Arkiv för Matematik

  • Ark. Mat.
  • Volume 37, Number 2 (1999), 275-289.

Singular solutions to p-Laplacian type equations

Tero Kilpeläinen

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We construct singular solutions to equations $div\mathcal{A}(x,\nabla u) = 0,$ similar to the p-Laplacian, that tend to ∞ on a given closed set of p-capacity zero. Moreover, we show that every Gδ-set of vanishing p-capacity is the infinity set of some A-superharmonic function.


The research is financed by the Academy of Finland (Project #8597).

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Ark. Mat. Volume 37, Number 2 (1999), 275-289.

Received: 13 October 1997
First available in Project Euclid: 31 January 2017

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1999 © Institut Mittag-Leffler


Kilpeläinen, Tero. Singular solutions to p -Laplacian type equations. Ark. Mat. 37 (1999), no. 2, 275--289. doi:10.1007/BF02412215.

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  • Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M. and Vazquez, J. L., An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1995), 241–273.
  • Choquet, G., Potentiels sur un ensemble de capacité nulle, C. R. Acad. Sci. Paris Sér. I Math. 244 (1957), 1707–1710.
  • Dolzmann, G., Hungerbühler, N. and Müller, S., Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right hand side, Preprint, 1998.
  • Evans, G. C., Potentials and positively infinite singularities of harmonic functions, Monatsh. Math. Phys. 43 (1936), 419–424.
  • Heinonen, J. and Kilpeläinen, T., Polar sets for supersolutions of degenerate elliptic equations, Math. Scand. 63 (1988), 136–150.
  • Heinonen, J., Kilpeläinen, T. and Martio, O., Nonliner Potential Theory of Degenerate Elliptic Equations, Oxford Univ. Press, Oxford, 1993.
  • Holopainen, I., Nonlinear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 74 (1990), 1–45.
  • Kilpeläinen, T., Potential theory for supersolutions of degenerate elliptic equations, Indiana Univ. Math. J. 38 (1989), 253–275.
  • Kilpeläinen, T. and Malý, J., Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 19 (1992), 591–613.
  • Kilpeläinen, T. and Malý, J., The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137–161.
  • Lindqvist, P. and Martio, O., Regularity and polar sets of supersolutions of certain degenerate elliptic equations, J. Anal. Math. 50 (1988), 1–17.
  • Mikkonen, P., On the Wolff potential and quasilinear elliptic equations involving measures, Ann. Acad. Sci. Fenn. Math. Dissertationes 104 (1996), 1–71.
  • Reshetnyak, Yu. G., Space Mappings with Bounded Distortion, Transl. Math. Monogr. 73, Amer. Math. Soc., Providence, R. I., 1989.