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

Potential Analysis volume 50, pages 521–539 (2019)Cite this article

Abstract

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.

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References

  1. Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math. 35, 209–273 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benilan, P., Brezis, H., Crandall, M.: A semilinear equation in L 1(R N). Ann. Scuola Norm. Sup. Pisa 2, 523–555 (1975)

    MathSciNet  MATH  Google Scholar 

  3. Biroli, M.: Nonlinear Kato measures and nonlinear subelliptic Schrödinger problems. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 21(5), 235–252 (1997)

    MathSciNet  Google Scholar 

  4. Biroli, M.: Schrödinger type and relaxed Dirichlet problems for the subelliptic p-Laplacian. Potential Anal. 15, 1–16 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer-Verlag, Berlin (1983)

    Book  MATH  Google Scholar 

  9. Giusti, E.: Metodi diretti nel calcolo delle variazioni. Unione Matematica Italiana, Bologna (1994)

    MATH  Google Scholar 

  10. Gutierrez, C.E.: Harnack’s inequality for degenerate Schrödinger operators. Trans. Amer. Math. Soc. 112, 403–419 (1989)

    Article  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  12. Julin, V.: Generalized Harnack inequality for nonhomogeneous elliptic equations. Arch. Rat. Mech. Anal. 216, 673–702 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Julin, V.: Generalized Harnack inequality for semilinear elliptic equations. J. Math. Pures Appl. 106, 877–904 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Kilpeläinen, T., Malý, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kon’kov, A.A.: On comparison theorems for elliptic inequalities. J. Math. Anal. Appl. 388, 102–124 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)

    MATH  Google Scholar 

  22. Marcus, M., Veron, L.: Nonlinear Second Order Elliptic Equations Involving Measures. Walter de Gruyter GmbH & Co KG, Berlin (2014)

    MATH  Google Scholar 

  23. Mohammed, A.: Harnack’s inequality for solutions of some degenerate elliptic equations. Rev. Mat. Iberoamericana 18, 325–354 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Moser, J.: On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  25. Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101–134 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nash, J.: Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 931–954 (1958)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  28. Pucci, P., Serrin, J.: The Harnack inequality in R 2 for quasilinear elliptic equations. J. d’Anal. Math. 85, 307–321 (2001)

    Article  MATH  Google Scholar 

  29. Pucci, P., Serrin, J.: The strong maximum principle revisted. J. Diff. Equat. 196, 1–66 (2004)

    Article  MATH  Google Scholar 

  30. Pucci, P., Serrin, J.: A note on the strong maximum principle for elliptic differential inequalities. J. Math. Pures Appl. 79, 57–71 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  32. Serrin, J.: Local behaviour of solutions of quasilinear equations. Acta Math. 111, 247–302 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  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 appear

  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)

    MathSciNet  MATH  Google Scholar 

  35. Trudinger, N.: Pointwise estimates and quasilinear parabolic equations. Comm. Pure Appl. Math. 21, 205–226 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vazquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  37. Veron, L.: Singularities of Solution of Second Order Quasilinear Equations. Pitman Research Notes in Mathematics Series, Longman (1996)

    MATH  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of NAS of Ukraine, G. Batiouk str. 19, 84100, Slovjansjk, Ukraine

    I. I. Skrypnik

  2. Vasyl’ Stus Donetsk National University, 600-richya str. 21, 21021, Vinnytsia, Ukraine

    I. I. Skrypnik

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  1. I. I. Skrypnik
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Correspondence to I. I. Skrypnik.

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Skrypnik, I.I. Harnack’s Inequality for Quasilinear Elliptic Equations with Singular Absorption Term. Potential Anal 50, 521–539 (2019). https://doi.org/10.1007/s11118-018-9691-9

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  • DOI: https://doi.org/10.1007/s11118-018-9691-9

Keywords

  • Quasilinear elliptic equations
  • Singular absorption term
  • Harnack’s inequality

Mathematics Subject Classification (2010)

  • 35B09
  • 35B40
  • 35B45
  • 35B65