Pac. J. Math. 7, 1641–1647 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pigola, S., Rigoli, M., Setti, A.G.: Maximum principles on Riemannian manifolds and applications. Mem. Am. Math. Soc. 174(822), x+99 (2005)

    Google Scholar 

  24. Redheffer, R.: On the inequality \(\varDelta u \geqq f(u,\,\vert \,\mathrm{grad}\,u\vert )\). J. Math. Anal. Appl. 1, 277–299 (1960)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria, Università degli studi di Bergamo, Dalmine, (BG), Italy

    Luca Brandolini

  2. Dipartimento di Matematica Informatica ed Economia, Università degli Studi della Basilicata, Potenza, Italy

    Marco Magliaro

Authors
  1. Luca Brandolini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Magliaro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luca Brandolini .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Bologna, Bologna, Italy

    Giovanna Citti

  2. Dipartimento di Matematica, Università di Bologna, Bologna, Bologna, Italy

    Maria Manfredini

  3. Dipartimento di Matematica, Università di Bologna, Bologna, Bologna, Italy

    Daniele Morbidelli

  4. Dipartimento FIM, Università di Modena e Reggio Emilia, Modena, Italy

    Sergio Polidoro

  5. Dipartimento di Matematica, Università di Bologna, Bologna, Italy

    Francesco Uguzzoni

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Brandolini, L., Magliaro, M. (2015). Liouville Type Theorems for Non-linear Differential Inequalities on Carnot Groups. In: Citti, G., Manfredini, M., Morbidelli, D., Polidoro, S., Uguzzoni, F. (eds) Geometric Methods in PDE’s. Springer INdAM Series, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-02666-4_11

Download citation

Publish with us

Policies and ethics

Search

Navigation