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



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 


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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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer-Verlag, Berlin (1983)CrossRefMATHGoogle Scholar
  9. 9.
    Giusti, E.: Metodi diretti nel calcolo delle variazioni. Unione Matematica Italiana, Bologna (1994)MATHGoogle Scholar
  10. 10.
    Gutierrez, C.E.: Harnack’s inequality for degenerate Schrödinger operators. Trans. Amer. Math. Soc. 112, 403–419 (1989)CrossRefMATHGoogle 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)MATHGoogle Scholar
  12. 12.
    Julin, V.: Generalized Harnack inequality for nonhomogeneous elliptic equations. Arch. Rat. Mech. Anal. 216, 673–702 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Julin, V.: Generalized Harnack inequality for semilinear elliptic equations. J. Math. Pures Appl. 106, 877–904 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Keller, J.B.: On solutions of u = f(u).. Comm. Pure Appl. Math. 10, 503–510 (1957)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kilpeläinen, T., Malý, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kon’kov, A.A.: On comparison theorems for elliptic inequalities. J. Math. Anal. Appl. 388, 102–124 (2012)MathSciNetCrossRefMATHGoogle 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)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)MATHGoogle Scholar
  22. 22.
    Marcus, M., Veron, L.: Nonlinear Second Order Elliptic Equations Involving Measures. Walter de Gruyter GmbH & Co KG, Berlin (2014)MATHGoogle Scholar
  23. 23.
    Mohammed, A.: Harnack’s inequality for solutions of some degenerate elliptic equations. Rev. Mat. Iberoamericana 18, 325–354 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Moser, J.: On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591 (1961)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101–134 (1964)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Nash, J.: Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 931–954 (1958)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Osserman, R.: On the inequality u f(u). Pac. J. Math. 7, 1641–1647 (1957)CrossRefMATHGoogle 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)CrossRefMATHGoogle Scholar
  29. 29.
    Pucci, P., Serrin, J.: The strong maximum principle revisted. J. Diff. Equat. 196, 1–66 (2004)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MATHGoogle Scholar
  32. 32.
    Serrin, J.: Local behaviour of solutions of quasilinear equations. Acta Math. 111, 247–302 (1964)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar
  35. 35.
    Trudinger, N.: Pointwise estimates and quasilinear parabolic equations. Comm. Pure Appl. Math. 21, 205–226 (1968)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Vazquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Veron, L.: Singularities of Solution of Second Order Quasilinear Equations. Pitman Research Notes in Mathematics Series, Longman (1996)MATHGoogle Scholar

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Authors and Affiliations

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

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