## Arkiv för Matematik

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

### Singular solutions to p-Laplacian type equations

Tero Kilpeläinen

#### Abstract

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.

#### Note

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

#### Article information

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

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485898635

Digital Object Identifier
doi:10.1007/BF02412215

Mathematical Reviews number (MathSciNet)
MR1714768

Zentralblatt MATH identifier
1018.35028

Rights

#### Citation

Kilpeläinen, Tero. Singular solutions to p -Laplacian type equations. Ark. Mat. 37 (1999), no. 2, 275--289. doi:10.1007/BF02412215. https://projecteuclid.org/euclid.afm/1485898635.

#### References

• 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.