Boundary Value Problems for the Singular p- and p(x)-Laplacian Equations in a Cone

  • Yury Alkhutov
  • Mikhail Borsuk
  • Sebastian Jankowski
Conference paper
Part of the Trends in Mathematics book series (TM)


In this paper we describe briefly recent new results about the degenerate equations of the p-Laplacian type in a bounded cone. We shall consider the Dirichlet problem for such equation with the strong nonlinear right part as well as the Robin problem for such equation with singular nonlinearity in the right part. Such problems are mathematical models occurring in reaction-diffusion theory, non-Newtonian fluid theory, non-Newtonian filtration, the turbulent flow of a gas in porous medium, in electromagnetic problems, in heat transfer problems, in Fick’s law of diffusion et al. The aim of our investigations is the behavior of week solutions to the problem in the neighborhood of an angular or conical boundary point of the bounded cone. We establish sharp estimates of the type |u(x)| = O(|x|ϰ) for the weak solutions u of the problems under consideration.


Degenerate equations p-Laplacian type bvp with strong nonlinear right part Angular or conical boundary point of the bounded cone Sharp estimate for week solution 

Mathematics Subject Classification (2010)

Primary 35J92 35G30; Secondary 35P20 76A05 


  1. 1.
    Yu. Alkhutov, The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with nonstandard growth condition. Differ. Equ. 33(12), 1653–1663 (1997)Google Scholar
  2. 2.
    Yu. Alkhutov, M.V. Borsuk, The behavior of solutions to the Dirichlet problem for second order elliptic equations with variable nonlinearity exponent in a neighborhood of a conical boundary point. J. Math. Sci. 210(4), 341–370 (2015)Google Scholar
  3. 3.
    Yu. Alkhutov, M. Borsuk, The Dirichlet problem in a cone for second order elliptic quasi-linear equation with the p-Laplacian. J. Math. Anal. Appl. 449, 1351–1367 (2017)Google Scholar
  4. 4.
    Yu. Alkhutov, O. Krasheninnikova, Continuity at boundary points of solutions of quasilinear elliptic equations with a non-standard growth condition. Izv. Math. 68(6), 1063–1117 (2004)Google Scholar
  5. 5.
    M. Borsuk, Transmission Problems for Elliptic Second-Order Equations in Non-smooth Domains. Frontiers in Mathematics (Birkhäuser, Boston, 2010), p. 218Google Scholar
  6. 6.
    M. Borsuk, S. Jankowski, The Robin problem for singular p-Laplacian equation in a cone. Complex Variables Elliptic Equ. (2017).
  7. 7.
    M. Borsuk, V. Kondratiev, Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains. North-Holland Mathematical Library, vol. 69 (Elsevier, Amsterdam, 2006), p. 530Google Scholar
  8. 8.
    M. Dobrowolski, On quasilinear elliptic equations in domains with conical boundary points. J. Reine Angew. Math. 394, 186–195 (1989)MathSciNetMATHGoogle Scholar
  9. 9.
    X. Fan, Global C 1, α regularity for variable exponent elliptic equations in divergence form. J. Differ. Equ. 235(2), 397–417 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    X. Fan, D. Zhao, A class of De Giorgi type and Hölder continuity. Nonlinear Anal. Theory Methods Appl. 36(3), 295–318 (1999)CrossRefMATHGoogle Scholar
  11. 11.
    I.N. Krol’, V.G. Maz’ya, The absence of the continuity and Hölder continuity of the solutions of quasilinear elliptic equations near a nonregular boundary. Trudy Moskov. Mat. Obšč (Russian). 26, 75–94 (1972)Google Scholar
  12. 12.
    O.A. Ladyzhenskaya, N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations (Academic Press, New York, 1968)MATHGoogle Scholar
  13. 13.
    P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points. Commun. Partial Differ. Equ. 8, 773–817 (1983)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    V.V. Zhikov, On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 13(2), 249–269 (1994)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Yury Alkhutov
    • 1
  • Mikhail Borsuk
    • 2
  • Sebastian Jankowski
    • 2
  1. 1.A. G. and N. G. Stoletov Vladimir State UniversityVladimirRussia
  2. 2.Department of Mathematics and InformaticsUniversity of Warmia and Mazury in OlsztynOlsztyn-KortowoPoland

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