This article is part of the series Harnack's Estimates, Positivity and Local Behavior of Degenerate and Singular Parabolic Equations.

Open Access Research Article

Unbounded Supersolutions of Nonlinear Equations with Nonstandard Growth

Petteri Harjulehto1*, Juha Kinnunen2 and Teemu Lukkari3

Author Affiliations

1 Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, Helsinki 00014, Finland

2 Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu 90014, Finland

3 Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, Espoo 02015, Finland

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Boundary Value Problems 2007, 2007:048348 doi:10.1155/2007/48348


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/048348


Received:3 March 2006
Revisions received:16 May 2006
Accepted:28 May 2006
Published:30 October 2006

© 2007 Harjulehto et al.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

References

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

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

  3. Acerbi, E, Fusco, N: Partial regularity under anisotropic growth conditions. Journal of Differential Equations. 107(1), 46–67 (1994). Publisher Full Text OpenURL

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

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

  6. Marcellini, P: Regularity and existence of solutions of elliptic equations with -growth conditions. Journal of Differential Equations. 90(1), 1–30 (1991). Publisher Full Text OpenURL

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

  8. Fan, X, Zhao, D: A class of De Giorgi type and Hölder continuity. Nonlinear Analysis. 36(3), 295–318 (1999). Publisher Full Text OpenURL

  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. 68(6), 3–60 English translation in Izvestiya: Mathematics 68 (2004), no. 6, 1063–1117 (2004)

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

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

  12. Lindqvist, P: On the definition and properties of -superharmonic functions. Journal für die reine und angewandte Mathematik. 365, 67–79 (1986). PubMed Abstract | Publisher Full Text | PubMed Central Full Text OpenURL

  13. Musielak, J: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics,p. iii+222. Springer, Berlin (1983)

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

  15. Fan, X, Zhao, D: On the spaces and . Journal of Mathematical Analysis and Applications. 263(2), 424–446 (2001). Publisher Full Text OpenURL

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

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

  18. Diening, L: Maximal function on generalized Lebesgue spaces . Mathematical Inequalities & Applications. 7(2), 245–253 (2004). PubMed Abstract | Publisher Full Text OpenURL

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

  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 webcite

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