Skip to content


  • Research Article
  • Open Access

Unbounded Supersolutions of Nonlinear Equations with Nonstandard Growth

Boundary Value Problems20062007:048348

  • Received: 3 March 2006
  • Accepted: 28 May 2006
  • Published:


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.


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


Authors’ Affiliations

Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, Helsinki, 00014, Finland
Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu, 90014, Finland
Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, Espoo, 02015, Finland


  1. Zhikov VV: On some variational problems. Russian Journal of Mathematical Physics 1997,5(1):105–116 (1998).MATHMathSciNetGoogle Scholar
  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/BF01234312MATHMathSciNetView ArticleGoogle Scholar
  3. Acerbi E, Fusco N: Partial regularity under anisotropic growth conditions. Journal of Differential Equations 1994,107(1):46–67. 10.1006/jdeq.1994.1002MATHMathSciNetView ArticleGoogle Scholar
  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/s002050100117MATHMathSciNetView ArticleGoogle Scholar
  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.MATHMathSciNetView ArticleGoogle Scholar
  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)90158-6MATHMathSciNetView ArticleGoogle Scholar
  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.MathSciNetGoogle Scholar
  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)00628-7MATHMathSciNetView ArticleGoogle Scholar
  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, 1063–1117MathSciNetView ArticleGoogle Scholar
  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.MATHMathSciNetView ArticleGoogle Scholar
  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. Lindqvist P: On the definition and properties of-superharmonic functions. Journal für die reine und angewandte Mathematik 1986, 365: 67–79.MATHMathSciNetGoogle Scholar
  13. Musielak J: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics. Volume 1034. Springer, Berlin; 1983:iii+222.Google Scholar
  14. Kováčik O, Rákosník J: On spacesand. Czechoslovak Mathematical Journal 1991,41(116)(4):592–618.Google Scholar
  15. Fan X, Zhao D: On the spacesand. Journal of Mathematical Analysis and Applications 2001,263(2):424–446. 10.1006/jmaa.2000.7617MATHMathSciNetView ArticleGoogle Scholar
  16. Harjulehto P: Variable exponent Sobolev spaces with zero boundary values. preprint,
  17. Hästö P: On the density of smooth functions in variable exponent Sobolev space. to appear in Revista Matemática IberoamericanaGoogle Scholar
  18. Diening L: Maximal function on generalized Lebesgue spaces. Mathematical Inequalities & Applications 2004,7(2):245–253.MATHMathSciNetView ArticleGoogle Scholar
  19. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. Springer, Berlin; 1977:x+401.MATHView ArticleGoogle Scholar
  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,
  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.MATHView ArticleGoogle Scholar


© Petteri Harjulehto et al. 2007

This article is published under license to BioMed Central Ltd. 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.