Skip to main content

Unbounded Supersolutions of Nonlinear Equations with Nonstandard Growth

Abstract

We show that every weak supersolution of a variable exponent-Laplace equation is lower semicontinuous and that the singular set of such a function is of zero capacity if the exponent is logarithmically Hölder continuous. As a technical tool we derive Harnack-type estimates for possibly unbounded supersolutions.

[123456789101112131415161718192021]

References

  1. 1.

    Zhikov VV: On some variational problems. Russian Journal of Mathematical Physics 1997,5(1):105–116 (1998).

    MATH  MathSciNet  Google Scholar 

  2. 2.

    Acerbi E, Fusco N: A transmission problem in the calculus of variations. Calculus of Variations and Partial Differential Equations 1994,2(1):1–16. 10.1007/BF01234312

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    Acerbi E, Fusco N: Partial regularity under anisotropic growth conditions. Journal of Differential Equations 1994,107(1):46–67. 10.1006/jdeq.1994.1002

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Acerbi E, Mingione G: Regularity results for a class of functionals with non-standard growth. Archive for Rational Mechanics and Analysis 2001,156(2):121–140. 10.1007/s002050100117

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    Marcellini P: Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions. Archive for Rational Mechanics and Analysis 1989,105(3):267–284.

    MATH  MathSciNet  Article  Google Scholar 

  6. 6.

    Marcellini P: Regularity and existence of solutions of elliptic equations with -growth conditions. Journal of Differential Equations 1991,90(1):1–30. 10.1016/0022-0396(91)90158-6

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Alkhutov YuA: The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition. Differential Equations 1997,33(12):1651–1660, 1726.

    MathSciNet  Google Scholar 

  8. 8.

    Fan X, Zhao D: A class of De Giorgi type and Hölder continuity. Nonlinear Analysis 1999,36(3):295–318. 10.1016/S0362-546X(97)00628-7

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    Alkhutov YuA, Krasheninnikova OV: Continuity at boundary points of solutions of quasilinear elliptic equations with a nonstandard growth condition. Izvestiya Rossijskoj Akademii Nauk. Seriya Matematicheskaya 2004,68(6):3–60. English translation in Izvestiya: Mathematics 68 (2004), no. 6, 1063–1117

    MathSciNet  Article  Google Scholar 

  10. 10.

    Harjulehto P, Hästö P, Koskenoja M, Varonen S: Sobolev capacity on the space. Journal of Function Spaces and Applications 2003,1(1):17–33.

    MATH  MathSciNet  Article  Google Scholar 

  11. 11.

    Heinonen J, Kilpeläinen T, Martio O: Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York; 1993:vi+363.

    Google Scholar 

  12. 12.

    Lindqvist P: On the definition and properties of-superharmonic functions. Journal für die reine und angewandte Mathematik 1986, 365: 67–79.

    MATH  MathSciNet  Google Scholar 

  13. 13.

    Musielak J: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics. Volume 1034. Springer, Berlin; 1983:iii+222.

    Google Scholar 

  14. 14.

    Kováčik O, Rákosník J: On spacesand. Czechoslovak Mathematical Journal 1991,41(116)(4):592–618.

    Google Scholar 

  15. 15.

    Fan X, Zhao D: On the spacesand. Journal of Mathematical Analysis and Applications 2001,263(2):424–446. 10.1006/jmaa.2000.7617

    MATH  MathSciNet  Article  Google Scholar 

  16. 16.

    Harjulehto P: Variable exponent Sobolev spaces with zero boundary values. preprint, http://www.math.helsinki.fi/analysis/varsobgroup

  17. 17.

    Hästö P: On the density of smooth functions in variable exponent Sobolev space. to appear in Revista Matemática Iberoamericana

  18. 18.

    Diening L: Maximal function on generalized Lebesgue spaces. Mathematical Inequalities & Applications 2004,7(2):245–253.

    MATH  MathSciNet  Article  Google Scholar 

  19. 19.

    Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. Springer, Berlin; 1977:x+401.

    Google Scholar 

  20. 20.

    Harjulehto P, Hästö P, Koskenoja M, Varonen S: The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values. to appear in Potential Analysis, http://www.math.helsinki.fi/analysis/varsobgroup

  21. 21.

    Harjulehto P, Hästö P, Koskenoja M: The Dirichlet energy integral on intervals in variable exponent Sobolev spaces. Zeitschrift für Analysis und ihre Anwendungen 2003,22(4):911–923.

    MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Petteri Harjulehto.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Harjulehto, P., Kinnunen, J. & Lukkari, T. Unbounded Supersolutions of Nonlinear Equations with Nonstandard Growth. Bound Value Probl 2007, 048348 (2006). https://doi.org/10.1155/2007/48348

Download citation

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Nonlinear Equation