Geodesic
Generating singularities of solutions of quasilinear elliptic equations using Wolff’s potential
Czechoslovak Mathematical Journal, Tome 53 (2003) no. 2, pp. 429-435
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of $p$-Laplacian type. If $p\gamma
We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of $p$-Laplacian type. If $p\gamma $ and the right-hand side is a Radon measure with singularity of order $\gamma $ at $x_0\in \Omega $, then any supersolution in $W_{\mathrm loc}^{1,p}(\Omega )$ has singularity of order at least $\frac{(\gamma -p)}{(p-1)}$ at $x_0$. In the proof we exploit a pointwise estimate of $\mathcal A$-superharmonic solutions, due to Kilpeläinen and Malý, which involves Wolff’s potential of Radon’s measure.
Classification : 31B05, 35A20, 35B05, 35J60
Keywords: quasilinear elliptic; singularity; Sobolev function 
@article{CMJ_2003_53_2_a16,
     author = {\v{Z}ubrini\'c, Darko},
     title = {Generating singularities of solutions of quasilinear elliptic equations using {Wolff{\textquoteright}s} potential},
     journal = {Czechoslovak Mathematical Journal},
     pages = {429--435},
     year = {2003},
     volume = {53},
     number = {2},
     mrnumber = {1983463},
     zbl = {1022.31005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2003_53_2_a16/}
}
TY  - JOUR
AU  - Žubrinić, Darko
TI  - Generating singularities of solutions of quasilinear elliptic equations using Wolff’s potential
JO  - Czechoslovak Mathematical Journal
PY  - 2003
SP  - 429
EP  - 435
VL  - 53
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMJ_2003_53_2_a16/
LA  - en
ID  - CMJ_2003_53_2_a16
ER  - 
%0 Journal Article
%A Žubrinić, Darko
%T Generating singularities of solutions of quasilinear elliptic equations using Wolff’s potential
%J Czechoslovak Mathematical Journal
%D 2003
%P 429-435
%V 53
%N 2
%U http://geodesic.mathdoc.fr/item/CMJ_2003_53_2_a16/
%G en
%F CMJ_2003_53_2_a16
Žubrinić, Darko. Generating singularities of solutions of quasilinear elliptic equations using Wolff’s potential. Czechoslovak Mathematical Journal, Tome 53 (2003) no. 2, pp. 429-435. http://geodesic.mathdoc.fr/item/CMJ_2003_53_2_a16/

[1] G.  Díaz and R. Leletier: Explosive solutions of quasilinear elliptic equations: existence and uniqueness. Nonlinear Anal. 20 (1993), 97–125. | DOI | MR

[2] M.  Giaquinta: Multiple Integrals in the Calculus of Variations and Elliptic Systems. Princeton University Press, Princeton, New Jersey, 1983. | MR

[3] M. Grillot: Prescribed singular submanifolds of some quasilinear elliptic equations. Nonlinear Anal. 34 (1998), 839–856. | DOI | MR | Zbl

[4] J. Heinonen, T.  Kilpeläinen and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford, 1993. | MR

[5] T. Kilpeläinen: Singular solutions to $p$-Laplacian type equations. Ark. Mat. 37 (1999), 275–289. | DOI | MR

[6] T. Kilpeläinen and J. Malý: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Scuola Norm. Sup. Pisa 19 (1992), 591–613. | MR

[7] L.  Korkut, M. Pašić and D. Žubrinić: Control of essential infimum and supremum of solutions of quasilinear elliptic equations. C.  R.  Acad. Sci. Paris t. 329, Série  I (1999), 269–274. | DOI | MR

[8] L.  Korkut, M. Pašić and D. Žubrinić: Some qualitative properties of solutions of quasilinear elliptic equations and applications. J.  Differential Equations 170 (2001), 247–280. | DOI | MR

[9] J.  Leray and J.-L. Lions: Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France 93 (1965), 97–107. | DOI | MR

[10] L.  Mou: Removability of singular sets of harmonic maps. Arch. Rational Mech. Anal. 127 (1994), 199–217. | MR | Zbl

[11] L.  Simon: Singularities of Geometric Variational Problems. IAS/Park City Math. Ser. Vol. 2. 1992, pp. 183–219. | MR

[12] D.  Žubrinić: Generating singularities of solutions of quasilinear elliptic equations. J.  Math. Anal. Appl. 244 (2000), 10–16. | DOI | MR