Skip to main content

Advertisement

Unbounded Supersolutions of Nonlinear Equations with Nonstandard Growth

Article metrics

  • 1447 Accesses

  • 32 Citations

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).

  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

  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

  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

  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.

  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

  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.

  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

  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

  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.

  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.

  12. 12.

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

  13. 13.

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

  14. 14.

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

  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

  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.

  19. 19.

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

  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.

Download references

Author information

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) doi:10.1155/2007/48348

Download citation

Keywords

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