Potential Analysis

, Volume 50, Issue 4, pp 521–539 | Cite as

Harnack’s Inequality for Quasilinear Elliptic Equations with Singular Absorption Term

  • I. I. SkrypnikEmail author


In this article we study nonnegative solutions of quasilinear equation model of which is
$$-\triangle_{p} u+V(x) f(u)= h(x)|\nabla u|^{p-1}+g(x), \,\,\,\, p>1.$$
Under the natural assumptions on the functions f, V, h and g we prove the Harnack inequality with constant independent of the solution. In the case g(x) ≡ V (x) we obtain an analogue of the well known Kilpeläinen-Malý sub-bound.


Quasilinear elliptic equations Singular absorption term Harnack’s inequality 

Mathematics Subject Classification (2010)

35B09 35B40 35B45 35B65 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work is supported by grant of Ministry of Education and Science of Ukraine (grant number is 0118U003138).


  1. 1.
    Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math. 35, 209–273 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benilan, P., Brezis, H., Crandall, M.: A semilinear equation in L 1(R N). Ann. Scuola Norm. Sup. Pisa 2, 523–555 (1975)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Biroli, M.: Nonlinear Kato measures and nonlinear subelliptic Schrödinger problems. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 21(5), 235–252 (1997)MathSciNetGoogle Scholar
  4. 4.
    Biroli, M.: Schrödinger type and relaxed Dirichlet problems for the subelliptic p-Laplacian. Potential Anal. 15, 1–16 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chiarenza, F., Fabes, E., Garofalo, N.: Harnack’s inequality for Schrödinger operators and the continuity of solutions. Proc. Amer. Math. Soc. 98, 415–425 (1986)MathSciNetzbMATHGoogle Scholar
  6. 6.
    De Giorgi, E.: Sulla differenziabilit’e l’analicit delle estremali degli integrali multipli regolari. Mem. Accad. Sci Torino, cl. Sci. Fis. Mat. Nat. 3, 25–43 (1957)MathSciNetGoogle Scholar
  7. 7.
    Felmer, P., Montenegro, M., Quaas, A.: A note on the strong maximum principle and the compact support principle. J. Diff. Equat. 246, 39–49 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer-Verlag, Berlin (1983)CrossRefzbMATHGoogle Scholar
  9. 9.
    Giusti, E.: Metodi diretti nel calcolo delle variazioni. Unione Matematica Italiana, Bologna (1994)zbMATHGoogle Scholar
  10. 10.
    Gutierrez, C.E.: Harnack’s inequality for degenerate Schrödinger operators. Trans. Amer. Math. Soc. 112, 403–419 (1989)CrossRefzbMATHGoogle Scholar
  11. 11.
    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Math. Monographs. Clarendon Press, Oxford Univ. Press, New York (1993)zbMATHGoogle Scholar
  12. 12.
    Julin, V.: Generalized Harnack inequality for nonhomogeneous elliptic equations. Arch. Rat. Mech. Anal. 216, 673–702 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Julin, V.: Generalized Harnack inequality for semilinear elliptic equations. J. Math. Pures Appl. 106, 877–904 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Keller, J.B.: On solutions of u = f(u).. Comm. Pure Appl. Math. 10, 503–510 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kilpeläinen, T., Malý, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kon’kov, A.A.: Comparison theorems for elliptic inequalities with a non-linearity in the principal part. J. Math. Anal. Appl. 325, 1013–1041 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kon’kov, A.A.: On comparison theorems for elliptic inequalities. J. Math. Anal. Appl. 388, 102–124 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kovalevsky, A.A., Skrypnik, I.I., Shishkov, A.E.: Singular Solutions of Nonlinear Elliptic and Prabolic Equations. De Gruyter, Series in Nonl. Analysis and Applications, Berlin (2016)CrossRefzbMATHGoogle Scholar
  19. 19.
    Krylov, N.V., Safonov, M.V.: A certain property of solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44:1, 161–175 (1980)Google Scholar
  20. 20.
    Kurata, K.: Continuity and Harnack’s inequality for solutions of elliptic partial differential equations of second order. Indiana Univ. Math. J. 43, 411–440 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)zbMATHGoogle Scholar
  22. 22.
    Marcus, M., Veron, L.: Nonlinear Second Order Elliptic Equations Involving Measures. Walter de Gruyter GmbH & Co KG, Berlin (2014)zbMATHGoogle Scholar
  23. 23.
    Mohammed, A.: Harnack’s inequality for solutions of some degenerate elliptic equations. Rev. Mat. Iberoamericana 18, 325–354 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Moser, J.: On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101–134 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nash, J.: Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 931–954 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Osserman, R.: On the inequality u f(u). Pac. J. Math. 7, 1641–1647 (1957)CrossRefzbMATHGoogle Scholar
  28. 28.
    Pucci, P., Serrin, J.: The Harnack inequality in R 2 for quasilinear elliptic equations. J. d’Anal. Math. 85, 307–321 (2001)CrossRefzbMATHGoogle Scholar
  29. 29.
    Pucci, P., Serrin, J.: The strong maximum principle revisted. J. Diff. Equat. 196, 1–66 (2004)CrossRefzbMATHGoogle Scholar
  30. 30.
    Pucci, P., Serrin, J.: A note on the strong maximum principle for elliptic differential inequalities. J. Math. Pures Appl. 79, 57–71 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Radulescu, V.D.: Singular Phenomena in Nonlinear Elliptic Problems: From Blow-Up Boundary Solutions to Equations with Singular Nonlinearities. Handb. Differ. Equat., North-Holland (2007)zbMATHGoogle Scholar
  32. 32.
    Serrin, J.: Local behaviour of solutions of quasilinear equations. Acta Math. 111, 247–302 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Shan, M.O., Skrypnik, I.I.: Keller-Osserman a priori estimates and the Harnack inequality for quasilinear elliptic and parabolic equations with absorption term. Nonlinear Anal., to appearGoogle Scholar
  34. 34.
    Skrypnik, I.I.: The Harnack inequality for a nonlinear elliptic equation with coefficients from the Kato class. Ukr. Math. Visn. 2, 219–235 (2005). (in Russian); transl. in: Ukr. Math. Bull. 2(2), 223-238 (2005)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Trudinger, N.: Pointwise estimates and quasilinear parabolic equations. Comm. Pure Appl. Math. 21, 205–226 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Vazquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Veron, L.: Singularities of Solution of Second Order Quasilinear Equations. Pitman Research Notes in Mathematics Series, Longman (1996)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and Mechanics of NAS of UkraineSlovjansjkUkraine
  2. 2.Vasyl’ Stus Donetsk National UniversityVinnytsiaUkraine

Personalised recommendations